Authors’ Note
With the exception of these notes, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited. We are releasing this manuscript on Hexagon to disseminate the results as quickly as possible because they may be of interest to the mathematical community. A more carefully written version of this manuscript is in preparation and will be submitted for publication. We welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas.
P. L. was partially supported by NSF grant DMS-2450004.
Introduction
Background
Let n ≥ 1 n \ge1 n ≥ 1 q ≥ 2 q \ge2 q ≥ 2 log \log log P \mathbb{P} P E \mathbb{E} E Var \operatorname{Var} Var k ≥ 1 k \ge1 k ≥ 1 [ k ] = { 1 , … , k } [k] = \{1, \ldots, k\} [ k ] = { 1 , … , k } σ : [ n ] → [ q ] \sigma: [n] \to[q] σ : [ n ] → [ q ] q q q [ n ] [n] [ n ] σ ( u ) ≠ σ ( v ) \sigma(u) \ne\sigma(v) σ ( u ) = σ ( v ) { u , v } \{u, v\} { u , v } q q q q q q 0 ≤ m ≤ ( n 2 ) 0 \le m \le\binom{n}{2} 0 ≤ m ≤ ( 2 n ) G ( n , m ) G(n,m) G ( n , m ) [ n ] [n] [ n ] m m m p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] G ( n , p ) G(n,p) G ( n , p ) [ n ] [n] [ n ] p p p G ( n , ⌊ c n ⌋ ) G(n,\lfloor cn \rfloor) G ( n , ⌊ c n ⌋) G ( n , d / n ) G(n,d/n) G ( n , d / n ) c , d ≥ 0 c,d \ge0 c , d ≥ 0 n n n 2 c 2c 2 c d d d
The study of the chromatic number of sparse random graphs goes back to Erdős and Rényi, who asked in [30 Section 10, p. 59] for the chromatic number of a random graph on n n n c n cn c n c > 1 / 2 c > 1/2 c > 1/2 q ≥ 3 q \ge3 q ≥ 3 q q q [2 Theorem 1.1] . More precisely, there exists a sequence ( d ^ n ) (\hat d_n) ( d ^ n ) q q q ω > 0 \omega> 0 ω > 0
lim n → ∞ P ( G ( n , min { 1 , max { 0 , d ^ n − ω n } } ) is q -colorable ) = 1 , lim n → ∞ P ( G ( n , min { 1 , d ^ n + ω n } ) is q -colorable ) = 0. (1) \begin{aligned}
\lim_{n \to\infty} \mathbb{P}\left(G\left(n,\min\left\{1,\max\left\{0,\frac{\hat d_n-\omega}{n}\right\}\right\}\right)\text{ is }q\text{-colorable}\right) &= 1,\\
\lim_{n \to\infty} \mathbb{P}\left(G\left(n,\min\left\{1,\frac{\hat d_n+\omega}{n}\right\}\right)\text{ is }q\text{-colorable}\right) &= 0. \tag*{(1)}
\end{aligned} n → ∞ lim P ( G ( n , min { 1 , max { 0 , n d ^ n − ω } } ) is q -colorable ) n → ∞ lim P ( G ( n , min { 1 , n d ^ n + ω } ) is q -colorable ) = 1 , = 0. (1) Their proof rests on Friedgut’s criterion for sharp thresholds [32 ] , which relates a coarse threshold to the approximation of the property by the appearance of bounded subgraphs, and it leaves the dependence of d ^ n \hat d_n d ^ n n n n d ^ n \hat d_n d ^ n [2 Remark 1.2] . Achlioptas and Molloy had already noted that this convergence was widely conjectured [3 p. 204] , and later works describe it as a well-known open problem [21 Section 1.1] , [9 Section 1] . The goal of this paper is to prove that d ^ n \hat d_n d ^ n q ≥ 3 q \ge3 q ≥ 3
We first recall the known bounds on the location of the transition. For q = 3 q = 3 q = 3 G ( n , d / n ) G(n,d/n) G ( n , d / n ) d ≤ 4.03 d \le4.03 d ≤ 4.03 [4 Corollary 2] , and with probability tending to zero for d > 4.697 d > 4.697 d > 4.697 [8 Section 1.3] (see [3 , 18 , 27 , 41 ] for earlier bounds). For general q q q γ q = log q / log ( q / ( q − 1 ) ) \gamma_q = \log q/\log(q/(q-1)) γ q = log q / log ( q / ( q − 1 )) o q ( 1 ) o_q(1) o q ( 1 ) q → ∞ q \to\infty q → ∞ n n n q q q G ( n , ⌊ c n ⌋ ) G(n,\lfloor cn\rfloor) G ( n , ⌊ c n ⌋) c > γ q c > \gamma_q c > γ q G ( n , d / n ) G(n,d/n) G ( n , d / n ) d > 2 γ q d > 2\gamma_q d > 2 γ q 2 γ q = 2 q log q − log q + o q ( 1 ) 2\gamma_q = 2q\log q - \log q + o_q(1) 2 γ q = 2 q log q − log q + o q ( 1 ) G ( n , d / n ) G(n,d/n) G ( n , d / n ) q q q d < 2 ( q − 1 ) log ( q − 1 ) d < 2(q-1)\log(q-1) d < 2 ( q − 1 ) log ( q − 1 ) [5 Theorem 1] . Coja-Oghlan and Vilenchik [21, Theorem 1.1} and Coja-Oghlan [20 Theorem 1] then proved that
lim inf n → ∞ d ^ n ≥ 2 q log q − log q − 2 log 2 − o q ( 1 ) , lim sup n → ∞ d ^ n ≤ 2 q log q − log q − 1 + o q ( 1 ) , (2) \liminf_{n\to\infty}\hat d_n \ge2q\log q-\log q-2\log2-o_q(1), \qquad\limsup_{n\to\infty}\hat d_n \le2q\log q-\log q-1+o_q(1), \tag*{(2)} n → ∞ lim inf d ^ n ≥ 2 q log q − log q − 2 log 2 − o q ( 1 ) , n → ∞ lim sup d ^ n ≤ 2 q log q − log q − 1 + o q ( 1 ) , (2) and Coja-Oghlan and Vilenchik also showed that the chromatic number of G ( n , d / n ) G(n,d/n) G ( n , d / n ) d d d [21 Theorem 1.2] . These results locate the transition. However, they do not show that d ^ n \hat d_n d ^ n
Statistical physics gives precise, non-rigorous predictions. Statistical-physics computations place the 3-colorability threshold near average degree 4.69 [49 ] , refined numerically to 4.687(2) [63 Table II] , and give the asymptotic form 2 q log q − log q − 1 + o q ( 1 ) 2q\log q-\log q-1+o_q(1) 2 q log q − log q − 1 + o q ( 1 ) q q q [42 ] , which agrees with the upper bound in (2) . The same theory predicts further transitions in the set of colorings below the colorability threshold [43 , 63 ] . Rigorous results on the number of colorings below that threshold were obtained for large q q q [9 Theorem 2.1] and later for all q ≥ 3 q \ge3 q ≥ 3 [23 arXiv:1611.00814v4, Theorem 1.2] .
For other random constraint satisfaction problems, the existence of a limiting threshold is known in several cases. The threshold of random 2-SAT is at one clause per variable [19 , 36 ] , with a window of order n 2 / 3 n^{2/3} n 2/3 [11 Theorem 1.1] . The threshold of random k k k k = 3 k = 3 k = 3 [28 ] and for all k ≥ 3 k \ge3 k ≥ 3 [56 ] . Ding, Sly, and Sun proved the existence of the threshold for random k k k k k k [26 Theorem 1] . For random d d d q q q q q q [22 Theorem 1.1 and Corollary 1.2] . Since the degree is an integer, this leaves at most one degree at which the limiting behavior is not determined. This argument does not apply to G ( n , d / n ) G(n,d/n) G ( n , d / n ) d d d
Recently, two preprints claimed the existence of a limiting threshold for random k k k k ≥ 3 k \ge3 k ≥ 3 [16 Theorem 1.1 and Corollary 1.2] claims an upper bound of order n 1 / 2 + 1 / k n^{1/2+1/k} n 1/2 + 1/ k O ( n / log n ) O(n/\log n) O ( n / log n ) [17 Corollary 7.10] . She obtains the polynomial bound from a variance bound for the hitting time of unsatisfiability and deduces the existence of the threshold from a criterion of Abbe and Montanari [1 ] , in the form stated in [55 Proposition 14] . The preprint [51 Theorem 1.1] claims the same conclusion, combining a variance bound for a capped hitting time with an interpolation inequality and comparisons between systems of different sizes. The latter preprint explicitly credits Carenini with priority for the threshold-existence result [51 abstract and Section 1] . These preprints are unrefereed. We give self-contained proofs of the adaptations used here, while drawing on their arguments as described below. To the best of our knowledge, the convergence of d ^ n \hat{d}_n d ^ n q ≥ 3 q \ge3 q ≥ 3
Main results
From now on, we fix an integer q ≥ 3 q \ge3 q ≥ 3
γ q = log q log ( q / ( q − 1 ) ) . (3) \gamma_q = \frac{\log q}{\log(q/(q-1))}. \tag*{(3)} γ q = log ( q / ( q − 1 )) log q . (3) Our main result is the existence of the q q q
Theorem 1.1. Let q ≥ 3 q \ge3 q ≥ 3 α q ∈ [ 1 / 2 , γ q ] \alpha_q \in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ] c ≥ 0 c \ge0 c ≥ 0
lim n → ∞ P ( G ( n , ⌊ c n ⌋ ) is q -colorable ) = { 1 if 0 ≤ c < α q , 0 if c > α q , \lim_{n \to\infty} \mathbb{P}\left(G(n,\lfloor cn\rfloor)\text{ is }q\text{-colorable}\right)
=
\begin{cases}
1 & \text{if }0 \le c < \alpha_q,\\
0 & \text{if }c > \alpha_q,
\end{cases} n → ∞ lim P ( G ( n , ⌊ c n ⌋) is q -colorable ) = { 1 0 if 0 ≤ c < α q , if c > α q , and for every fixed real d ≥ 0 d \ge0 d ≥ 0
lim n → ∞ P ( G ( n , d / n ) is q -colorable ) = { 1 if 0 ≤ d < 2 α q , 0 if d > 2 α q , \lim_{n \to\infty} \mathbb{P}\left(G(n,d/n)\text{ is }q\text{-colorable}\right)
=
\begin{cases}
1 & \text{if }0 \le d < 2\alpha_q,\\
0 & \text{if }d > 2\alpha_q,
\end{cases} n → ∞ lim P ( G ( n , d / n ) is q -colorable ) = { 1 0 if 0 ≤ d < 2 α q , if d > 2 α q , where G ( n , d / n ) G(n,d/n) G ( n , d / n ) n ≥ d n \ge d n ≥ d
We define d q = 2 α q d_q = 2\alpha_q d q = 2 α q d ^ n → d q \hat{d}_n \to d_q d ^ n → d q 0 < ω < d q 0 < \omega< d_q 0 < ω < d q d ^ n ≥ d q + ω \hat{d}_n \ge d_q + \omega d ^ n ≥ d q + ω (1) , with shift ω / 2 \omega/2 ω /2 G ( n , ( d q + ω / 2 ) / n ) G(n,(d_q + \omega/2)/n) G ( n , ( d q + ω /2 ) / n ) d ^ n ≤ d q − ω \hat{d}_n \le d_q - \omega d ^ n ≤ d q − ω G ( n , ( d q − ω / 2 ) / n ) G(n,(d_q - \omega/2)/n) G ( n , ( d q − ω /2 ) / n )
The bounds on the limit inferior and limit superior in (2) now transfer directly to d q d_q d q d q d_q d q 4.03 ≤ d 3 ≤ 4.697 4.03 \le d_3 \le4.697 4.03 ≤ d 3 ≤ 4.697 d q d_q d q (2) .
The bounds 1 / 2 ≤ α q ≤ γ q 1/2 \le\alpha_q \le\gamma_q 1/2 ≤ α q ≤ γ q c < 1 / 2 c < 1/2 c < 1/2 G ( n , ⌊ c n ⌋ ) G(n, \lfloor cn \rfloor) G ( n , ⌊ c n ⌋) [30 ] , [40 Theorem 5.5, p. 112] , and such graphs are 3-colorable. Theorem 1.1 is an asymptotic statement. As we explain in Remark 7.4, our proof gives no further information on the value of α q \alpha_q α q
Our second result is a bound on the width of the transition in a random graph process. For a finite set S S S S S S multigraph on S S S occurrences . We define a coloring of a multigraph on S S S S S S [ q ] [q] [ q ] n ≥ 2 n \ge2 n ≥ 2 e 1 , e 2 , … e_1,e_2,\ldots e 1 , e 2 , … [ n ] [n] [ n ] ( n 2 ) \binom{n}{2} ( 2 n ) ( e i ) i ≥ 1 (e_i)_{i\ge1} ( e i ) i ≥ 1 edge process . For each integer m ≥ 0 m \ge0 m ≥ 0 G m G_m G m time-m m m , the multigraph on [ n ] [n] [ n ] e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m n ≥ 2 q n \ge2q n ≥ 2 q H H H m ≥ 0 m \ge0 m ≥ 0 G m G_m G m q q q
H = min { m ≥ 0 : G m is not q -colorable } . H = \min\{m \ge0 : G_m \text{ is not } q\text{-colorable}\}. H = min { m ≥ 0 : G m is not q -colorable } . By Lemma 3.10 , H H H
Theorem 1.2. Let q ≥ 3 q \ge3 q ≥ 3 α q \alpha_q α q Theorem 1.1 , and let ϵ ∈ ( 0 , 1 ) \epsilon\in(0,1) ϵ ∈ ( 0 , 1 ) C > 0 C > 0 C > 0 n 0 ≥ 2 q n_0 \ge2q n 0 ≥ 2 q q q q ϵ \epsilon ϵ m n m_n m n m n / n → α q m_n/n \to\alpha_q m n / n → α q n → ∞ n \to\infty n → ∞
P ( ∣ H − m n ∣ ≤ C n ( log n ) − 3 / 2 ) ≥ 1 − ϵ for all n ≥ n 0 . (4) \mathbb{P}\left(\lvert H-m_n\rvert\le Cn(\log n)^{-3/2}\right) \ge1-\epsilon\qquad\text{for all } n \ge n_0. \tag*{(4)} P ( ∣ H − m n ∣ ≤ C n ( log n ) − 3/2 ) ≥ 1 − ϵ for all n ≥ n 0 . (4) Since P ( H > m ) = P ( G m is q -colorable ) \mathbb{P}(H > m) = \mathbb{P}(G_m \text{ is } q\text{-colorable}) P ( H > m ) = P ( G m is q -colorable ) Theorem 1.2 implies that for all n ≥ n 0 n \ge n_0 n ≥ n 0 m ≥ 0 m \ge0 m ≥ 0
P ( G m is q -colorable ) ≥ 1 − ϵ if m < m n − C n ( log n ) − 3 / 2 , \mathbb{P}(G_m \text{ is } q\text{-colorable}) \ge1-\epsilon\qquad\text{if } m < m_n-Cn(\log n)^{-3/2}, P ( G m is q -colorable ) ≥ 1 − ϵ if m < m n − C n ( log n ) − 3/2 , P ( G m is q -colorable ) ≤ ϵ if m ≥ m n + C n ( log n ) − 3 / 2 . \mathbb{P}(G_m \text{ is } q\text{-colorable}) \le\epsilon\qquad\text{if } m \ge m_n+Cn(\log n)^{-3/2}. P ( G m is q -colorable ) ≤ ϵ if m ≥ m n + C n ( log n ) − 3/2 . Thus the probability that the process is colorable drops from 1 − ϵ 1-\epsilon 1 − ϵ ϵ \epsilon ϵ O ( n ( log n ) − 3 / 2 ) O(n(\log n)^{-3/2}) O ( n ( log n ) − 3/2 ) m / n m/n m / n O ( ( log n ) − 3 / 2 ) O((\log n)^{-3/2}) O (( log n ) − 3/2 ) § 2.3 . We have not tried to optimize the exponent 3 / 2 3/2 3/2 Theorem 4.1 requires the relative window widths to have a finite sum along exponentially growing sizes. The choice ( log n ) − 3 / 2 (\log n)^{-3/2} ( log n ) − 3/2
Theorem 1.1 is false for q = 2 q = 2 q = 2 bipartite if it admits a 2-coloring. Pittel and Yeum showed that for every d ∈ ( 0 , 1 ) d \in(0,1) d ∈ ( 0 , 1 )
lim n → ∞ P ( G ( n , d / n ) is bipartite ) = e d / 2 ( 1 − d 1 + d ) 1 / 4 ∈ ( 0 , 1 ) . \lim_{n\to\infty}\mathbb{P}(G(n,d/n)\text{ is bipartite}) = e^{d/2}\left(\frac{1-d}{1+d}\right)^{1/4} \in(0,1). n → ∞ lim P ( G ( n , d / n ) is bipartite ) = e d /2 ( 1 + d 1 − d ) 1/4 ∈ ( 0 , 1 ) . [57 Section 1, equation (1.5)] . Hence there is no constant d 2 d_2 d 2 d 2 d_2 d 2 [2 p. 64] ). Our proof uses q ≥ 3 q \ge3 q ≥ 3 Lemma 5.3 that, if a vertex receives the same color as another vertex in every coloring, then its neighbors use at least q − 1 ≥ 2 q - 1 \ge2 q − 1 ≥ 2
Our proof has two parts, which we describe in more detail in § 3 . We begin by showing that the existence of α q \alpha_q α q [10 Theorem 5 and Proposition 2] . We then prove the window bound of Theorem 1.2 . We define the expected lifetime of a colorable graph to be the expected number of independent uniformly random pair additions needed to make it non-colorable, including the addition that destroys colorability. A variance argument reduces the window bound to controlling this lifetime for G H − 1 G_{H-1} G H − 1
We call e H e_H e H final edge . For a colorable graph, we call two distinct vertices a frozen pair if they receive the same color in every coloring, and we call each equivalence class under equality in every coloring a frozen class . The final edge joins a frozen pair of G H − 1 G_{H-1} G H − 1 ⌊ ( log n ) 1 / 4 ⌋ \lfloor(\log n)^{1/4}\rfloor ⌊( log n ) 1/4 ⌋
Related work
We now discuss the sources of the methods that we use, and the known bounds on the width of the transition.
The comparison of different system sizes in Appendix § 9 uses the interpolation method. System-splitting interpolation was used by Guerra and Toninelli to prove the existence of the limiting free energy of the Sherrington–Kirkpatrick model, by interpolating between a large system and two independent subsystems [38 Section 3] . Franz and Leone [31 ] and Panchenko and Talagrand [54 ] developed free-energy bounds for diluted models with Poisson numbers of interactions, including random k k k [10 ] . They proved the convergence of the normalized optimal values of several problems and of the normalized logarithm of the probability of q q q [10 Theorem 1] . However, such a limit does not by itself locate a transition from probability one to probability zero, since it is consistent with probabilities that tend to zero more slowly than exponentially. For coloring, they also compared the probability that a graph is colorable with the probability that two smaller disjoint graphs are colorable [10 Theorem 5 and Proposition 2] , and their proof of this comparison uses the classes of vertices that receive the same color in every coloring. Our product inequality, Lemma A.2, is its form for a Poisson number of edges.
To pass from comparisons of system sizes to convergence, one uses a principle of near-superadditivity. De Bruijn and Erdős proved such a principle for sequences that are subadditive up to an error satisfying an integrability condition, for comparable sizes [25 Theorem 23] , and Abbe and Montanari gave a version with summable errors [1 arXiv:1006.3786v1, Section 4.3, Lemma 8 and Remark 3] . In [51 Section 5] , the product inequality [51 Lemma 5.2] is used to compare the normalized center of the transition for the combined system with the minimum of the centers for the two parts, with polynomially small errors, and a balanced tree of blocks yields convergence [51 Lemma 5.4] . Appendix A adapts this argument to quantiles of the hitting time and to errors that are only summable along n = ⌊ e k ⌋ n = \lfloor e^k \rfloor n = ⌊ e k ⌋ n / log n n / \log n n / log n
The reduction of the window to the expected lifetime of the last colorable graph, in § 5.2 , has several antecedents. Aldous [7 Theorem 1(b)] considered the cover time of a finite set by independent copies of a random subset, the first time at which their union is the whole set. He bounded the variance of the cover time by an absolute constant times its mean times the expected time that a fresh sequence of such subsets needs to cover the set left uncovered just before the cover time. Coloring fits this framework, if each pair { u , v } \{u,v\} { u , v } σ : [ n ] → [ q ] \sigma: [n] \to[q] σ : [ n ] → [ q ] σ ( u ) = σ ( v ) \sigma(u) = \sigma(v) σ ( u ) = σ ( v ) H H H H − 1 H - 1 H − 1 G H − 1 G_{H-1} G H − 1 k k k [16 Theorem 4.1] . In [51 Lemmas 3.1 and 3.2] , the variance of a capped hitting time is bounded through the Efron–Stein inequality [29 , 61 ] , in the form [12 (3.6)] in which one variable is omitted, by deleting one clause at a time without renumbering the other clauses, and a later preprint combines expected lifetimes with such deletions [53 Section 3.3, Lemma 3.4] . Lemma 4.3 follows the deletion argument of [51 ] and evaluates the sum of the squared deletion delays exactly, in terms of the expected lifetime of the last colorable graph.
The estimates specific to k k k [16 , 51 ] bound the sum over time of the fraction of variables that take the same value in every satisfying assignment [16 Proposition 3.2, 51, Proposition 2.2] . These proofs delete a variable and compare the clauses that contained it, which become clauses of length k − 1 k - 1 k − 1 k k k e H e_H e H u u u v v v σ ( u ) = σ ( v ) \sigma(u) = \sigma(v) σ ( u ) = σ ( v ) σ \sigma σ G H − 1 G_{H-1} G H − 1
Culberson and Gent studied frozen pairs and their development as edges are added to a graph, and used frozen-pair counts to compute conditional one-edge survival probabilities in a process without replacement [24 Sections 2.1 and 3.3] . The hazard identity in Lemma 3.7 is the version for independent pairs sampled with replacement. For a satisfiable Boolean formula, we call the literals true in every satisfying assignment its backbone [47 , 48 ] . Following [11 ] , we call the literals whose addition makes some satisfiable subformula unsatisfiable its spine . These notions record an absolute Boolean value, whereas a frozen pair records equality without specifying a named color. In [11 Section 3.3] , the probability that the next clause makes a 2-SAT formula unsatisfiable is expressed through the spine, and Lemma 3.7 is the analogous identity for frozen pairs. Other notions of freezing concern a variable that has one value throughout a specified restricted family of solutions [6 , 46 , 60 ] . Frozen pairs here quantify over all colorings. Our estimates for the frozen pairs of the last colorable graph, in Sections 5 and § 7 , replace the estimates specific to k k k
To treat final edges with an endpoint of low degree, we identify vertices. Let Z Z Z Z Z Z Z Z Z [58 Section 3, Theorem 1] . We combine it with the splitting of a Poisson process of edges at a vertex [44 Theorems 5.2, 5.6, and 5.8, Corollary 5.9] , and we prove in Proposition 5.6 that identifying at most ( log n ) 1 / 4 (\log n)^{1/4} ( log n ) 1/4 O ( n 3 / 8 ) O(n^{3/8}) O ( n 3/8 )
For final edges between vertices of high degree, we instead delete edge occurrences. The deletion identities are the analogue, for a fixed number of independent points, of the Mecke equation for Poisson processes [45 ] , [44 Theorem 4.1] . We call an occurrence at one endpoint of a frozen pair pivotal for that pair if deleting it removes the forced equality; otherwise we call it non-pivotal . This is the usual notion of sensitivity to one coordinate [59 ] , applied to the event that adding the given pair destroys colorability. The main estimate of § 7 bounds the total, along the process, of the density of a relation that describes how the frozen class of a vertex grows when an edge at this vertex is added. We prove it by charging this density to the growth of the logarithm of the size of the frozen class. Bounds on hitting and occupation times through a function whose expected increments can be controlled are classical [39 Sections 2–3] . In the k k k [16 Lemma 3.1] , and a later preprint charges the visits of the process above a level of the probability of becoming unsatisfiable to the expected lifetime at the first visit [53 Section 4.1, Lemma 4.1] . A related bounded-potential occupation argument appears in [52 Lemmas 2.2 and 4.1] : expected increments of a bounded monotone potential control accumulated powers of short-clause killing probabilities. The inequality that we need between the density of the relation and the expected growth of the frozen class, Lemma 6.9, is specific to colorings.
Finally, we compare Theorem 1.2 with known bounds on the width of transitions. At fixed probability levels, for every nontrivial increasing graph property that is invariant under relabeling, Friedgut and Kalai proved that the transition in G ( n , p ) G(n,p) G ( n , p ) p p p O ( 1 / log n ) O(1/\log n) O ( 1/ log n ) [33 Theorem 1.1, p. 2993] , and Bourgain and Kalai improved this to O ( ( log n ) − 2 + δ ) O((\log n)^{-2+\delta}) O (( log n ) − 2 + δ ) δ > 0 \delta> 0 δ > 0 O ( ( log n ) − 2 ) O((\log n)^{-2}) O (( log n ) − 2 ) [50 Theorem 1.1] . These additive bounds do not give a shrinking relative window when p p p 1 / n 1/n 1/ n p p p p p p [33 Corollary 3.3, p. 2996] gives an upper bound O ( 1 / n ) O(1/n) O ( 1/ n ) p = Θ ( 1 / n ) p = \Theta(1/n) p = Θ ( 1/ n ) O ( n ) O(n) O ( n ) q q q o ( n ) o(n) o ( n ) O ( n ( log n ) − 3 / 2 ) O(n(\log n)^{-3/2}) O ( n ( log n ) − 3/2 ) n \sqrt{n} n q ≥ 3 q \ge3 q ≥ 3 [62 Corollary 6] .
We do not know the true order of the window. For comparison, the window of random 2-SAT has order n 2 / 3 n^{2/3} n 2/3 [11 Theorem 1.1] . For random k k k n 1 / 2 + 1 / k n^{1/2+1/k} n 1/2 + 1/ k [16 Theorem 1.1] , and two further preprints claim that the hitting time of unsatisfiability has variance of order n n n k ≥ 4 k \ge4 k ≥ 4 [53 Theorem 1.1] and for k = 3 k = 3 k = 3 [52 Theorem 1.1] . As we explain in Remark 7.3, when the probability that H H H n \sqrt{n} n
Organization
Section 2 gives an overview of the proof. Section 3 collects definitions and elementary facts on the edge process, frozen pairs, and expected lifetimes. Section 4 states the convergence criterion and reduces the window bound to two estimates for the final edge, one for endpoints of low degree and one for endpoints of high degree. Section 5 proves the first estimate. This is the main place where the assumption q ≥ 3 q \ge3 q ≥ 3 q = 2 q = 2 q = 2
Proof overview
Notation. We fix an integer q ≥ 3 q \ge3 q ≥ 3 k ≥ 1 k \ge1 k ≥ 1 [ k ] = { 1 , … , k } [k] = \{1, \ldots, k\} [ k ] = { 1 , … , k } P \mathbb{P} P E \mathbb{E} E Var \operatorname{Var} Var 1 A \mathbf{1}_A 1 A A A A 1 { ⋅ } \mathbf{1}\{\cdot\} 1 { ⋅ } A A A X X X A A A E [ X ; A ] \mathbb{E}[X; A] E [ X ; A ] X X X A A A E [ X 1 A ] \mathbb{E}[X\mathbf{1}_A] E [ X 1 A ] X X X A A A Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) μ ≥ 0 \mu\ge0 μ ≥ 0 Bin ( m , p ) \operatorname{Bin}(m,p) Bin ( m , p ) m ≥ 0 m \ge0 m ≥ 0 p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] q q q n n n C C C C ′ C' C ′ for all large n n n when a statement holds for every n ≥ n 0 n \ge n_0 n ≥ n 0 n 0 n_0 n 0
Our proof has four steps. Step 1 deduces a limiting threshold from a short interval containing the loss of colorability with high probability. Step 2 bounds the length of this interval through the expected number of further pair additions needed to destroy colorability of the last colorable graph. Step 3 bounds the advance of non-colorability under random vertex identification to treat final edges at vertices of low degree. Step 4 controls the growth of classes of vertices equal in every coloring to treat final edges at vertices of high degree. Steps 1 and 2 adapt the k k k [16 , 51 ] ; our new estimates in Steps 3 and 4 bound the expected lifetime of the last colorable graph at the final edge. In the k k k k − 1 k - 1 k − 1 k k k [16 , 51 , 53 ] ; this comparison has no direct analogue for colorings, since permuting the colors shows that no vertex has the same color in every coloring. The dependencies between these steps are recorded in Figure 2.1 . Remarks labeled heuristic describe the motivation for the estimates.
Figure 2.1. The main steps of the proof. An arrow means that the first result is used to prove the second. The left branch is the low-degree estimate of Section 5, and the right branch is the high-degree estimate of § 7 . The convergence criterion is proved in Appendix § 9 .
We begin by defining the edge process. For a finite set S S S S S S multigraph on S S S occurrences . We call a map σ : S → [ q ] \sigma: S \to[q] σ : S → [ q ] colorable when there exists such a map. For a multigraph G G G e e e G + e G + e G + e e e e e e e G G G G − e G - e G − e deg G ( u ) \operatorname{deg}_G(u) deg G ( u ) u u u uniform pair .
Let n ≥ 2 q n \ge2q n ≥ 2 q e 1 , e 2 , … e_1,e_2,\ldots e 1 , e 2 , … [ n ] [n] [ n ] edge process . For each integer m ≥ 0 m \ge0 m ≥ 0 G m G_m G m m m m H H H m m m G m G_m G m F m \mathcal{F}_m F m σ \sigma σ e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m H H H
Step 1: the window determines the threshold
We set γ q = log q / log ( q / ( q − 1 ) ) \gamma_q = \log q/\log(q/(q-1)) γ q = log q / log ( q / ( q − 1 )) Λ = γ q + 2 \Lambda= \gamma_q + 2 Λ = γ q + 2 M = ⌊ Λ n ⌋ M = \lfloor\Lambda n \rfloor M = ⌊ Λ n ⌋ V V V M M M V = min ( H − 1 , M ) V = \min(H-1,M) V = min ( H − 1 , M ) V V V e 1 , … , e M e_1,\ldots,e_M e 1 , … , e M η ∈ ( 0 , 1 / 2 ] \eta\in(0,1/2] η ∈ ( 0 , 1/2 ] L n ( η ) L_n(\eta) L n ( η ) m m m P ( V ≤ m ) ≥ η \mathbb{P}(V \le m) \ge\eta P ( V ≤ m ) ≥ η U n ( η ) U_n(\eta) U n ( η ) m ∈ { 0 , … , M } m \in\{0,\ldots,M\} m ∈ { 0 , … , M } P ( V ≥ m ) ≥ η \mathbb{P}(V \ge m) \ge\eta P ( V ≥ m ) ≥ η w n ( η ) = U n ( η ) − L n ( η ) w_n(\eta)=U_n(\eta)-L_n(\eta) w n ( η ) = U n ( η ) − L n ( η ) window at level η \eta η
L n ( η ) = min { m ≥ 0 : P ( V ≤ m ) ≥ η } , U n ( η ) = max { m ∈ { 0 , … , M } : P ( V ≥ m ) ≥ η } . (5) \begin{aligned}
L_n(\eta)&=\min\{m\ge0:\mathbb{P}(V\le m)\ge\eta\},\\
U_n(\eta)&=\max\{m\in\{0,\ldots,M\}:\mathbb{P}(V\ge m)\ge\eta\}. \tag*{(5)}
\end{aligned} L n ( η ) U n ( η ) = min { m ≥ 0 : P ( V ≤ m ) ≥ η } , = max { m ∈ { 0 , … , M } : P ( V ≥ m ) ≥ η } . (5) Here the extrema are over nonnegative integers. By Lemma 3.4 , G m G_m G m 1 − η 1-\eta 1 − η 0 ≤ m ≤ L n ( η ) 0\le m\le L_n(\eta) 0 ≤ m ≤ L n ( η ) η \eta η U n ( η ) < m ≤ M U_n(\eta)<m\le M U n ( η ) < m ≤ M Lemma 3.17 (b) with c 0 = 1 / 4 c_0=1/4 c 0 = 1/4 c 1 = ( γ q + Λ ) / 2 c_1=(\gamma_q+\Lambda)/2 c 1 = ( γ q + Λ ) /2 n / 4 < L n ( η ) ≤ U n ( η ) < M n/4<L_n(\eta)\le U_n(\eta)<M n /4 < L n ( η ) ≤ U n ( η ) < M n n n q ≥ 3 q\ge3 q ≥ 3
We define η 0 \eta_0 η 0
η 0 = e − 2 Λ 8 . \eta_0=\frac{e^{-2\Lambda}}{8}. η 0 = 8 e − 2Λ . We will use the following convergence criterion, proved in Theorem 4.1 . Let ( δ n ) n ≥ 1 (\delta_n)_{n\ge1} ( δ n ) n ≥ 1 ( 0 , 1 ] (0,1] ( 0 , 1 ] ∑ k ≥ 1 δ ⌊ e k ⌋ < ∞ \sum_{k\ge1}\delta_{\lfloor e^k\rfloor}<\infty ∑ k ≥ 1 δ ⌊ e k ⌋ < ∞ η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] K ′ ( η ) > 0 K'(\eta)>0 K ′ ( η ) > 0
w n ( η ) ≤ K ′ ( η ) n δ n for all large n . (6) w_n(\eta)\le K'(\eta)n\delta_n\qquad\text{for all large }n. \tag*{(6)} w n ( η ) ≤ K ′ ( η ) n δ n for all large n . (6) Then there exists α q ∈ [ 1 / 2 , γ q ] \alpha_q\in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ] L n ( η ) / n → α q L_n(\eta)/n\to\alpha_q L n ( η ) / n → α q U n ( η ) / n → α q U_n(\eta)/n\to\alpha_q U n ( η ) / n → α q η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] c ≥ 0 c\ge0 c ≥ 0 c ≠ α q c\ne\alpha_q c = α q
lim n → ∞ P ( H > ⌊ c n ⌋ ) = { 1 if c < α q , 0 if c > α q . \lim_{n\to\infty}\mathbb{P}(H>\lfloor cn\rfloor)=
\begin{cases}
1 & \text{if }c<\alpha_q,\\
0 & \text{if }c>\alpha_q.
\end{cases} n → ∞ lim P ( H > ⌊ c n ⌋) = { 1 0 if c < α q , if c > α q . The same limits hold for the probability of colorability of a uniformly random simple graph on [ n ] [n] [ n ] ⌊ c n ⌋ \lfloor cn\rfloor ⌊ c n ⌋ [ n ] [n] [ n ] 2 c / n 2c/n 2 c / n n n n δ n = ( log n ) − 3 / 2 \delta_n=(\log n)^{-3/2} δ n = ( log n ) − 3/2 n ≥ 3 n\ge3 n ≥ 3 δ 1 = δ 2 = 1 \delta_1=\delta_2=1 δ 1 = δ 2 = 1 δ ⌊ e k ⌋ ≤ ( k − 1 ) − 3 / 2 \delta_{\lfloor e^k\rfloor}\le(k-1)^{-3/2} δ ⌊ e k ⌋ ≤ ( k − 1 ) − 3/2 k ≥ 2 k\ge2 k ≥ 2 Theorem 1.1 . For the error probability ϵ ∈ ( 0 , 1 ) \epsilon\in(0,1) ϵ ∈ ( 0 , 1 ) Theorem 1.2 , we set η 1 = min ( ϵ / 2 , η 0 ) \eta_1=\min(\epsilon/2,\eta_0) η 1 = min ( ϵ /2 , η 0 ) m n = L n ( η 1 ) m_n=L_n(\eta_1) m n = L n ( η 1 ) η 1 \eta_1 η 1 m n m_n m n m n / n → α q m_n/n\to\alpha_q m n / n → α q § 8.2 .
To prove Theorem 4.1 , we compare quantiles at different numbers of vertices through an auxiliary Poisson model. For every integer n ≥ 1 n\ge1 n ≥ 1 c ≥ 0 c\ge0 c ≥ 0 A n ( c ) A_n(c) A n ( c ) Pois ( c n ) \operatorname{Pois}(cn) Pois ( c n ) [ n ] × [ n ] [n]\times[n] [ n ] × [ n ] loop . On a finite vertex set, we call a finite multiset of unordered pairs, allowing coincident endpoints, a multigraph with loops . We require different colors at the endpoints of every occurrence; a loop prevents colorability. We let ρ n ( c ) \rho_n(c) ρ n ( c ) A n ( c ) A_n(c) A n ( c )
ρ N 1 + N 2 ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) for all integers N 1 , N 2 ≥ 1 and all c ≥ 0. (7) \rho_{N_1+N_2}(c) \ge\rho_{N_1}(c)\rho_{N_2}(c) \quad\text{for all integers } N_1,N_2 \ge1 \text{ and all } c \ge0. \tag*{(7)} ρ N 1 + N 2 ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) for all integers N 1 , N 2 ≥ 1 and all c ≥ 0. (7) Let N = N 1 + N 2 N=N_1+N_2 N = N 1 + N 2 S 1 = [ N 1 ] S_1=[N_1] S 1 = [ N 1 ] S 2 = [ N ] ∖ [ N 1 ] S_2=[N]\setminus[N_1] S 2 = [ N ] ∖ [ N 1 ] G G G u u u frozen class C G ( u ) C_G(u) C G ( u ) u u u [24 ] . For 0 ≤ υ ≤ c 0\leq\upsilon\leq c 0 ≤ υ ≤ c A ( υ ) A^{(\upsilon)} A ( υ ) Pois ( N υ ) \operatorname{Pois}(N\upsilon) Pois ( N υ ) [ N ] 2 [N]^2 [ N ] 2 j = 1 , 2 j=1,2 j = 1 , 2 Pois ( N j ( c − υ ) ) \operatorname{Pois}(N_j(c-\upsilon)) Pois ( N j ( c − υ )) S j 2 S_j^2 S j 2 χ ( υ ) = P ( A ( υ ) is colorable ) \chi(\upsilon)=\mathbb{P}(A^{(\upsilon)}\text{ is colorable}) χ ( υ ) = P ( A ( υ ) is colorable ) A ( υ ) A^{(\upsilon)} A ( υ ) C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k
χ ′ ( υ ) = E [ 1 { A ( υ ) is colorable } ∑ i = 1 k ( ∣ C i ∩ S 1 ∣ 2 N 1 + ∣ C i ∩ S 2 ∣ 2 N 2 − ∣ C i ∣ 2 N ) ] . \chi'(\upsilon)=\mathbb{E}\left[\mathbf{1}\{A^{(\upsilon)}\text{ is colorable}\}\sum_{i=1}^{k}\left(\frac{\lvert C_i\cap S_1\rvert^2}{N_1}+\frac{\lvert C_i\cap S_2\rvert^2}{N_2}-\frac{\lvert C_i\rvert^2}{N}\right)\right]. χ ′ ( υ ) = E [ 1 { A ( υ ) is colorable } i = 1 ∑ k ( N 1 ∣ C i ∩ S 1 ∣ 2 + N 2 ∣ C i ∩ S 2 ∣ 2 − N ∣ C i ∣ 2 ) ] . For each such class,
∣ C i ∩ S 1 ∣ 2 N 1 + ∣ C i ∩ S 2 ∣ 2 N 2 − ∣ C i ∣ 2 N 1 + N 2 = ( N 2 ∣ C i ∩ S 1 ∣ − N 1 ∣ C i ∩ S 2 ∣ ) 2 N 1 N 2 ( N 1 + N 2 ) ≥ 0. (8) \frac{\lvert C_i\cap S_1\rvert^2}{N_1}+\frac{\lvert C_i\cap S_2\rvert^2}{N_2}-\frac{\lvert C_i\rvert^2}{N_1+N_2}=\frac{(N_2\lvert C_i\cap S_1\rvert-N_1\lvert C_i\cap S_2\rvert)^2}{N_1N_2(N_1+N_2)}\geq0. \tag*{(8)} N 1 ∣ C i ∩ S 1 ∣ 2 + N 2 ∣ C i ∩ S 2 ∣ 2 − N 1 + N 2 ∣ C i ∣ 2 = N 1 N 2 ( N 1 + N 2 ) ( N 2 ∣ C i ∩ S 1 ∣ − N 1 ∣ C i ∩ S 2 ∣ ) 2 ≥ 0. (8) It follows that ρ N ( c ) = χ ( c ) ≥ χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) \rho_N(c)=\chi(c)\geq\chi(0)=\rho_{N_1}(c)\rho_{N_2}(c) ρ N ( c ) = χ ( c ) ≥ χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) [10 Theorem 5 and Proposition 2] , in Poisson form as in [51 Lemma 5.2] .
Fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] ξ n = L n ( η ) / n \xi_n=L_n(\eta)/n ξ n = L n ( η ) / n Δ n = ⌈ n log n ⌉ \Delta_n=\lceil\sqrt{n}\log n\rceil Δ n = ⌈ n log n ⌉ n ≥ 2 q n\geq2q n ≥ 2 q c ≥ 0 c\geq0 c ≥ 0 J J J Pois ( c ( n − 1 ) ) \operatorname{Pois}(c(n-1)) Pois ( c ( n − 1 )) H H H
ρ n ( c ) = e − c P ( H > J ) . \rho_n(c)=e^{-c}\mathbb{P}(H>J). ρ n ( c ) = e − c P ( H > J ) . For 0 < c ≤ Λ 0<c\leq\Lambda 0 < c ≤ Λ n n n
c ( n − 1 ) ≤ L n ( η ) − Δ n ⟹ ρ n ( c ) ≥ e − Λ ( 1 − η ) ( 1 − e − ( log n ) 2 / ( 2 Λ ) ) , c(n-1)\leq L_n(\eta)-\Delta_n\quad\Longrightarrow\quad\rho_n(c)\geq e^{-\Lambda}(1-\eta)\left(1-e^{-(\log n)^2/(2\Lambda)}\right), c ( n − 1 ) ≤ L n ( η ) − Δ n ⟹ ρ n ( c ) ≥ e − Λ ( 1 − η ) ( 1 − e − ( l o g n ) 2 / ( 2Λ ) ) , c ( n − 1 ) ≥ U n ( η ) + Δ n ⟹ ρ n ( c ) ≤ η + e − ( log n ) 2 / ( 2 Λ ) . c(n-1)\geq U_n(\eta)+\Delta_n\quad\Longrightarrow\quad\rho_n(c)\leq\eta+e^{-(\log n)^2/(2\Lambda)}. c ( n − 1 ) ≥ U n ( η ) + Δ n ⟹ ρ n ( c ) ≤ η + e − ( l o g n ) 2 / ( 2Λ ) . For large integers N 1 , N 2 N_1,N_2 N 1 , N 2 N 1 , N 2 , N = N 1 + N 2 N_1,N_2,N=N_1+N_2 N 1 , N 2 , N = N 1 + N 2
c = min j ∈ { 1 , 2 } ( ξ N j − 2 Δ N j N j ) . c=\min_{j\in\{1,2\}}\left(\xi_{N_j}-\frac{2\Delta_{N_j}}{N_j}\right). c = j ∈ { 1 , 2 } min ( ξ N j − N j 2 Δ N j ) . By the product inequality, we have ρ N ( c ) > e − 2 Λ / 2 > 2 η \rho_N(c)>e^{-2\Lambda/2}>2\eta ρ N ( c ) > e − 2Λ/2 > 2 η c ( N − 1 ) < U N ( η ) + Δ N c(N-1)<U_N(\eta)+\Delta_N c ( N − 1 ) < U N ( η ) + Δ N w N ( η ) w_N(\eta) w N ( η ) u ≥ 1 u \ge1 u ≥ 1 g ( u ) = K ( η ) δ ⌊ u ⌋ + ( Λ + 6 ) ( log u ) / u g(u) = K^{(\eta)}\delta_{\lfloor u\rfloor} + (\Lambda+ 6)(\log u)/\sqrt{u} g ( u ) = K ( η ) δ ⌊ u ⌋ + ( Λ + 6 ) ( log u ) / u Lemma A.4 , for all sufficiently large integers N 1 , N 2 N_1,N_2 N 1 , N 2
ζ ~ N 1 + N 2 ≥ min ( ζ ~ N 1 , ζ ~ N 2 ) − g ( min ( N 1 , N 2 ) ) , ζ ~ N 1 + 1 ≥ ζ ~ N 1 − g ( N 1 ) . (9) \widetilde{\zeta}_{N_1+N_2} \ge\min(\widetilde{\zeta}_{N_1},\widetilde{\zeta}_{N_2}) - g(\min(N_1,N_2)), \qquad\widetilde{\zeta}_{N_1+1} \ge\widetilde{\zeta}_{N_1} - g(N_1). \tag*{(9)} ζ N 1 + N 2 ≥ min ( ζ N 1 , ζ N 2 ) − g ( min ( N 1 , N 2 )) , ζ N 1 + 1 ≥ ζ N 1 − g ( N 1 ) . (9) For the adjacent-size inequality, we apply the product inequality with N 2 = 1 N_2 = 1 N 2 = 1 c = ζ ~ N 1 − 2 Δ N 1 / N 1 c = \widetilde{\zeta}_{N_1} - 2\Delta_{N_1}/N_1 c = ζ N 1 − 2 Δ N 1 / N 1 ρ 1 ( c ) = e − c \rho_1(c) = e^{-c} ρ 1 ( c ) = e − c N 2 = 1 N_2 = 1 N 2 = 1
Fix a sufficiently large integer ℓ \ell ℓ n ≥ ℓ 2 n \ge\ell^2 n ≥ ℓ 2 n n n ℓ \ell ℓ ℓ + 1 \ell+ 1 ℓ + 1 ( N 1 , N 2 ) (N_1,N_2) ( N 1 , N 2 ) ℓ + 1 \ell+ 1 ℓ + 1 N 1 = ℓ N_1 = \ell N 1 = ℓ 4 / 3 4/3 4/3 g g g [ e 2 , ∞ ) [e^2,\infty) [ e 2 , ∞ ) ∑ k ≥ 1 g ( e k ) < ∞ \sum_{k\ge1} g(e^k) < \infty ∑ k ≥ 1 g ( e k ) < ∞ Lemma A.5 to this g g g ( ζ ~ n ) (\widetilde{\zeta}_n) ( ζ n ) u 0 = e 2 u_0 = e^2 u 0 = e 2
ζ ~ n ≥ ζ ~ ℓ − g ( ℓ ) − ∑ i ≥ 1 g ( ℓ 4 ( 4 3 ) i ) (10) \widetilde{\zeta}_n \ge\widetilde{\zeta}_\ell- g(\ell) - \sum_{i\ge1} g\left(\frac{\ell}{4}\left(\frac{4}{3}\right)^i\right) \tag*{(10)} ζ n ≥ ζ ℓ − g ( ℓ ) − i ≥ 1 ∑ g ( 4 ℓ ( 3 4 ) i ) (10) for all n ≥ ℓ 2 n \ge\ell^2 n ≥ ℓ 2
The right side differs from ζ ~ ℓ \widetilde{\zeta}_\ell ζ ℓ ℓ → ∞ \ell\to\infty ℓ → ∞ ∑ k g ( e k ) < ∞ \sum_k g(e^k) < \infty ∑ k g ( e k ) < ∞ lim inf n ζ ~ n ≥ lim sup ℓ ζ ~ ℓ \liminf_n \widetilde{\zeta}_n \ge\limsup_\ell\widetilde{\zeta}_\ell lim inf n ζ n ≥ lim sup ℓ ζ ℓ ζ ~ n \widetilde{\zeta}_n ζ n n / log n n/\log n n / log n
Finally, for 0 < η − < η + ≤ η 0 0 < \eta_- < \eta_+ \le\eta_0 0 < η − < η + ≤ η 0 L n ( η − ) ≤ L n ( η + ) ≤ U n ( η − ) L_n(\eta_-) \le L_n(\eta_+) \le U_n(\eta_-) L n ( η − ) ≤ L n ( η + ) ≤ U n ( η − ) α q \alpha_q α q η → 0 \eta\to0 η → 0 n → ∞ n \to\infty n → ∞ P ( H > ⌊ c n ⌋ ) \mathbb{P}(H > \lfloor cn\rfloor) P ( H > ⌊ c n ⌋) Lemma A.7 (a) with m = ⌊ c n ⌋ m = \lfloor cn\rfloor m = ⌊ c n ⌋ 0 ≤ c < Λ 0 \le c < \Lambda 0 ≤ c < Λ c c c d ≥ 0 d \ge0 d ≥ 0 p = d / n p = d/n p = d / n m = ⌊ c n ⌋ m = \lfloor cn\rfloor m = ⌊ c n ⌋ d / 2 < c < α q d/2 < c < \alpha_q d /2 < c < α q α q < c < d / 2 \alpha_q < c < d/2 α q < c < d /2 [51 Section 5] , which compares expectations rather than quantiles and has polynomially small errors. Summable errors appear in the classical theory of nearly superadditive sequences [25 Theorem 23] , [1 arXiv:1006.3786v1, Section 4.3, Lemma 8 and Remark 3] .
Step 2: the window and the last colorable graph
We fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L L L U U U w w w L n ( η ) L_n(\eta) L n ( η ) U n ( η ) U_n(\eta) U n ( η ) w n ( η ) w_n(\eta) w n ( η ) Γ = G H − 1 \Gamma= G_{H-1} Γ = G H − 1 e H e_H e H G G G T G T_G T G T G = 0 T_G = 0 T G = 0 G G G T G = ∞ T_G = \infty T G = ∞ b ≥ 0 b \ge0 b ≥ 0 expected lifetime and its cap by
r ( G ) = E T G , r b ( G ) = E min ( T G , b ) . r(G) = \mathbb{E}T_G,\qquad r_b(G) = \mathbb{E}\min(T_G,b). r ( G ) = E T G , r b ( G ) = E min ( T G , b ) . We define Y = min ( U , max ( L , V ) ) \mathcal{Y} = \min(U,\max(L,V)) Y = min ( U , max ( L , V )) V V V [ L , U ] [L,U] [ L , U ]
Y = min ( U , max ( L , V ) ) = L + ∑ m = L + 1 U 1 { G m is colorable } . \mathcal{Y} = \min(U,\max(L,V)) = L + \sum_{m=L+1}^{U} \boldsymbol{1}\{G_m\text{ is colorable}\}. Y = min ( U , max ( L , V )) = L + m = L + 1 ∑ U 1 { G m is colorable } . For 1 ≤ i ≤ U 1 \le i \le U 1 ≤ i ≤ U Y ( − i ) \mathcal{Y}^{(-i)} Y ( − i ) e i e_i e i G m G_m G m [12 (3.6)] states that, for a square-integrable function Z Z Z X 1 , … , X U X_1,\ldots,X_U X 1 , … , X U Z i Z_i Z i X i X_i X i
Var ( Z ) ≤ ∑ i = 1 U E [ ( Z − Z i ) 2 ] . \operatorname{Var}(Z) \le\sum_{i=1}^{U}\mathbb{E}[(Z-Z_i)^2]. Var ( Z ) ≤ i = 1 ∑ U E [( Z − Z i ) 2 ] . The variables Y \mathcal{Y} Y Y ( − i ) \mathcal{Y}^{(-i)} Y ( − i ) X i = e i X_i=e_i X i = e i Z = Y Z=\mathcal{Y} Z = Y Z i = Y ( − i ) Z_i=\mathcal{Y}^{(-i)} Z i = Y ( − i )
Var ( Y ) ≤ ∑ i = 1 U E [ ( Y ( − i ) − Y ) 2 ] . \operatorname{Var}(\mathcal{Y}) \le\sum_{i=1}^{U}\mathbb{E}[(\mathcal{Y}^{(-i)}-\mathcal{Y})^2]. Var ( Y ) ≤ i = 1 ∑ U E [( Y ( − i ) − Y ) 2 ] . We evaluate this sum exactly in Lemma 4.3:
∑ i = 1 U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . (11) \sum_{i=1}^{U}\mathbb{E}\left[(\mathcal{Y}^{(-i)}-\mathcal{Y})^2\right]
= \mathbb{E}\left[H(2r_{U-H+1}(\Gamma)-1);L < H \le U\right]. \tag*{(11)} i = 1 ∑ U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . (11) The difference Y ( − i ) − Y \mathcal{Y}^{(-i)}-\mathcal{Y} Y ( − i ) − Y m ∈ ( L , U ] m \in(L,U] m ∈ ( L , U ] G m G_m G m G m G_m G m e i e_i e i { u , v } \{u,v\} { u , v } frozen pair of a colorable multigraph G G G u u u v v v G G G S S S ∣ S ∣ ≥ 2 |S| \ge2 ∣ S ∣ ≥ 2 κ ( G ) \kappa(G) κ ( G ) ( ∣ S ∣ 2 ) \binom{|S|}{2} ( 2 ∣ S ∣ ) κ ( G ) = 0 \kappa(G)=0 κ ( G ) = 0 G G G m m m e i e_i e i G G G κ ( G ) \kappa(G) κ ( G ) G G G 2 r U − m + 1 ( G ) − 1 2r_{U-m+1}(G)-1 2 r U − m + 1 ( G ) − 1 H H H h h h [ n ] [n] [ n ] m ≥ 0 m \ge0 m ≥ 0
E [ h ( Γ ) ; H = m + 1 ] = E [ κ ( G m ) h ( G m ) ] . (12) \mathbb{E}[h(\Gamma);H=m+1]=\mathbb{E}[\kappa(G_m)h(G_m)]. \tag*{(12)} E [ h ( Γ ) ; H = m + 1 ] = E [ κ ( G m ) h ( G m )] . (12) More generally, let E \mathcal{E} E ( G , e ) (G,e) ( G , e ) G G G [ n ] [n] [ n ] e e e G G G κ E ( G ) \kappa_{\mathcal{E}}(G) κ E ( G ) e e e ( G , e ) ∈ E (G,e) \in\mathcal{E} ( G , e ) ∈ E
E [ h ( Γ ) ; H = m + 1 , ( Γ , e H ) ∈ E ] = E [ κ E ( G m ) h ( G m ) ] . \mathbb{E}[h(\Gamma); H = m+1, (\Gamma,e_H) \in\mathcal{E}] = \mathbb{E}[\kappa_{\mathcal{E}}(G_m)h(G_m)]. E [ h ( Γ ) ; H = m + 1 , ( Γ , e H ) ∈ E ] = E [ κ E ( G m ) h ( G m )] . Taking E \mathcal{E} E (12) . By Lemmas 3.7 and 3.8 , this identity follows from the fact that P ( H = m + 1 ∣ F m ) = κ ( G m ) \mathbb{P}(H = m+1 \mid\mathcal{F}_m) = \kappa(G_m) P ( H = m + 1 ∣ F m ) = κ ( G m ) (12) repeatedly, also for final edges of a prescribed kind, to convert statements about Γ \Gamma Γ G m G_m G m [51 Lemmas 3.1 and 3.2] , and the role of the lifetime of the last state goes back to Aldous [7, Theorem 1(b)] and its application by Carenini [16, Theorem 4.1]. The related argument with weighted deletion appears in [53 Section 3.3, Lemma 3.4] ; see § 2.3 .
Lemma 4.4 uses an independent copy of Y Y Y Var ( Y ) ≥ P ( Y = L ) P ( Y = U ) w 2 ≥ η 2 w 2 \operatorname{Var}(Y) \ge\mathbb{P}(Y = L)\mathbb{P}(Y = U)w^2 \ge\eta^2w^2 Var ( Y ) ≥ P ( Y = L ) P ( Y = U ) w 2 ≥ η 2 w 2 { L < H ≤ U } \{L < H \le U\} { L < H ≤ U } r U − H + 1 ( Γ ) ≤ min ( w , r ( Γ ) ) r_{U-H+1}(\Gamma) \le\min(w,r(\Gamma)) r U − H + 1 ( Γ ) ≤ min ( w , r ( Γ )) (11) , we find
η 2 w 2 ≤ Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ ) ) ; L < H ≤ U ] . (13) \eta^2w^2 \le\operatorname{Var}(Y) \le2U\mathbb{E}\left[\min(w,r(\Gamma));L < H \le U\right]. \tag*{(13)} η 2 w 2 ≤ Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ )) ; L < H ≤ U ] . (13) To bound the window, it suffices to bound the expected lifetime at final edges inside it. We set ϵ n = ( log n ) − 3 / 2 \epsilon_n = (\log n)^{-3/2} ϵ n = ( log n ) − 3/2 n ≥ 3 n \ge3 n ≥ 3 ϵ 1 = ϵ 2 = 1 \epsilon_1 = \epsilon_2 = 1 ϵ 1 = ϵ 2 = 1 x n = n ϵ n 2 x_n = n\epsilon_n^2 x n = n ϵ n 2 n ≥ 2 q n \ge2q n ≥ 2 q x n = n ( log n ) − 3 x_n = n(\log n)^{-3} x n = n ( log n ) − 3 K 0 ≥ 0 K_0 \ge0 K 0 ≥ 0 Proposition 4.5
E [ min ( n ϵ n , r ( Γ ) ) ; L < H ≤ U ] ≤ K 0 n ϵ n 2 ⟹ w ≤ max ( 1 , 2 Λ K 0 / η 2 ) n ϵ n . \mathbb{E}\left[\min(n\epsilon_n,r(\Gamma));L < H \le U\right] \le K_0n\epsilon_n^2 \quad\Longrightarrow\quad w \le\max(1,2\Lambda K_0/\eta^2)n\epsilon_n. E [ min ( n ϵ n , r ( Γ )) ; L < H ≤ U ] ≤ K 0 n ϵ n 2 ⟹ w ≤ max ( 1 , 2Λ K 0 / η 2 ) n ϵ n . Since
min ( n ϵ n , r ( Γ ) ) ≤ x n + n ϵ n 1 { r ( Γ ) > x n } , \min(n\epsilon_n,r(\Gamma)) \le x_n+n\epsilon_n\mathbf{1}\{r(\Gamma)>x_n\}, min ( n ϵ n , r ( Γ )) ≤ x n + n ϵ n 1 { r ( Γ ) > x n } , it suffices to prove
P ( L < H ≤ U , r ( Γ ) > x n ) ≤ 2 ϵ n . (14) \mathbb{P}(L < H \le U,\ r(\Gamma)>x_n) \le2\epsilon_n. \tag*{(14)} P ( L < H ≤ U , r ( Γ ) > x n ) ≤ 2 ϵ n . (14) and Proposition 4.6 gives the conclusion with K 0 = 3 K_0 = 3 K 0 = 3
However, the elementary lifetime bound for a single graph does not give (14) . For a colorable multigraph G G G [ n ] [n] [ n ] κ ( G ) ≤ 1 / r ( G ) \kappa(G) \le1/r(G) κ ( G ) ≤ 1/ r ( G ) Lemma 3.10 . We apply (12) with h ( G ) = 1 { r ( G ) > x n } h(G) = \mathbf{1}\{r(G)>x_n\} h ( G ) = 1 { r ( G ) > x n } L ≤ m < U L \le m < U L ≤ m < U κ ( G ) r ( G ) ≤ 1 \kappa(G)r(G) \le1 κ ( G ) r ( G ) ≤ 1 1 / x n 1/x_n 1/ x n (14) , and summing over the M M M M / x n ≈ Λ ( log n ) 3 M/x_n \approx\Lambda(\log n)^3 M / x n ≈ Λ ( log n ) 3 D n = ⌊ ( log n ) 1 / 4 ⌋ D_n = \lfloor(\log n)^{1/4}\rfloor D n = ⌊( log n ) 1/4 ⌋ low-degree if its degree in Γ \Gamma Γ D n D_n D n high-degree otherwise. We split according to these degrees and prove that, for all large n n n
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n , (15) \mathbb{P}\left(H < \infty,\ \min_{v\in e_H}\deg_{\Gamma}(v) \le D_n,\ r(\Gamma) > x_n\right) \le\varepsilon_n, \tag*{(15)} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n , (15) P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ ε n . (16) \mathbb{P}\left(H \le M,\ \max_{v\in e_H}\deg_{\Gamma}(v) > D_n\right) \le\varepsilon_n. \tag*{(16)} P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ ε n . (16) These are Corollary 5.8 and Theorem 6.13 . Each final edge in the event of (14) belongs to one of these events, since U ≤ M U \le M U ≤ M M M M Proposition 4.6 , they give w n ( η ) ≤ ( 6 Λ / η 2 ) n ( log n ) − 3 / 2 w_n(\eta) \le(6\Lambda/\eta^2)n(\log n)^{-3/2} w n ( η ) ≤ ( 6Λ/ η 2 ) n ( log n ) − 3/2 n n n Theorem 7.1 .
Step 3: final edges at a vertex of low degree
We now explain the proof of (15) . Theorem 5.7 shows that there exists C l o w > 0 C_{\mathrm{low}} > 0 C low > 0 q q q n n n y > 0 y > 0 y > 0
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ C l o w y − 1 / 2 n 3 / 8 , (17) \mathbb{P}\left(H < \infty,\ \min_{v\in e_H}\deg_{\Gamma}(v) \le D_n,\ r(\Gamma) > y\right) \le C_{\mathrm{low}}y^{-1/2}n^{3/8}, \tag*{(17)} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ C low y − 1/2 n 3/8 , (17) and with y = x n y = x_n y = x n C l o w ( log n ) 3 / 2 n − 1 / 8 C_{\mathrm{low}}(\log n)^{3/2}n^{-1/8} C low ( log n ) 3/2 n − 1/8 ε n \varepsilon_n ε n n n n
We use the restriction on the lifetime to bound the sizes of the frozen classes. We set A q = 2 q log q + 2 q + 1 A_q = 2q\log q + 2q + 1 A q = 2 q log q + 2 q + 1 G G G [ n ] [n] [ n ] κ ( G ) r ( G ) ≤ 1 \kappa(G)r(G) \le1 κ ( G ) r ( G ) ≤ 1 r ( G ) ≤ A q n r(G) \le A_qn r ( G ) ≤ A q n Lemma 3.10 . Applying Lemma 3.11 with vertex set [ n ] [n] [ n ] n ≥ 2 q n \ge2q n ≥ 2 q y > 0 y > 0 y > 0 r ( G ) > y r(G) > y r ( G ) > y C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k G G G
max 1 ≤ i ≤ k ∣ C i ∣ 2 n 2 ≤ ∑ i = 1 k ∣ C i ∣ 2 n 2 = ( 1 − 1 n ) κ ( G ) + 1 n ≤ 1 + A q r ( G ) ≤ 1 + A q y , (18) \max_{1\le i\le k}\frac{|C_i|^2}{n^2} \le\sum_{i=1}^{k}\frac{|C_i|^2}{n^2} = \left(1-\frac{1}{n}\right)\kappa(G)+\frac{1}{n} \le\frac{1+A_q}{r(G)} \le\frac{1+A_q}{y}, \tag*{(18)} 1 ≤ i ≤ k max n 2 ∣ C i ∣ 2 ≤ i = 1 ∑ k n 2 ∣ C i ∣ 2 = ( 1 − n 1 ) κ ( G ) + n 1 ≤ r ( G ) 1 + A q ≤ y 1 + A q , (18) so every frozen class of G G G n ( ( 1 + A q ) / y ) 1 / 2 n\bigl((1+A_q)/y\bigr)^{1/2} n ( ( 1 + A q ) / y ) 1/2 Γ \Gamma Γ E \mathcal{E} E D n D_n D n h ( G ) = 1 { r ( G ) > y } h(G)=1\{r(G)>y\} h ( G ) = 1 { r ( G ) > y } m ≥ 0 m\ge0 m ≥ 0 ( Π t ) t ≥ 0 (\Pi_t)_{t\ge0} ( Π t ) t ≥ 0 G ( t ) = G Π t G(t)=G_{\Pi_t} G ( t ) = G Π t t ≥ 0 t\ge0 t ≥ 0 Poissonized process . On any other finite vertex set, the same term means the edge process evaluated at an independent Poisson clock of rate one. Applying Lemma 3.13 with θ = 1 \theta=1 θ = 1 G m G_m G m a 0 = n a_0 = n a 0 = n (17) by ( 2 n / ( n − 1 ) ) ( ( 1 + A q ) / y ) 1 / 2 (2n/(n-1))((1+A_q)/y)^{1/2} ( 2 n / ( n − 1 )) (( 1 + A q ) / y ) 1/2
∫ 0 ∞ P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) d t . (19) \int_0^\infty\mathbb{P}\bigl(\deg_{G(t)}(a_0) \le D_n \text{ and } a_0 \text{ lies in a frozen pair of } G(t)\bigr)\,\mathrm{d}t. \tag*{(19)} ∫ 0 ∞ P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) d t . (19) It remains to show that this integral, the expected total time that a 0 a_0 a 0 deg G ( t ) ( a 0 ) ≤ D n \deg_{G(t)}(a_0) \le D_n deg G ( t ) ( a 0 ) ≤ D n O ( n 3 / 8 ) O(n^{3/8}) O ( n 3/8 )
Let G G G [ n ] [n] [ n ] a , b ∈ [ n ] a,b \in[n] a , b ∈ [ n ] B B B [ n ] ∖ { a } [n] \setminus\{a\} [ n ] ∖ { a } a a a x x x a a a neighbor list of a a a G G G B B B G G G b b b a a a σ \sigma σ B B B
b ∈ C G ( a ) ∖ { a } ⟹ ∣ { σ ( v ) : v is an entry of x } ∣ ≥ q − 1 ≥ 2. b \in C_G(a) \setminus\{a\} \quad\Longrightarrow\quad\left|\{\sigma(v): v\text{ is an entry of }x\}\right| \ge q-1 \ge2. b ∈ C G ( a ) ∖ { a } ⟹ ∣ { σ ( v ) : v is an entry of x } ∣ ≥ q − 1 ≥ 2. We call a list never monochromatic in a colorable multigraph if no coloring of that multigraph gives the same color to all entries of the list; a coloring is constant on an empty list by convention. The displayed inequality says that x x x B B B q ≥ 3 q \ge3 q ≥ 3 q = 2 q=2 q = 2 a , v , b a,v,b a , v , b { a , v } \{a,v\} { a , v } { v , b } \{v,b\} { v , b } a , b a,b a , b ( v ) (v) ( v )
For a multigraph B B B Z Z Z B / Z B/Z B / Z Z Z Z Z Z Z Lemma 5.3 (iv) with F = B F=B F = B [58 ] , we have
B / Z is colorable ⟺ some coloring of B is constant on Z . B/Z\text{ is colorable}\quad\Longleftrightarrow\quad\text{some coloring of }B\text{ is constant on }Z. B / Z is colorable ⟺ some coloring of B is constant on Z . Taking Z Z Z x x x B / Z B/Z B / Z
We call the multigraph on [ n − 1 ] [n-1] [ n − 1 ] G ( t ) G(t) G ( t ) a 0 a_0 a 0 t t t x x x a 0 a_0 a 0 t t t S = [ n ] S=[n] S = [ n ] a = a 0 = n a=a_0=n a = a 0 = n [44 Theorems 5.2 and 5.8] . Before conditioning on the event in (19) , the background graph and this list are independent. The background graph has the law of the Poissonized process on [ n − 1 ] [n-1] [ n − 1 ] ( n − 2 ) t / n (n-2)t/n ( n − 2 ) t / n Pois ( 2 t / n ) \operatorname{Pois}(2t/n) Pois ( 2 t / n ) [ n − 1 ] [n-1] [ n − 1 ] D n D_n D n D n D_n D n Z Z Z D n D_n D n ( B m ) m ≥ 0 (B_m)_{m\ge0} ( B m ) m ≥ 0 m m m [ n − 1 ] [n-1] [ n − 1 ] D n D_n D n Z Z Z H ( n − 1 ) H^{(n-1)} H ( n − 1 ) H Z H_Z H Z B m B_m B m B m / Z B_m/Z B m / Z
H ( n − 1 ) − H Z = ∑ m ≥ 0 1 { B m is colorable , B m / Z is not colorable } . H^{(n-1)}-H_Z=\sum_{m\ge0}\mathbf{1}\{B_m\text{ is colorable},\ B_m/Z\text{ is not colorable}\}. H ( n − 1 ) − H Z = m ≥ 0 ∑ 1 { B m is colorable , B m / Z is not colorable } . We apply Lemma 3.13 on [ n − 1 ] [n-1] [ n − 1 ] θ = ( n − 2 ) / n \theta=(n-2)/n θ = ( n − 2 ) / n D n D_n D n
∫ 0 ∞ P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) d t ≤ n n − 2 E [ H ( n − 1 ) − H Z ] . (20) \int_0^\infty\mathbb{P}\bigl(\deg_{G(t)}(a_0)\le D_n\text{ and }a_0\text{ lies in a frozen pair of }G(t)\bigr)\,\mathrm{d}t
\le\frac{n}{n-2}\mathbb{E}\bigl[H^{(n-1)}-H_Z\bigr]. \tag*{(20)} ∫ 0 ∞ P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) d t ≤ n − 2 n E [ H ( n − 1 ) − H Z ] . (20) The right side is n / ( n − 2 ) n/(n-2) n / ( n − 2 ) D n D_n D n
Proposition 5.6 bounds this advance. Let N ≥ ⌈ 4 q log q ⌉ + 1 N\ge\lceil4q\log q\rceil+1 N ≥ ⌈ 4 q log q ⌉ + 1 1 ≤ D ≤ N / 4 1\le D\le N/4 1 ≤ D ≤ N /4 z 0 = 4 q log q + 1 z_0=4q\log q+1 z 0 = 4 q log q + 1 C q = 2 e z 0 C_q=2e^{z_0} C q = 2 e z 0 k > e D z 0 k>eDz_0 k > eD z 0 ( B m ) m ≥ 0 (B_m)_{m\ge0} ( B m ) m ≥ 0 m m m [ N ] [N] [ N ] Z Z Z D D D [ N ] [N] [ N ] H ( N ) H^{(N)} H ( N ) H Z H_Z H Z B m B_m B m B m / Z B_m/Z B m / Z
E [ H ( N ) − H Z ] ≤ C q ( D k + N ( e D z 0 k ) k + D 2 + 1 ) . (21) \mathbb{E}\bigl[H^{(N)}-H_Z\bigr]\le C_q\left(D^k+N\left(\frac{eDz_0}{k}\right)^k+D^2+1\right). \tag*{(21)} E [ H ( N ) − H Z ] ≤ C q ( D k + N ( k eD z 0 ) k + D 2 + 1 ) . (21) For all sufficiently large n n n k n = ⌈ log n / log log n ⌉ k_n=\lceil\log n/\log\log n\rceil k n = ⌈ log n / log log n ⌉ (21) with ( N , D , k ) = ( n − 1 , D n , k n ) (N,D,k)=(n-1,D_n,k_n) ( N , D , k ) = ( n − 1 , D n , k n )
D n k n ≤ n 1 / 4 ( log n ) 1 / 4 , ( n − 1 ) ( e D n z 0 k n ) k n ≤ n 3 / 8 . D_n^{k_n}\le n^{1/4}(\log n)^{1/4},\qquad(n-1)\left(\frac{eD_nz_0}{k_n}\right)^{k_n}\le n^{3/8}. D n k n ≤ n 1/4 ( log n ) 1/4 , ( n − 1 ) ( k n e D n z 0 ) k n ≤ n 3/8 . Using D n 2 ≤ ( log n ) 1 / 2 D_n^2\le(\log n)^{1/2} D n 2 ≤ ( log n ) 1/2
E [ H ( n − 1 ) − H Z ] ≤ 2 C q n 3 / 8 . \mathbb{E}\bigl[H^{(n-1)}-H_Z\bigr]\le2C_qn^{3/8}. E [ H ( n − 1 ) − H Z ] ≤ 2 C q n 3/8 . Together with the class-mass bound, the reduction to (19) , and (20) , this proves (17) .
To prove (21) , we compare the lists of neighbors before and after identification. After Poissonizing the edge process on [ N ] [N] [ N ] 2 q ( log q ) N 2q(\log q)N 2 q ( log q ) N 2 q 2q 2 q 0 ≤ s ≤ 2 q ( log q ) N 0\le s\le2q(\log q)N 0 ≤ s ≤ 2 q ( log q ) N D D D N − D N-D N − D Pois ( s / ( N 2 ) ) \operatorname{Pois}(s/\binom{N}{2}) Pois ( s / ( 2 N ) ) z ^ \hat z z ^
z ^ = ( N − D ) s ( N 2 ) ≤ z 0 . \hat z=\frac{(N-D)s}{\binom{N}{2}}\le z_0. z ^ = ( 2 N ) ( N − D ) s ≤ z 0 . The neighbor lists have independent Pois ( z ^ ) \operatorname{Pois}(\hat z) Pois ( z ^ ) O q ( D 2 ) O_q(D^2) O q ( D 2 ) Pois ( D z ^ ) \operatorname{Pois}(D\hat z) Pois ( D z ^ )
For comparison, replace the new vertex’s neighbor list by an independent list of Pois ( z ^ ) \operatorname{Pois}(\hat z) Pois ( z ^ ) F ( t ) F(t) F ( t ) N − D N-D N − D λ ′ = ( N − D + 1 2 ) / ( N 2 ) \lambda' = \binom{N-D+1}{2}/\binom{N}{2} λ ′ = ( 2 N − D + 1 ) / ( 2 N ) F ( λ ′ s ) F(\lambda' s) F ( λ ′ s ) s s s
V ′ = remaining vertex set , N ′ = N − D , o = new vertex , λ = 1 / ( N 2 ) , B ′ = background graph . V' = \text{remaining vertex set}, \qquad N' = N-D, \qquad o = \text{new vertex}, \\
\lambda= 1/\binom{N}{2}, \qquad B' = \text{background graph}. V ′ = remaining vertex set , N ′ = N − D , o = new vertex , λ = 1/ ( 2 N ) , B ′ = background graph . It bounds the probability that this replacement list destroys colorability of a colorable background graph by ( e z ^ − 1 ) E κ ( F ( λ ′ s ) ) (e^{\hat z}-1)\mathbb{E}\kappa(F(\lambda' s)) ( e z ^ − 1 ) E κ ( F ( λ ′ s ))
∫ 0 ∞ E κ ( F ( λ ′ s ) ) d s ≤ 1 λ ′ ≤ 16 9 . \int_0^\infty\mathbb{E}\kappa(F(\lambda' s))\,\mathrm{d}s \le\frac{1}{\lambda'} \le\frac{16}{9}. ∫ 0 ∞ E κ ( F ( λ ′ s )) d s ≤ λ ′ 1 ≤ 9 16 . We use the convention 0 0 = 1 0^0=1 0 0 = 1 z ≥ 0 z\ge0 z ≥ 0 j ≥ 0 j\ge0 j ≥ 0 π z ( j ) = e − z z j / j ! \pi_z(j)=e^{-z}z^j/j! π z ( j ) = e − z z j / j !
π D z ^ ( j ) = D j e − ( D − 1 ) z ^ π z ^ ( j ) ≤ D k π z ^ ( j ) for integers 0 ≤ j ≤ k . (22) \pi_{D\hat z}(j)=D^j e^{-(D-1)\hat z}\pi_{\hat z}(j)\le D^k\pi_{\hat z}(j) \qquad\text{for integers }0\le j\le k. \tag*{(22)} π D z ^ ( j ) = D j e − ( D − 1 ) z ^ π z ^ ( j ) ≤ D k π z ^ ( j ) for integers 0 ≤ j ≤ k . (22) We bound the probability that the identified vertex has more than k k k ( e D z 0 / k ) k (eDz_0/k)^k ( eD z 0 / k ) k (22) , this accounts for the terms in (2.17) from the length comparison, long lists, occurrences inside the identified set, and large times. The limitation of this comparison is quantified in Remark 5.9: at degree of order $\sqrt{\log n}$, its two principal terms cannot both be $o(\sqrt n)$, whereas the factor $((1+A_q)/y)^{1/2}$ used before {xref}(2.15)` is of order n − 1 / 2 ( log n ) 3 / 2 n^{-1/2}(\log n)^{3/2} n − 1/2 ( log n ) 3/2 y = x n y=x_n y = x n
Step 4: final edges at vertices of high degree
We now prove (16) . For a colorable multigraph G G G [ n ] [n] [ n ] j ≥ 0 j\ge0 j ≥ 0 κ j ( G ) \kappa_j(G) κ j ( G ) ( u , v ) (u,v) ( u , v ) ordered frozen pairs , with deg G ( u ) = j \deg_G(u)=j deg G ( u ) = j n ( n − 1 ) n(n-1) n ( n − 1 ) κ j ( G ) = 0 \kappa_j(G)=0 κ j ( G ) = 0 G G G E \mathcal{E} E D n D_n D n h = 1 h=1 h = 1 0 ≤ m < M 0 \le m < M 0 ≤ m < M
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ 2 ∑ m = 0 M − 1 ∑ j = D n + 1 m E κ j ( G m ) . \mathbb{P}\left(H \le M,\ \max_{v \in e_H} \deg_{\Gamma}(v) > D_n\right) \le2\sum_{m=0}^{M-1}\sum_{j=D_n+1}^{m}\mathbb{E}\kappa_j(G_m). P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ 2 m = 0 ∑ M − 1 j = D n + 1 ∑ m E κ j ( G m ) . The identification estimate above does not cover all these degrees. We instead delete occurrences. For a colorable G G G ( u , v ) (u,v) ( u , v ) e e e u u u non-pivotal if the pair remains frozen in G − e G-e G − e pivotal otherwise. We set K = ⌈ 8 Λ ⌉ K=\lceil8\Lambda\rceil K = ⌈ 8Λ ⌉ κ j p i v ( G ) \kappa_j^{\mathrm{piv}}(G) κ j piv ( G ) κ j ( G ) \kappa_j(G) κ j ( G ) K − 1 K-1 K − 1 u u u G G G
Let f f f 1 ≤ i ≤ m 1 \le i \le m 1 ≤ i ≤ m ( G m − e i , e i ) (G_m-e_i,e_i) ( G m − e i , e i ) ( G m − 1 , f ) (G_{m-1},f) ( G m − 1 , f ) [44 Theorem 4.1] . Deleting one of at least K K K u u u ( u , v ) (u,v) ( u , v ) u u u v v v u u u u u u 2 / n 2/n 2/ n
E κ j ( G m ) ≤ E κ j p i v ( G m ) + 2 m n K E κ j − 1 ( G m − 1 ) ≤ E κ j p i v ( G m ) + 1 4 E κ j − 1 ( G m − 1 ) (23) \mathbb{E}\kappa_j(G_m) \le\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m) + \frac{2m}{nK}\mathbb{E}\kappa_{j-1}(G_{m-1}) \le\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m) + \frac{1}{4}\mathbb{E}\kappa_{j-1}(G_{m-1}) \tag*{(23)} E κ j ( G m ) ≤ E κ j piv ( G m ) + n K 2 m E κ j − 1 ( G m − 1 ) ≤ E κ j piv ( G m ) + 4 1 E κ j − 1 ( G m − 1 ) (23) for $1 \le j \le m \le M$, where we used m ≤ M ≤ Λ n m \le M \le\Lambda n m ≤ M ≤ Λ n K ≥ 8 Λ K \ge8\Lambda K ≥ 8Λ w 0 = ⌊ D n / 2 ⌋ w_0=\lfloor D_n/2\rfloor w 0 = ⌊ D n /2 ⌋ j j j w 0 w_0 w 0 w 0 < j ≤ m < M w_0<j\le m<M w 0 < j ≤ m < M
E κ j ( G m ) ≤ ∑ j ′ = w 0 + 1 j 4 − ( j − j ′ ) E κ j ′ p i v ( G m − j + j ′ ) + 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) . (24) \mathbb{E}\kappa_j(G_m) \le\sum_{j'=w_0+1}^{j}4^{-(j-j')}\mathbb{E}\kappa_{j'}^{\mathrm{piv}}(G_{m-j+j'}) + 4^{-(j-w_0)}\mathbb{E}\kappa_{w_0}(G_{m-j+w_0}). \tag*{(24)} E κ j ( G m ) ≤ j ′ = w 0 + 1 ∑ j 4 − ( j − j ′ ) E κ j ′ piv ( G m − j + j ′ ) + 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) . (24) Summing (2.8) with h = 1 h=1 h = 1 m m m ∑ m E κ ( G m ) ≤ 1 \sum_m \mathbb{E}\kappa(G_m)\le1 ∑ m E κ ( G m ) ≤ 1 ( 8 / 3 ) 4 − ( D n + 1 − w 0 ) (8/3)4^{-(D_n+1-w_0)} ( 8/3 ) 4 − ( D n + 1 − w 0 ) w 0 w_0 w 0
If a pivotal occurrence for ( u , v ) (u,v) ( u , v ) u u u y y y u u u v v v u ∈ [ n ] u\in[n] u ∈ [ n ] root . For a colorable multigraph G G G [ n ] [n] [ n ] W = [ n ] ∖ C G ( u ) W=[n]\setminus C_G(u) W = [ n ] ∖ C G ( u ) R G ( u ) R_G(u) R G ( u ) ( y , v ) ∈ W 2 (y,v)\in W^2 ( y , v ) ∈ W 2 u u u y y y v v v β G ( u ) = ∣ R G ( u ) ∣ / n 2 \beta_G(u)=|R_G(u)|/n^2 β G ( u ) = ∣ R G ( u ) ∣/ n 2 y ∈ [ n ] y\in[n] y ∈ [ n ] section R G ( u ; y ) = { v ∈ [ n ] : ( y , v ) ∈ R G ( u ) } R_G(u;y)=\{v\in[n]:(y,v)\in R_G(u)\} R G ( u ; y ) = { v ∈ [ n ] : ( y , v ) ∈ R G ( u )} G G G u u u R ( y ) R(y) R ( y ) R G ( u ; y ) R_G(u;y) R G ( u ; y ) R G ( u ) = ∅ R_G(u)=\varnothing R G ( u ) = ∅ β G ( u ) = 0 \beta_G(u)=0 β G ( u ) = 0 G G G
R G ( u ) = { ( y , v ) ∈ W × W : σ ( u ) ∈ { σ ( y ) , σ ( v ) } for every coloring σ of G } , β G ( u ) = ∣ R G ( u ) ∣ n 2 , R ( y ) = { v : ( y , v ) ∈ R G ( u ) } . (25) \begin{aligned}
R_G(u) &= \{(y,v)\in W\times W:\sigma(u)\in\{\sigma(y),\sigma(v)\}\text{ for every coloring }\sigma\text{ of }G\},\\
\beta_G(u) &= \frac{|R_G(u)|}{n^2}, \qquad R(y)=\{v:(y,v)\in R_G(u)\}. \tag*{(25)}
\end{aligned} R G ( u ) β G ( u ) = {( y , v ) ∈ W × W : σ ( u ) ∈ { σ ( y ) , σ ( v )} for every coloring σ of G } , = n 2 ∣ R G ( u ) ∣ , R ( y ) = { v : ( y , v ) ∈ R G ( u )} . (25) The relation is symmetric. For y ∈ W y \in W y ∈ W Lemma 6.3 shows that G + { u , y } G + \{u,y\} G + { u , y }
C G + { u , y } ( u ) = C G ( u ) ∪ R ( y ) for all y ∈ W . (26) C_{G+\{u,y\}}(u)=C_G(u)\cup R(y)\qquad\text{for all }y\in W. \tag*{(26)} C G + { u , y } ( u ) = C G ( u ) ∪ R ( y ) for all y ∈ W . (26) Indeed, the colorings of G + { u , y } G + \{u,y\} G + { u , y } σ \sigma σ G G G σ ( y ) ≠ σ ( u ) \sigma(y) \ne\sigma(u) σ ( y ) = σ ( u ) v ∈ W v \in W v ∈ W σ ( v ) = σ ( u ) \sigma(v) = \sigma(u) σ ( v ) = σ ( u ) ( y , v ) ∈ R G ( u ) (y,v) \in R_G(u) ( y , v ) ∈ R G ( u ) n β G ( u ) n\beta_G(u) n β G ( u ) ∣ R ( y ) ∣ |R(y)| ∣ R ( y ) ∣ y ∈ [ n ] y \in[n] y ∈ [ n ] y ∈ W y \in W y ∈ W ∣ R ( y ) ∣ |R(y)| ∣ R ( y ) ∣ u u u { u , y } \{u,y\} { u , y } κ j p i v ( G m ) \kappa_j^{\mathrm{piv}}(G_m) κ j piv ( G m ) j − K + 1 j-K+1 j − K + 1 u u u Lemma 6.4 (b) for integers m ≥ 1 m \ge1 m ≥ 1 j ≥ K j \ge K j ≥ K u ∈ [ n ] u \in[n] u ∈ [ n ]
E κ j p i v ( G m ) ≤ 2 m n ( n − 1 ) 2 ( j − K + 1 ) E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . (27) \mathbb{E}\kappa_j^{\mathrm{piv}}(G_m)\le\frac{2mn}{(n-1)^2(j-K+1)}\mathbb{E}\left[1\{\deg_{G_{m-1}}(u)=j-1\}\beta_{G_{m-1}}(u)\right]. \tag*{(27)} E κ j piv ( G m ) ≤ ( n − 1 ) 2 ( j − K + 1 ) 2 mn E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . (27) Suppose w 0 ≥ K w_0 \ge K w 0 ≥ K (27) over w 0 < j ≤ m < M w_0 < j \le m < M w 0 < j ≤ m < M
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ 8 3 4 − ( D n + 1 − w 0 ) + 16 Λ 3 ( w 0 − K + 2 ) ( n n − 1 ) 2 M − 1 ∑ m = 0 M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] . \mathbb{P}\left(H\le M,\ \max_{v\in e_H}\deg_{\Gamma}(v)>D_n\right)
\le\frac{8}{3}4^{-(D_n+1-w_0)}
+\frac{16\Lambda}{3(w_0-K+2)}\left(\frac{n}{n-1}\right)^{2M-1}\sum_{m=0}^{M-1}\mathbb{E}\left[1\{\deg_{G_m}(u)\ge w_0\}\beta_{G_m}(u)\right]. P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ 3 8 4 − ( D n + 1 − w 0 ) + 3 ( w 0 − K + 2 ) 16Λ ( n − 1 n ) 2 M − 1 m = 0 ∑ M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] . It therefore suffices to bound the sum
∑ m = 0 M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] . (28) \sum_{m=0}^{M-1}\mathbb{E}\left[1\{\deg_{G_m}(u)\ge w_0\}\beta_{G_m}(u)\right]. \tag*{(28)} m = 0 ∑ M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] . (28) The main difficulty is that β G ( u ) \beta_G(u) β G ( u ) Remark 6.11 , take q = 3 q=3 q = 3 n ≥ 7 n\ge7 n ≥ 7 n ≡ 1 ( m o d 3 ) n\equiv1\pmod{3} n ≡ 1 ( mod 3 ) G G G u u u ( n − 4 ) / 3 (n-4)/3 ( n − 4 ) /3 u u u C G ( u ) = { u } C_G(u)=\{u\} C G ( u ) = { u }
β G ( u ) = 2 ( n − 1 ) 2 9 n 2 ⟶ 2 9 . \beta_G(u)=\frac{2(n-1)^2}{9n^2}\longrightarrow\frac{2}{9}. β G ( u ) = 9 n 2 2 ( n − 1 ) 2 ⟶ 9 2 . Adding occurrences from u u u β G ( u ) \beta_G(u) β G ( u ) 0 ≤ m < M 0\le m<M 0 ≤ m < M
P ( deg G m ( u ) ≥ w 0 ) ≤ ( 2 e Λ w 0 ) w 0 . \mathbb{P}\left(\deg_{G_m}(u)\ge w_0\right)\le\left(\frac{2e\Lambda}{w_0}\right)^{w_0}. P ( deg G m ( u ) ≥ w 0 ) ≤ ( w 0 2 e Λ ) w 0 . Step 4: final edges at vertices of high degree
Bounding β G m ( u ) \beta_{G_m}(u) β G m ( u ) n ( 2 e Λ / w 0 ) w 0 n(2e\Lambda/w_0)^{w_0} n ( 2 e Λ/ w 0 ) w 0
For a colorable G G G f = { a , b } f=\{a,b\} f = { a , b } credited set A G ( f ) \mathcal{A}_G(f) A G ( f ) a , b ∈ W a,b\in W a , b ∈ W A G ( f ) = R ( a ) ∩ R ( b ) \mathcal{A}_G(f)=R(a)\cap R(b) A G ( f ) = R ( a ) ∩ R ( b ) W W W A G ( f ) \mathcal{A}_G(f) A G ( f ) C G ( u ) C_G(u) C G ( u ) A G ( f ) = ∅ \mathcal{A}_G(f)=\varnothing A G ( f ) = ∅ G + f G+f G + f ∪ ˙ \mathbin{\dot\cup} ∪ ˙ G + f G+f G + f G G G u u u f f f
C G ( u ) ∪ ˙ A G ( f ) ⊆ C G + f ( u ) , ∣ C G + f ( u ) ∣ ≥ ∣ C G ( u ) ∣ + ∣ A G ( f ) ∣ . C_G(u)\mathbin{\dot\cup}\mathcal{A}_G(f)\subseteq C_{G+f}(u),\qquad|C_{G+f}(u)|\ge|C_G(u)|+|\mathcal{A}_G(f)|. C G ( u ) ∪ ˙ A G ( f ) ⊆ C G + f ( u ) , ∣ C G + f ( u ) ∣ ≥ ∣ C G ( u ) ∣ + ∣ A G ( f ) ∣. For example, if a , b ∈ W a,b\in W a , b ∈ W y ∈ R ( a ) ∩ R ( b ) y\in R(a)\cap R(b) y ∈ R ( a ) ∩ R ( b ) a , b ∈ R ( y ) a,b\in R(y) a , b ∈ R ( y ) (26) , adding { u , y } \{u,y\} { u , y } a , b a,b a , b u u u G + { u , y } + { a , b } G+\{u,y\}+\{a,b\} G + { u , y } + { a , b } G + { a , b } G+\{a,b\} G + { a , b } y y y u u u
We now take f f f [ n ] [n] [ n ] E f \mathbb{E}_f E f
β G ( u ) ≤ 2 log 2 E f log ( 1 + ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ ) . (29) \beta_G(u)\le\frac{2}{\log2}\mathbb{E}_f\log\left(1+\frac{|\mathcal{A}_G(f)|}{|C_G(u)|}\right). \tag*{(29)} β G ( u ) ≤ log 2 2 E f log ( 1 + ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ ) . (29) which bounds the density by the expected credited logarithmic increment. The increment is bounded by the actual growth of the logarithm of the class size when the added pair preserves colorability, and remains defined when it destroys colorability. To prove this estimate, we use the following inequality from Lemma 6.8, with the current graph G G G u u u
∣ R G ( u ) ∣ ≤ 2 ∑ ( a , b ) ∈ W × W min ( ∣ R ( a ) ∩ R ( b ) ∣ ∣ C G ( u ) ∣ , 1 ) + 2 ( ∣ C G ( u ) ∣ − 1 ) ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) . |R_G(u)|\le2\sum_{(a,b)\in W\times W}\min\left(\frac{|R(a)\cap R(b)|}{|C_G(u)|},1\right)+2(|C_G(u)|-1)\sum_{a\in W}\min\left(\frac{|R(a)|}{|C_G(u)|},1\right). ∣ R G ( u ) ∣ ≤ 2 ( a , b ) ∈ W × W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∩ R ( b ) ∣ , 1 ) + 2 ( ∣ C G ( u ) ∣ − 1 ) a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) . Its proof uses symmetry to count relation pairs incident to a section of size at least ∣ C G ( u ) ∣ |C_G(u)| ∣ C G ( u ) ∣ A G ( f ) \mathcal{A}_G(f) A G ( f )
E f min ( ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ , 1 ) ≥ ∣ R G ( u ) ∣ 2 n ( n − 1 ) ≥ β G ( u ) 2 . \mathbb{E}_f\min\left(\frac{|\mathcal{A}_G(f)|}{|C_G(u)|},1\right)\ge\frac{|R_G(u)|}{2n(n-1)}\ge\frac{\beta_G(u)}{2}. E f min ( ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ , 1 ) ≥ 2 n ( n − 1 ) ∣ R G ( u ) ∣ ≥ 2 β G ( u ) . Here pairs with one endpoint in C G ( u ) C_G(u) C G ( u ) W × W W\times W W × W log ( 1 + z ) ≥ ( log 2 ) min ( z , 1 ) \log(1+z)\ge(\log2)\min(z,1) log ( 1 + z ) ≥ ( log 2 ) min ( z , 1 ) z ≥ 0 z\ge0 z ≥ 0 z = ∣ A G ( f ) ∣ / ∣ C G ( u ) ∣ z=|\mathcal{A}_G(f)|/|C_G(u)| z = ∣ A G ( f ) ∣/∣ C G ( u ) ∣
We define τ = min ( H , M ) \tau= \min(H, M) τ = min ( H , M ) 0 ≤ m < H 0 \le m < H 0 ≤ m < H potential Φ m = log ∣ C G m ( u ) ∣ \Phi_m = \log|C_{G_m}(u)| Φ m = log ∣ C G m ( u ) ∣ credited logarithmic increment Ψ m = log ( 1 + ∣ A G m ( e m + 1 ) ∣ / ∣ C G m ( u ) ∣ ) \Psi_m = \log(1 + |A_{G_m}(e_{m+1})|/|C_{G_m}(u)|) Ψ m = log ( 1 + ∣ A G m ( e m + 1 ) ∣/∣ C G m ( u ) ∣ ) Φ m = Ψ m = 0 \Phi_m = \Psi_m = 0 Φ m = Ψ m = 0 m ≥ H m \ge H m ≥ H [ 0 , log n ] [0, \log n] [ 0 , log n ]
Φ m + 1 − Φ m ≥ Ψ m when m + 1 < H . \Phi_{m+1} - \Phi_m \ge\Psi_m \qquad\text{when } m+1 < H. Φ m + 1 − Φ m ≥ Ψ m when m + 1 < H . At the last index before τ \tau τ [ n ] [n] [ n ]
Ψ τ − 1 ≤ log n − Φ τ − 1 . \Psi_{\tau-1} \le\log n - \Phi_{\tau-1}. Ψ τ − 1 ≤ log n − Φ τ − 1 . The pair e τ e_\tau e τ τ = H \tau= H τ = H τ = M < H \tau= M < H τ = M < H τ 0 \tau_0 τ 0 ( F m ) m ≥ 0 (\mathcal{F}_m)_{m\ge0} ( F m ) m ≥ 0 { 0 , 1 , … } ∪ { ∞ } \{0,1,\ldots\} \cup\{\infty\} { 0 , 1 , … } ∪ { ∞ }
∑ m : τ 0 ≤ m < τ Ψ m ≤ ( log n ) 1 { τ 0 < τ } . \sum_{m:\,\tau_0\le m<\tau} \Psi_m \le(\log n)\mathbf{1}_{\{\tau_0<\tau\}}. m : τ 0 ≤ m < τ ∑ Ψ m ≤ ( log n ) 1 { τ 0 < τ } . For 0 ≤ m < M 0 \le m < M 0 ≤ m < M (29) conditionally on F m \mathcal{F}_m F m G = G m G = G_m G = G m f = e m + 1 f = e_{m+1} f = e m + 1 m < H m < H m < H F m \mathcal{F}_m F m F m \mathcal{F}_m F m { τ 0 ≤ m < τ } \{\tau_0 \le m < \tau\} { τ 0 ≤ m < τ }
E ∑ m : τ 0 ≤ m < τ β G m ( u ) ≤ 2 log 2 ( log n ) P ( τ 0 < τ ) . (30) \mathbb{E}\sum_{m:\,\tau_0\le m<\tau} \beta_{G_m}(u) \le\frac{2}{\log2}(\log n)\mathbb{P}(\tau_0<\tau). \tag*{(30)} E m : τ 0 ≤ m < τ ∑ β G m ( u ) ≤ log 2 2 ( log n ) P ( τ 0 < τ ) . (30) This estimate controls the total occupation of the process. The final edge must be credited. In the example above, pairs f f f G + f G+f G + f O ( ( log n ) / n ) O((\log n)/n) O (( log n ) / n ) (29) , although β G ( u ) → 2 / 9 \beta_G(u) \to2/9 β G ( u ) → 2/9 Remark 6.11 .
We now take τ 0 = min { m ≥ 0 : deg G m ( u ) ≥ w 0 } \tau_0 = \min\{m \ge0 : \deg_{G_m}(u) \ge w_0\} τ 0 = min { m ≥ 0 : deg G m ( u ) ≥ w 0 } ∞ \infty ∞ β G m ( u ) = 0 \beta_{G_m}(u) = 0 β G m ( u ) = 0 H H H (30) is (28) . For w 0 ≥ K w_0 \ge K w 0 ≥ K M − 1 M-1 M − 1 2 / n 2/n 2/ n w 0 w_0 w 0
P ( τ 0 < τ ) ≤ P ( Bin ( M − 1 , 2 / n ) ≥ w 0 ) ≤ ( 2 e Λ w 0 ) w 0 . \mathbb{P}(\tau_0 < \tau) \le\mathbb{P}(\operatorname{Bin}(M-1,2/n) \ge w_0) \le\left(\frac{2e\Lambda}{w_0}\right)^{w_0}. P ( τ 0 < τ ) ≤ P ( Bin ( M − 1 , 2/ n ) ≥ w 0 ) ≤ ( w 0 2 e Λ ) w 0 . Inserting this bound into (30) gives the estimate of Corollary 6.12 with j = w 0 j = w_0 j = w 0 w 0 ≥ K w_0 \ge K w 0 ≥ K
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ 8 3 4 − ( D n + 1 − w 0 ) + 32 Λ 3 log 2 ( n n − 1 ) 2 log n w 0 − K + 2 ( 2 e Λ w 0 ) w 0 . (31) \mathbb{P}\left(H \le M,\ \max_{v\in e_H}\deg_{\Gamma}(v) > D_n\right)
\le\frac{8}{3}4^{-(D_n+1-w_0)}+\frac{32\Lambda}{3\log2}\left(\frac{n}{n-1}\right)^2\frac{\log n}{w_0-K+2}\left(\frac{2e\Lambda}{w_0}\right)^{w_0}. \tag*{(31)} P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ 3 8 4 − ( D n + 1 − w 0 ) + 3 log 2 32Λ ( n − 1 n ) 2 w 0 − K + 2 log n ( w 0 2 e Λ ) w 0 . (31) Since w 0 ≥ ( log n ) 1 / 4 / 2 − 1 w_0 \ge(\log n)^{1/4}/2 - 1 w 0 ≥ ( log n ) 1/4 /2 − 1 ε n \varepsilon_n ε n n n n (16) . The argument also holds for q = 2 q = 2 q = 2
Preliminaries
This section gives the definitions and elementary facts used throughout the paper. We introduce the random graph process and its quantiles, record identities for the step at which colorability is lost, and prove the elementary estimates used below. The assumption q ≥ 3 q \ge3 q ≥ 3 q = 2 q = 2 q = 2
Throughout the paper, we fix an integer q ≥ 3 q \ge3 q ≥ 3 k ≥ 1 k \ge1 k ≥ 1 [ k ] = { 1 , … , k } [k] = \{1,\ldots,k\} [ k ] = { 1 , … , k } log \log log q q q n n n C C C C ′ C' C ′ for all large n n n if there exists an integer n 0 n_0 n 0 q q q n ≥ n 0 n \ge n_0 n ≥ n 0
The edge process
We begin with multigraphs and their colorings. We will sample pairs of vertices with replacement and count degrees with multiplicity.
Definition 3.1. Let S S S { u , v } \{u,v\} { u , v } u , v ∈ S u,v \in S u , v ∈ S u ≠ v u \ne v u = v pair of S S S multigraph on S S S G G G S S S G G G occurrences of G G G e e e S S S G + e G+e G + e G G G e e e e e e G G G G − e G-e G − e G G G e e e p 1 , … , p k p_1,\ldots,p_k p 1 , … , p k S S S G + p 1 + ⋯ + p k G+p_1+\cdots+p_k G + p 1 + ⋯ + p k G G G p i p_i p i p 1 + ⋯ + p k p_1+\cdots+p_k p 1 + ⋯ + p k p 1 , … , p k p_1,\ldots,p_k p 1 , … , p k k = 0 k=0 k = 0 G ⊂ G ′ G \subset G' G ⊂ G ′ G G G G ′ G' G ′ u ∈ S u \in S u ∈ S degree deg G ( u ) \deg_G(u) deg G ( u ) G G G u u u S ′ ⊂ S S' \subset S S ′ ⊂ S G [ S ′ ] G[S'] G [ S ′ ] S ′ S' S ′ G G G S ′ S' S ′ ϕ : S → S ′ \phi:S \to S' ϕ : S → S ′ ϕ G \phi G ϕG S ′ S' S ′ { ϕ ( u ) , ϕ ( v ) } \{\phi(u),\phi(v)\} { ϕ ( u ) , ϕ ( v )} { u , v } \{u,v\} { u , v } G G G σ : S → [ q ] \sigma:S \to[q] σ : S → [ q ] coloring of G G G σ ( u ) ≠ σ ( v ) \sigma(u) \ne\sigma(v) σ ( u ) = σ ( v )
for every occurrence { u , v } \{u,v\} { u , v } G G G Col ( G ) \operatorname{Col}(G) Col ( G ) G G G G G G colorable if Col ( G ) ≠ ∅ \operatorname{Col}(G) \ne\varnothing Col ( G ) = ∅
A multigraph has the same colorings as the simple graph formed by its distinct pairs. Repeated occurrences therefore do not affect colorability. They do affect degrees, and in this paper degrees always count occurrences. Note also that Col ( G ′ ) ⊂ Col ( G ) \operatorname{Col}(G') \subset\operatorname{Col}(G) Col ( G ′ ) ⊂ Col ( G ) G ⊂ G ′ G \subset G' G ⊂ G ′
Definition 3.2. Let S S S edge process on S S S to be a sequence of independent random pairs of S S S ( ∣ S ∣ 2 ) \binom{|S|}{2} ( 2 ∣ S ∣ ) m ≥ 0 m \ge0 m ≥ 0 m m m time-m m m .
Let n ≥ 2 q n \ge2q n ≥ 2 q ( e i ) i ≥ 1 (e_i)_{i \ge1} ( e i ) i ≥ 1 [ n ] [n] [ n ] m ≥ 0 m \ge0 m ≥ 0 G m = e 1 + ⋯ + e m G_m = e_1 + \cdots+ e_m G m = e 1 + ⋯ + e m F m \mathcal{F}_m F m σ \sigma σ e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m u ∈ [ n ] u \in[n] u ∈ [ n ] e i e_i e i ( n − 1 ) / ( n 2 ) = 2 / n (n-1)/\binom{n}{2} = 2/n ( n − 1 ) / ( 2 n ) = 2/ n deg G m ( u ) \deg_{G_m}(u) deg G m ( u ) Bin ( m , 2 / n ) \operatorname{Bin}(m,2/n) Bin ( m , 2/ n )
H = min { m ≥ 0 : G m is not colorable } . (32) H = \min\{m \ge0 : G_m \text{ is not colorable}\}. \tag*{(32)} H = min { m ≥ 0 : G m is not colorable } . (32) We use the convention min ∅ = ∞ \min\varnothing= \infty min ∅ = ∞ A q = 2 q log q + 2 q + 1 A_q = 2q \log q + 2q + 1 A q = 2 q log q + 2 q + 1 G 0 G_0 G 0 H ≥ 1 H \ge1 H ≥ 1 Lemma 3.10 (d) that E H ≤ A q n \mathbb{E}H \le A_q n E H ≤ A q n H H H { H < ∞ } \{H < \infty\} { H < ∞ } Γ = G H − 1 \Gamma= G_{H-1} Γ = G H − 1 e H e_H e H final edge . Note that P ( H > m ) = P ( G m is colorable ) \mathbb{P}(H > m) = \mathbb{P}(G_m \text{ is colorable}) P ( H > m ) = P ( G m is colorable ) m ≥ 0 m \ge0 m ≥ 0 m m m
We set
γ q = log q log ( q / ( q − 1 ) ) . (33) \gamma_q = \frac{\log q}{\log(q/(q-1))}. \tag*{(33)} γ q = log ( q / ( q − 1 )) log q . (33) For every real c 1 > γ q c_1 > \gamma_q c 1 > γ q Lemma 3.17 (a) implies that the probability that G ⌊ c 1 n ⌋ G_{\lfloor c_1 n\rfloor} G ⌊ c 1 n ⌋ γ q \gamma_q γ q Λ > γ q + 1 \Lambda> \gamma_q + 1 Λ > γ q + 1 n n n Λ \Lambda Λ Λ = γ q + 2 \Lambda= \gamma_q + 2 Λ = γ q + 2 M = ⌊ Λ n ⌋ M = \lfloor\Lambda n\rfloor M = ⌊ Λ n ⌋
V = min ( H − 1 , M ) . (34) V = \min(H-1,M). \tag*{(34)} V = min ( H − 1 , M ) . (34) The cap M M M V V V e 1 , … , e M e_1,\ldots,e_M e 1 , … , e M Lemma 3.17 (a), the quantiles defined below are smaller than M M M n n n
Definition 3.3. For η ∈ ( 0 , 1 / 2 ] \eta\in(0,1/2] η ∈ ( 0 , 1/2 ] lower quantile L n ( η ) L_n(\eta) L n ( η ) upper quantile U n ( η ) U_n(\eta) U n ( η )
L n ( η ) = min { m ≥ 0 : P ( V ≤ m ) ≥ η } , L_n(\eta) = \min\{m \ge0 : \mathbb{P}(V \le m) \ge\eta\}, L n ( η ) = min { m ≥ 0 : P ( V ≤ m ) ≥ η } , U n ( η ) = max { m ∈ { 0 , … , M } : P ( V ≥ m ) ≥ η } . (35) U_n(\eta) = \max\{m \in\{0,\ldots,M\} : \mathbb{P}(V \ge m) \ge\eta\}. \tag*{(35)} U n ( η ) = max { m ∈ { 0 , … , M } : P ( V ≥ m ) ≥ η } . (35) where m m m w n ( η ) = U n ( η ) − L n ( η ) w_n(\eta)=U_n(\eta)-L_n(\eta) w n ( η ) = U n ( η ) − L n ( η ) η \eta η η 0 = e − 2 Λ / 8 \eta_0=e^{-2\Lambda/8} η 0 = e − 2Λ/8
The two sets in (3.4) are nonempty, since P ( V ≤ M ) = 1 \mathbb{P}(V\le M)=1 P ( V ≤ M ) = 1 P ( V ≥ 0 ) = 1 \mathbb{P}(V\ge0)=1 P ( V ≥ 0 ) = 1 L n ( η ) ≤ M L_n(\eta)\le M L n ( η ) ≤ M η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] η 0 \eta_0 η 0 n n n η \eta η L L L U U U w w w L n ( η ) L_n(\eta) L n ( η ) U n ( η ) U_n(\eta) U n ( η ) w n ( η ) w_n(\eta) w n ( η )
Lemma 3.4. Let η ∈ ( 0 , 1 / 2 ] \eta\in(0,1/2] η ∈ ( 0 , 1/2 ] L ≤ U ≤ M L\le U\le M L ≤ U ≤ M
P ( H > m ) > 1 − η for all integers 0 ≤ m ≤ L , P ( H > m ) < η for all integers U < m ≤ M . (36) \begin{aligned}
\mathbb{P}(H>m)&>1-\eta&&\text{for all integers }0\le m\le L,\\
\mathbb{P}(H>m)&<\eta&&\text{for all integers }U<m\le M. \tag*{(36)}
\end{aligned} P ( H > m ) P ( H > m ) > 1 − η < η for all integers 0 ≤ m ≤ L , for all integers U < m ≤ M . (36) Proof. By the minimality of L L L P ( V ≤ L − 1 ) < η \mathbb{P}(V\le L-1)<\eta P ( V ≤ L − 1 ) < η L = 0 L=0 L = 0 V ≥ 0 V\ge0 V ≥ 0
P ( V ≥ L ) = 1 − P ( V ≤ L − 1 ) > 1 − η ≥ η , \mathbb{P}(V\ge L)=1-\mathbb{P}(V\le L-1)>1-\eta\ge\eta, P ( V ≥ L ) = 1 − P ( V ≤ L − 1 ) > 1 − η ≥ η , where we used η ≤ 1 / 2 \eta\le1/2 η ≤ 1/2 L ≤ M L\le M L ≤ M L L L U U U L ≤ U L\le U L ≤ U U ≤ M U\le M U ≤ M U U U
Next, we observe that for every integer m ≤ M m\le M m ≤ M
{ V ≥ m } = { H − 1 ≥ m } = { H > m } . (37) \{V\ge m\}=\{H-1\ge m\}=\{H>m\}. \tag*{(37)} { V ≥ m } = { H − 1 ≥ m } = { H > m } . (37) Taking m = L m=L m = L (37) , we find that
P ( H > L ) = P ( V ≥ L ) > 1 − η , \mathbb{P}(H>L)=\mathbb{P}(V\ge L)>1-\eta, P ( H > L ) = P ( V ≥ L ) > 1 − η , and the first bound in (36) follows because P ( H > m ) \mathbb{P}(H>m) P ( H > m ) m m m U < m ≤ M U<m\le M U < m ≤ M U U U m m m U U U (37) implies that P ( H > m ) = P ( V ≥ m ) < η \mathbb{P}(H>m)=\mathbb{P}(V\ge m)<\eta P ( H > m ) = P ( V ≥ m ) < η
Frozen pairs
The final edge of the process joins two vertices that receive the same color in every coloring of the last colorable graph. We now introduce the corresponding notions.
Definition 3.5. Let S S S N ≥ 2 N\ge2 N ≥ 2 G G G S S S u , v ∈ S u,v\in S u , v ∈ S u ∼ G v u\sim_G v u ∼ G v G G G σ ( u ) = σ ( v ) \sigma(u)=\sigma(v) σ ( u ) = σ ( v ) σ ∈ Col ( G ) \sigma\in\operatorname{Col}(G) σ ∈ Col ( G ) { u , v } \{u,v\} { u , v } S S S frozen pair of G G G u ∼ G v u\sim_G v u ∼ G v G G G C G ( u ) = { v ∈ S : u ∼ G v } C_G(u)=\{v\in S:u\sim_G v\} C G ( u ) = { v ∈ S : u ∼ G v } frozen class of u ∈ S u\in S u ∈ S G G G C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k
κ ( G ) = ( N 2 ) − 1 # { frozen pairs of G } , Q ( G ) = ∑ i = 1 k ( ∣ C i ∣ N ) 2 , (38) \kappa(G)=\binom{N}{2}^{-1}\#\{\text{frozen pairs of }G\},\qquad Q(G)=\sum_{i=1}^{k}\left(\frac{|C_i|}{N}\right)^2, \tag*{(38)} κ ( G ) = ( 2 N ) − 1 # { frozen pairs of G } , Q ( G ) = i = 1 ∑ k ( N ∣ C i ∣ ) 2 , (38) and we set Q ( G ) = 0 Q(G)=0 Q ( G ) = 0 G G G
The relation ∼ G \sim_G ∼ G S S S G G G u ∈ C G ( u ) u \in C_G(u) u ∈ C G ( u ) u ∈ S u \in S u ∈ S u u u frozen pair of G G G ∣ C G ( u ) ∣ ≥ 2 |C_G(u)| \ge2 ∣ C G ( u ) ∣ ≥ 2 κ ( G ) = 0 \kappa(G) = 0 κ ( G ) = 0 G G G ∼ G \sim_G ∼ G κ ( G ) \kappa(G) κ ( G ) S S S G G G G G G G G G G G G
Throughout the paper, we define numerical statistics constructed from colorings or frozen classes to be 0 when the multigraph is not colorable, and set-valued statistics to be the empty set. We evaluate expressions involving frozen classes, including logarithms and quotients, on the colorable event, and define the entire expression to be 0 otherwise. In particular, Col ( G ) = ∅ \operatorname{Col}(G) = \varnothing Col ( G ) = ∅ G G G
Culberson and Gent studied frozen pairs in random graph coloring and the graph obtained by merging their equivalence classes [24 Sections 2.1 and 3.2] . Their tests for frozen and free pairs use edge addition and vertex identification, respectively [24 Section 2.2] . Our relation ∼ G \sim_G ∼ G G G G
Lemma 3.6. Let S S S N ≥ 2 N \ge2 N ≥ 2 G G G S S S
(a) For all distinct u , v ∈ S u,v \in S u , v ∈ S G + { u , v } G + \{u,v\} G + { u , v } G G G u ̸ ∼ G v u \not\sim_G v u ∼ G v f f f S S S
P ( G is colorable and G + f is not colorable ) = κ ( G ) . \mathbb{P}(G\text{ is colorable and }G+f\text{ is not colorable})=\kappa(G). P ( G is colorable and G + f is not colorable ) = κ ( G ) . (b) Let G ′ G' G ′ S S S G ⊂ G ′ G \subset G' G ⊂ G ′ G G G u ∼ G v u \sim_G v u ∼ G v u ∼ G ′ v u \sim_{G'} v u ∼ G ′ v u , v ∈ S u,v \in S u , v ∈ S C G ( u ) ⊂ C G ′ ( u ) C_G(u) \subset C_{G'}(u) C G ( u ) ⊂ C G ′ ( u ) u ∈ S u \in S u ∈ S κ ( G ) ≤ κ ( G ′ ) \kappa(G) \le\kappa(G') κ ( G ) ≤ κ ( G ′ )
(c) If G G G
Q ( G ) = ( 1 − 1 N ) κ ( G ) + 1 N , Q(G)=\left(1-\frac{1}{N}\right)\kappa(G)+\frac{1}{N}, Q ( G ) = ( 1 − N 1 ) κ ( G ) + N 1 , and ∣ C G ( u ) ∣ ≤ N Q ( G ) 1 / 2 |C_G(u)| \le NQ(G)^{1/2} ∣ C G ( u ) ∣ ≤ N Q ( G ) 1/2 u ∈ S u \in S u ∈ S
Proof. We begin with (a). Observe that
Col ( G + { u , v } ) = { σ ∈ Col ( G ) : σ ( u ) ≠ σ ( v ) } . \operatorname{Col}(G+\{u,v\})=\{\sigma\in\operatorname{Col}(G):\sigma(u)\ne\sigma(v)\}. Col ( G + { u , v }) = { σ ∈ Col ( G ) : σ ( u ) = σ ( v )} . This set is empty if and only if either Col ( G ) = ∅ \operatorname{Col}(G)=\varnothing Col ( G ) = ∅ Col ( G ) ≠ ∅ \operatorname{Col}(G)\ne\varnothing Col ( G ) = ∅ σ ( u ) = σ ( v ) \sigma(u)=\sigma(v) σ ( u ) = σ ( v ) σ ∈ Col ( G ) \sigma\in\operatorname{Col}(G) σ ∈ Col ( G ) ∼ G \sim_G ∼ G u ∼ G v u\sim_G v u ∼ G v ( N 2 ) \binom{N}{2} ( 2 N ) f f f
To prove (b), we use the inclusion Col ( G ′ ) ⊂ Col ( G ) \operatorname{Col}(G')\subset\operatorname{Col}(G) Col ( G ′ ) ⊂ Col ( G ) G G G u ∼ G v u\sim_G v u ∼ G v σ ∈ Col ( G ′ ) \sigma\in\operatorname{Col}(G') σ ∈ Col ( G ′ ) G G G σ ( u ) = σ ( v ) \sigma(u)=\sigma(v) σ ( u ) = σ ( v ) u ∼ G ′ v u\sim_{G'}v u ∼ G ′ v
::: Finally, let C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k G G G
∑ i = 1 k ∣ C i ∣ 2 = ∑ i = 1 k ∣ C i ∣ ( ∣ C i ∣ − 1 ) + ∑ i = 1 k ∣ C i ∣ = 2 # { frozen pairs of G } + N = N ( N − 1 ) κ ( G ) + N . \sum_{i=1}^{k} |C_i|^2 = \sum_{i=1}^{k} |C_i|(|C_i|-1) + \sum_{i=1}^{k} |C_i| = 2\#\{\text{frozen pairs of }G\} + N = N(N-1)\kappa(G)+N. i = 1 ∑ k ∣ C i ∣ 2 = i = 1 ∑ k ∣ C i ∣ ( ∣ C i ∣ − 1 ) + i = 1 ∑ k ∣ C i ∣ = 2# { frozen pairs of G } + N = N ( N − 1 ) κ ( G ) + N . Dividing by N 2 N^2 N 2 Q ( G ) Q(G) Q ( G ) ∣ C G ( u ) ∣ |C_G(u)| ∣ C G ( u ) ∣ ( ∣ C G ( u ) ∣ / N ) 2 ≤ Q ( G ) (|C_G(u)|/N)^2 \le Q(G) ( ∣ C G ( u ) ∣/ N ) 2 ≤ Q ( G )
Lemma 3.6(a) is the elementary case of the addition–identification relation for chromatic polynomials [58 Section 3, Theorem 1] . For nonadjacent u u u v v v G G G G + { u , v } G+\{u,v\} G + { u , v } { u , v } \{u,v\} { u , v } G G G u u u v v v
The final edge
We now express the law of the final edge e H e_H e H G m G_m G m [24 Section 3.3] ; here independent sampling with replacement gives the normalization ( n 2 ) − 1 \binom{n}{2}^{-1} ( 2 n ) − 1 [11 Section 3.3] ; our proofs follow the same idea. They are short applications of Lemma 3.6(a).
Lemma 3.7. For all integers m ≥ 0 m \ge0 m ≥ 0
P ( H = m + 1 ∣ F m ) = κ ( G m ) almost surely. (39) \mathbb{P}(H=m+1\mid\mathcal{F}_m)=\kappa(G_m)\quad\text{almost surely.} \tag*{(39)} P ( H = m + 1 ∣ F m ) = κ ( G m ) almost surely. (39) For every F m \mathcal{F}_m F m ζ ≥ 0 \zeta\ge0 ζ ≥ 0 E [ ζ κ ( G m ) ] = E [ ζ 1 { H = m + 1 } ] \mathbb{E}[\zeta\kappa(G_m)]=\mathbb{E}[\zeta\mathbf{1}\{H=m+1\}] E [ ζ κ ( G m )] = E [ ζ 1 { H = m + 1 }]
In particular, if J \mathcal{J} J ζ m \zeta_m ζ m F m \mathcal{F}_m F m [ 0 , 1 ] [0,1] [ 0 , 1 ] m ∈ J m\in\mathcal{J} m ∈ J
∑ m ∈ J E [ ζ m κ ( G m ) ] ≤ P ( H − 1 ∈ J ) ≤ 1. (40) \sum_{m\in\mathcal{J}}\mathbb{E}[\zeta_m\kappa(G_m)]\le\mathbb{P}(H-1\in\mathcal{J})\le1. \tag*{(40)} m ∈ J ∑ E [ ζ m κ ( G m )] ≤ P ( H − 1 ∈ J ) ≤ 1. (40) Proof. We have
{ H = m + 1 } = { G m is colorable and G m + e m + 1 is not colorable } , \{H=m+1\}=\{G_m\text{ is colorable and }G_m+e_{m+1}\text{ is not colorable}\}, { H = m + 1 } = { G m is colorable and G m + e m + 1 is not colorable } , where G m G_m G m e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m e m + 1 e_{m+1} e m + 1 [ n ] [n] [ n ] F m \mathcal{F}_m F m G = G m G=G_m G = G m f = e m + 1 f=e_{m+1} f = e m + 1 ζ \zeta ζ m ∈ J m\in\mathcal{J} m ∈ J ζ m ≤ 1 \zeta_m\le1 ζ m ≤ 1
∑ m ∈ J E [ ζ m κ ( G m ) ] = ∑ m ∈ J E [ ζ m 1 { H = m + 1 } ] ≤ ∑ m ∈ J P ( H = m + 1 ) = P ( H − 1 ∈ J ) , (41) \sum_{m\in\mathcal{J}}\mathbb{E}[\zeta_m\kappa(G_m)] = \sum_{m\in\mathcal{J}}\mathbb{E}[\zeta_m\mathbf{1}\{H=m+1\}] \le\sum_{m\in\mathcal{J}}\mathbb{P}(H=m+1)=\mathbb{P}(H-1\in\mathcal{J}), \tag*{(41)} m ∈ J ∑ E [ ζ m κ ( G m )] = m ∈ J ∑ E [ ζ m 1 { H = m + 1 }] ≤ m ∈ J ∑ P ( H = m + 1 ) = P ( H − 1 ∈ J ) , (41) where the last equality holds because the events { H = m + 1 } \{H = m + 1\} { H = m + 1 }
To isolate final edges with prescribed properties, we use sets of frozen pairs. Let E \mathcal{E} E ( G , e ) (G,e) ( G , e ) G G G [ n ] [n] [ n ] e e e G G G G G G [ n ] [n] [ n ]
κ E ( G ) = ( n 2 ) − 1 # { e : ( G , e ) ∈ E } . (42) \kappa_{\mathcal{E}}(G) = \binom{n}{2}^{-1}\#\{e : (G,e) \in\mathcal{E}\}. \tag*{(42)} κ E ( G ) = ( 2 n ) − 1 # { e : ( G , e ) ∈ E } . (42) Then 0 ≤ κ E ( G ) ≤ κ ( G ) 0 \le\kappa_{\mathcal{E}}(G) \le\kappa(G) 0 ≤ κ E ( G ) ≤ κ ( G ) κ E ( G ) = 0 \kappa_{\mathcal{E}}(G) = 0 κ E ( G ) = 0 G G G ϕ \phi ϕ [ n ] [n] [ n ] e = { u , v } e = \{u,v\} e = { u , v } ϕ ( e ) = { ϕ ( u ) , ϕ ( v ) } \phi(e) = \{\phi(u),\phi(v)\} ϕ ( e ) = { ϕ ( u ) , ϕ ( v )} E \mathcal{E} E invariant under relabeling if, for every permutation ϕ \phi ϕ [ n ] [n] [ n ] ( ϕ G , ϕ ( e ) ) ∈ E (\phi G,\phi(e)) \in\mathcal{E} ( ϕG , ϕ ( e )) ∈ E ( G , e ) ∈ E (G,e) \in\mathcal{E} ( G , e ) ∈ E h h h [ n ] [n] [ n ] invariant under relabeling if h ( ϕ G ) = h ( G ) h(\phi G) = h(G) h ( ϕG ) = h ( G ) G G G ϕ \phi ϕ [ n ] [n] [ n ] G G G [ n ] [n] [ n ] invariant under relabeling if ϕ G \phi G ϕG G G G ϕ \phi ϕ [ n ] [n] [ n ] G m G_m G m m ≥ 0 m \ge0 m ≥ 0
Lemma 3.8. Let E \mathcal{E} E ( G , e ) (G,e) ( G , e ) G G G [ n ] [n] [ n ] e e e G G G
(a) Let h 0 , h 1 , … h_0,h_1,\ldots h 0 , h 1 , … [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) [ n ] [n] [ n ] L ′ ≥ 0 L' \ge0 L ′ ≥ 0 U ′ U' U ′ U ′ > L ′ U' > L' U ′ > L ′ U ′ = ∞ U' = \infty U ′ = ∞ U ′ = ∞ U' = \infty U ′ = ∞ { L ′ < H ≤ U ′ } \{L' < H \le U'\} { L ′ < H ≤ U ′ } { L ′ < H < ∞ } \{L' < H < \infty\} { L ′ < H < ∞ }
E [ h H − 1 ( Γ ) 1 { L ′ < H ≤ U ′ , ( Γ , e H ) ∈ E } ] = ∑ m = L ′ U ′ − 1 E [ κ E ( G m ) h m ( G m ) ] . (43) \mathbb{E}\left[h_{H-1}(\Gamma)1\{L' < H \le U',\,(\Gamma,e_H) \in\mathcal{E}\}\right] = \sum_{m=L'}^{U'-1}\mathbb{E}\left[\kappa_{\mathcal{E}}(G_m)h_m(G_m)\right]. \tag*{(43)} E [ h H − 1 ( Γ ) 1 { L ′ < H ≤ U ′ , ( Γ , e H ) ∈ E } ] = m = L ′ ∑ U ′ − 1 E [ κ E ( G m ) h m ( G m ) ] . (43) If U ′ = ∞ U' = \infty U ′ = ∞ m ≥ L ′ m \ge L' m ≥ L ′ (43) holds with h m = h h_m = h h m = h m m m h ≥ 0 h \ge0 h ≥ 0 [ n ] [n] [ n ]
(b) Suppose that E \mathcal{E} E h ≥ 0 h \ge0 h ≥ 0 [ n ] [n] [ n ] G G G [ n ] [n] [ n ] a , b ∈ [ n ] a,b \in[n] a , b ∈ [ n ]
E [ κ E ( G ) h ( G ) ] = E [ 1 { ( G , { a , b } ) ∈ E } h ( G ) ] . (44) \mathbb{E}\left[\kappa_{\mathcal{E}}(G)h(G)\right] = \mathbb{E}\left[1\{(G,\{a,b\}) \in\mathcal{E}\}h(G)\right]. \tag*{(44)} E [ κ E ( G ) h ( G ) ] = E [ 1 {( G , { a , b }) ∈ E } h ( G ) ] . (44) Proof. We begin with (a). Fix m ≥ 0 m \ge0 m ≥ 0 ( G m , e m + 1 ) ∈ E (G_m,e_{m+1}) \in\mathcal{E} ( G m , e m + 1 ) ∈ E G m G_m G m e m + 1 e_{m+1} e m + 1 G m G_m G m G m + e m + 1 G_m + e_{m+1} G m + e m + 1 Lemma 3.6 (a), and H = m + 1 H = m + 1 H = m + 1 Γ = G m \Gamma= G_m Γ = G m e H = e m + 1 e_H = e_{m+1} e H = e m + 1 { H = m + 1 } \{H = m + 1\} { H = m + 1 }
{ H = m + 1 , ( Γ , e H ) ∈ E } = { ( G m , e m + 1 ) ∈ E } . (45) \{H = m + 1,\,(\Gamma,e_H) \in\mathcal{E}\} = \{(G_m,e_{m+1}) \in\mathcal{E}\}. \tag*{(45)} { H = m + 1 , ( Γ , e H ) ∈ E } = {( G m , e m + 1 ) ∈ E } . (45) The pair e m + 1 e_{m+1} e m + 1 [ n ] [n] [ n ] F m \mathcal{F}_m F m (45) given F m \mathcal{F}_m F m κ E ( G m ) \kappa_{\mathcal{E}}(G_m) κ E ( G m ) h m ( G m ) h_m(G_m) h m ( G m ) F m \mathcal{F}_m F m
E [ h H − 1 ( Γ ) 1 { H = m + 1 , ( Γ , e H ) ∈ E } ] = E [ h m ( G m ) 1 { ( G m , e m + 1 ) ∈ E } ] = E [ κ E ( G m ) h m ( G m ) ] . \mathbb{E}\left[h_{H-1}(\Gamma)\mathbf{1}\{H=m+1,(\Gamma,e_H)\in\mathcal{E}\}\right]
=
\mathbb{E}\left[h_m(G_m)\mathbf{1}\{(G_m,e_{m+1})\in\mathcal{E}\}\right]
=
\mathbb{E}\left[\kappa_{\mathcal{E}}(G_m)h_m(G_m)\right]. E [ h H − 1 ( Γ ) 1 { H = m + 1 , ( Γ , e H ) ∈ E } ] = E [ h m ( G m ) 1 {( G m , e m + 1 ) ∈ E } ] = E [ κ E ( G m ) h m ( G m ) ] . The event { L ′ < H ≤ U ′ } \{L'<H\le U'\} { L ′ < H ≤ U ′ } { H = m + 1 } \{H=m+1\} { H = m + 1 } L ′ ≤ m ≤ U ′ − 1 L'\le m\le U'-1 L ′ ≤ m ≤ U ′ − 1 m m m (43) . (All terms are nonnegative, and the exchange of sum and expectation is justified even when the sum is infinite.)
We now prove (b). By the definition (3.10), we have
κ E ( G ) h ( G ) = ( n 2 ) − 1 ∑ e 1 { ( G , e ) ∈ E } h ( G ) . \kappa_{\mathcal{E}}(G)h(G)=\binom{n}{2}^{-1}\sum_e\mathbf{1}\{(G,e)\in\mathcal{E}\}h(G). κ E ( G ) h ( G ) = ( 2 n ) − 1 e ∑ 1 {( G , e ) ∈ E } h ( G ) . where the sum runs over the pairs e e e [ n ] [n] [ n ] e e e ϕ \phi ϕ [ n ] [n] [ n ] ϕ ( e ) = { a , b } \phi(e)=\{a,b\} ϕ ( e ) = { a , b } E \mathcal{E} E h h h
1 { ( G , e ) ∈ E } h ( G ) = 1 { ( ϕ G , { a , b } ) ∈ E } h ( ϕ G ) , \mathbf{1}\{(G,e)\in\mathcal{E}\}h(G)=\mathbf{1}\{(\phi G,\{a,b\})\in\mathcal{E}\}h(\phi G), 1 {( G , e ) ∈ E } h ( G ) = 1 {( ϕG , { a , b }) ∈ E } h ( ϕG ) , and since ϕ G \phi G ϕG G G G (44) . Averaging over the ( n 2 ) \binom{n}{2} ( 2 n ) e e e
Lemma 3.8 (a) converts statements about the final edge into statements about frozen pairs at deterministic times. We use this identity to estimate probabilities involving the final edge.
Lifetime
We will control the window by the expected number of further uniformly random pairs that the last colorable graph Γ \Gamma Γ
Definition 3.9. Let S S S N ≥ 2 N\ge2 N ≥ 2 G G G S S S f 1 , f 2 , … f_1,f_2,\ldots f 1 , f 2 , … S S S
T G = min { k ≥ 0 : G + f 1 + ⋯ + f k is not colorable } ∈ { 0 , 1 , … } ∪ { ∞ } . (46) T_G=\min\{k\ge0:G+f_1+\cdots+f_k\text{ is not colorable}\}\in\{0,1,\ldots\}\cup\{\infty\}. \tag*{(46)} T G = min { k ≥ 0 : G + f 1 + ⋯ + f k is not colorable } ∈ { 0 , 1 , … } ∪ { ∞ } . (46) We call r ( G ) = E T G ∈ [ 0 , ∞ ] r(G)=\mathbb{E}T_G\in[0,\infty] r ( G ) = E T G ∈ [ 0 , ∞ ] expected lifetime of G G G b ≥ 0 b\ge0 b ≥ 0
r b ( G ) = E min ( T G , b ) . r_b(G)=\mathbb{E}\min(T_G,b). r b ( G ) = E min ( T G , b ) . In words, r ( G ) r(G) r ( G ) G G G T G T_G T G S S S r ( G ) r(G) r ( G ) r b ( G ) r_b(G) r b ( G ) G G G min ( T G , b ) = ∑ k = 0 b − 1 1 { T G > k } \min(T_G,b)=\sum_{k=0}^{b-1}\mathbf{1}\{T_G>k\} min ( T G , b ) = ∑ k = 0 b − 1 1 { T G > k }
r b ( G ) = ∑ k = 0 b − 1 P ( T G > k ) . (47) r_b(G)=\sum_{k=0}^{b-1}\mathbb{P}(T_G>k). \tag*{(47)} r b ( G ) = k = 0 ∑ b − 1 P ( T G > k ) . (47) Lemma 3.10. Let S S S N ≥ 2 N\ge2 N ≥ 2 G G G G ′ G' G ′ S S S
(a) If G G G r ( G ) = 0 r(G)=0 r ( G ) = 0 r b ( G ) = 0 r_b(G)=0 r b ( G ) = 0 b ≥ 0 b\ge0 b ≥ 0 G G G 1 ≤ r b ( G ) ≤ min ( b , r ( G ) ) 1\le r_b(G)\le\min(b,r(G)) 1 ≤ r b ( G ) ≤ min ( b , r ( G )) b ≥ 1 b\ge1 b ≥ 1 r b ( G ) r_b(G) r b ( G ) b b b G G G N ≥ 2 q N\ge2q N ≥ 2 q
r ( G ) ≤ A q N . (48) r(G)\le A_qN. \tag*{(48)} r ( G ) ≤ A q N . (48) (b) If G ⊂ G ′ G\subset G' G ⊂ G ′ r ( G ′ ) ≤ r ( G ) r(G')\le r(G) r ( G ′ ) ≤ r ( G ) r b ( G ′ ) ≤ r b ( G ) r_b(G')\le r_b(G) r b ( G ′ ) ≤ r b ( G ) b ≥ 0 b\ge0 b ≥ 0 ϕ : S → S ′ \phi:S\to S' ϕ : S → S ′ r ( ϕ G ) = r ( G ) r(\phi G)=r(G) r ( ϕG ) = r ( G ) r b ( ϕ G ) = r b ( G ) r_b(\phi G)=r_b(G) r b ( ϕG ) = r b ( G ) b ≥ 0 b\ge0 b ≥ 0
(c) If κ ( G ) > 0 \kappa(G)>0 κ ( G ) > 0 r ( G ) ≤ 1 / κ ( G ) r(G)\le1/\kappa(G) r ( G ) ≤ 1/ κ ( G )
(d) For the edge process on [ n ] [n] [ n ] m ≥ 0 m\ge0 m ≥ 0
r ( G m ) = E [ ( H − m ) 1 { H > m } ∣ F m ] almost surely . (49) r(G_m)=\mathbb{E}\left[(H-m)\mathbf{1}\{H>m\}\mid\mathcal{F}_m\right]\quad\text{almost surely}. \tag*{(49)} r ( G m ) = E [ ( H − m ) 1 { H > m } ∣ F m ] almost surely . (49) In particular, E H = r ( G 0 ) ≤ A q n \mathbb{E}H=r(G_0)\le A_qn E H = r ( G 0 ) ≤ A q n
Proof. We begin with (a). We have T G = 0 T_G=0 T G = 0 G G G T G ≥ 1 T_G\ge1 T G ≥ 1 G G G min ( T G , b ) ≥ 1 \min(T_G,b)\ge1 min ( T G , b ) ≥ 1 b ≥ 1 b\ge1 b ≥ 1 min ( T G , b ) ≤ b \min(T_G,b)\le b min ( T G , b ) ≤ b min ( T G , b ) ≤ T G \min(T_G,b)\le T_G min ( T G , b ) ≤ T G b b b
We now prove (3.16). Let ∅ S \varnothing_S ∅ S S S S T G T_G T G T ∅ S T_{\varnothing_S} T ∅ S G + f 1 + ⋯ + f k G+f_1+\cdots+f_k G + f 1 + ⋯ + f k f 1 + ⋯ + f k f_1+\cdots+f_k f 1 + ⋯ + f k T G ≤ T ∅ S T_G\le T_{\varnothing_S} T G ≤ T ∅ S r ( ∅ S ) r(\varnothing_S) r ( ∅ S )
Let σ : S → [ q ] \sigma:S\to[q] σ : S → [ q ] i ∈ [ q ] i\in[q] i ∈ [ q ] b i = ∣ σ − 1 ( { i } ) ∣ b_i=\lvert\sigma^{-1}(\{i\})\rvert b i = ∣ σ − 1 ({ i })∣ x ↦ x ( x − 1 ) / 2 x\mapsto x(x-1)/2 x ↦ x ( x − 1 ) /2
# { pairs { u , v } of S : σ ( u ) = σ ( v ) } = ∑ i = 1 q b i ( b i − 1 ) 2 ≥ N ( N / q − 1 ) 2 . \#\{\text{pairs }\{u,v\}\text{ of }S:\sigma(u)=\sigma(v)\}=\sum_{i=1}^{q}\frac{b_i(b_i-1)}{2}\ge\frac{N(N/q-1)}{2}. # { pairs { u , v } of S : σ ( u ) = σ ( v )} = i = 1 ∑ q 2 b i ( b i − 1 ) ≥ 2 N ( N / q − 1 ) . The fraction of pairs of S S S σ \sigma σ ( N / q − 1 ) / ( N − 1 ) (N/q-1)/(N-1) ( N / q − 1 ) / ( N − 1 ) 1 / ( 2 q ) 1/(2q) 1/ ( 2 q ) N ≥ 2 q N\ge2q N ≥ 2 q
It follows that σ \sigma σ f 1 + ⋯ + f k f_1+\cdots+f_k f 1 + ⋯ + f k ( 1 − 1 / ( 2 q ) ) k ≤ e − k / ( 2 q ) (1-1/(2q))^k\le e^{-k/(2q)} ( 1 − 1/ ( 2 q ) ) k ≤ e − k / ( 2 q ) q N q^N q N σ \sigma σ
P ( T ∅ S > k ) = P ( f 1 + ⋯ + f k is colorable ) ≤ min ( 1 , q N e − k / ( 2 q ) ) . \mathbb{P}(T_{\varnothing_S}>k)=\mathbb{P}(f_1+\cdots+f_k\text{ is colorable})\le\min\left(1,q^N e^{-k/(2q)}\right). P ( T ∅ S > k ) = P ( f 1 + ⋯ + f k is colorable ) ≤ min ( 1 , q N e − k / ( 2 q ) ) . Let k 0 = ⌈ 2 q N log q ⌉ k_0=\lceil2qN\log q\rceil k 0 = ⌈ 2 q N log q ⌉ k ≥ k 0 k\ge k_0 k ≥ k 0 q N e − k / ( 2 q ) ≤ e − ( k − k 0 ) / ( 2 q ) q^N e^{-k/(2q)}\le e^{-(k-k_0)/(2q)} q N e − k / ( 2 q ) ≤ e − ( k − k 0 ) / ( 2 q )
r ( ∅ S ) = ∑ k ≥ 0 P ( T ∅ S > k ) ≤ k 0 + ∑ j ≥ 0 e − j / ( 2 q ) = k 0 + 1 1 − e − 1 / ( 2 q ) r(\varnothing_S)=\sum_{k\ge0}\mathbb{P}(T_{\varnothing_S}>k)\le k_0+\sum_{j\ge0}e^{-j/(2q)}=k_0+\frac{1}{1-e^{-1/(2q)}} r ( ∅ S ) = k ≥ 0 ∑ P ( T ∅ S > k ) ≤ k 0 + j ≥ 0 ∑ e − j / ( 2 q ) = k 0 + 1 − e − 1/ ( 2 q ) 1 ≤ 2 q N log q + 2 q + 2 ≤ A q N . \le2qN\log q+2q+2\le A_qN. ≤ 2 q N log q + 2 q + 2 ≤ A q N . In the second line, we used 1 / ( 1 − e − x ) ≤ 1 + 1 / x 1/(1-e^{-x})\le1+1/x 1/ ( 1 − e − x ) ≤ 1 + 1/ x x > 0 x>0 x > 0 2 q + 2 ≤ ( 2 q + 1 ) N 2q+2\le(2q+1)N 2 q + 2 ≤ ( 2 q + 1 ) N N ≥ 2 N\ge2 N ≥ 2
For (b), we compute T G T_G T G T G ′ T_{G'} T G ′ G ′ + f 1 + ⋯ + f k G'+f_1+\cdots+f_k G ′ + f 1 + ⋯ + f k G + f 1 + ⋯ + f k G+f_1+\cdots+f_k G + f 1 + ⋯ + f k T G ′ ≤ T G T_{G'}\le T_G T G ′ ≤ T G
Now let ϕ : S → S ′ \phi:S\to S' ϕ : S → S ′ ( ϕ ( f k ) ) k ≥ 1 (\phi(f_k))_{k\ge1} ( ϕ ( f k ) ) k ≥ 1 S ′ S' S ′ σ ↦ σ ∘ ϕ − 1 \sigma\mapsto\sigma\circ\phi^{-1} σ ↦ σ ∘ ϕ − 1 Col ( G + f 1 + ⋯ + f k ) \operatorname{Col}(G+f_1+\cdots+f_k) Col ( G + f 1 + ⋯ + f k ) Col ( ϕ G + ϕ ( f 1 ) + ⋯ + ϕ ( f k ) ) \operatorname{Col}(\phi G+\phi(f_1)+\cdots+\phi(f_k)) Col ( ϕG + ϕ ( f 1 ) + ⋯ + ϕ ( f k )) T ϕ G T_{\phi G} T ϕG ( ϕ ( f k ) ) (\phi(f_k)) ( ϕ ( f k )) T G T_G T G
Next, we prove (c). Let Y \mathcal{Y} Y G G G ∣ Y ∣ = κ ( G ) ( N 2 ) > 0 |\mathcal{Y}|=\kappa(G)\binom{N}{2}>0 ∣ Y ∣ = κ ( G ) ( 2 N ) > 0
T ′ = min { k ≥ 1 : f k ∈ Y } . T'=\min\{k\ge1:f_k\in\mathcal{Y}\}. T ′ = min { k ≥ 1 : f k ∈ Y } . Then T ′ T' T ′ κ ( G ) \kappa(G) κ ( G ) { 1 , 2 , … } \{1,2,\ldots\} { 1 , 2 , … } E T ′ = 1 / κ ( G ) \mathbb{E}T'=1/\kappa(G) E T ′ = 1/ κ ( G ) T G ≤ T ′ T_G\le T' T G ≤ T ′
This holds if G + f 1 + ⋯ + f T ′ − 1 G+f_1+\cdots+f_{T'-1} G + f 1 + ⋯ + f T ′ − 1 G + f 1 + ⋯ + f T ′ − 1 G+f_1+\cdots+f_{T'-1} G + f 1 + ⋯ + f T ′ − 1 G G G f T ′ f_{T'} f T ′ G + f 1 + ⋯ + f T ′ G+f_1+\cdots+f_{T'} G + f 1 + ⋯ + f T ′
Finally, we prove (d). For a multigraph G G G [ n ] [n] [ n ] ( p k ) k ≥ 1 (p_k)_{k\ge1} ( p k ) k ≥ 1 [ n ] [n] [ n ]
Θ ( G , ( p k ) k ≥ 1 ) = min { k ≥ 0 : G + p 1 + ⋯ + p k is not colorable } . \Theta\left(G,(p_k)_{k\ge1}\right)=\min\{k\ge0:G+p_1+\cdots+p_k\text{ is not colorable}\}. Θ ( G , ( p k ) k ≥ 1 ) = min { k ≥ 0 : G + p 1 + ⋯ + p k is not colorable } . By the definition of T G T_G T G E Θ ( G , ( f k ) k ≥ 1 ) = r ( G ) \mathbb{E}\Theta\left(G,(f_k)_{k\ge1}\right)=r(G) E Θ ( G , ( f k ) k ≥ 1 ) = r ( G ) G G G [ n ] [n] [ n ] ( f k ) (f_k) ( f k ) [ n ] [n] [ n ]
If H > m H>m H > m H − m = Θ ( G m , ( e m + k ) k ≥ 1 ) H-m=\Theta\left(G_m,(e_{m+k})_{k\ge1}\right) H − m = Θ ( G m , ( e m + k ) k ≥ 1 ) H ≤ m H\le m H ≤ m G m G_m G m Θ ( G m , ( e m + k ) k ≥ 1 ) = 0 \Theta\left(G_m,(e_{m+k})_{k\ge1}\right)=0 Θ ( G m , ( e m + k ) k ≥ 1 ) = 0
( H − m ) 1 { H > m } = Θ ( G m , ( e m + k ) k ≥ 1 ) . (H-m)1_{\{H>m\}}=\Theta\left(G_m,(e_{m+k})_{k\ge1}\right). ( H − m ) 1 { H > m } = Θ ( G m , ( e m + k ) k ≥ 1 ) . The sequence ( e m + k ) k ≥ 1 (e_{m+k})_{k\ge1} ( e m + k ) k ≥ 1 [ n ] [n] [ n ] F m \mathcal{F}_m F m G m G_m G m e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m Θ ≥ 0 \Theta\ge0 Θ ≥ 0 F m \mathcal{F}_m F m r ( G m ) r(G_m) r ( G m ) m = 0 m=0 m = 0 N = n ≥ 2 q N=n\ge2q N = n ≥ 2 q E H = r ( G 0 ) ≤ A q n \mathbb{E}H=r(G_0)\le A_qn E H = r ( G 0 ) ≤ A q n
The identity (3.17) holds at deterministic times m m m H − 1 H-1 H − 1 Γ \Gamma Γ G m G_m G m Lemma 3.8 when we bound r ( Γ ) r(\Gamma) r ( Γ )
Lemma 3.11. Let S S S N ≥ 2 q N \ge2q N ≥ 2 q G G G S S S y > 0 y > 0 y > 0 r ( G ) > y r(G) > y r ( G ) > y
Q ( G ) ≤ 1 + A q y , ∣ C G ( u ) ∣ ≤ N ( 1 + A q y ) 1 / 2 for all u ∈ S . (50) Q(G) \le\frac{1 + A_q}{y}, \qquad|C_G(u)| \le N\left(\frac{1 + A_q}{y}\right)^{1/2} \qquad\text{for all } u \in S. \tag*{(50)} Q ( G ) ≤ y 1 + A q , ∣ C G ( u ) ∣ ≤ N ( y 1 + A q ) 1/2 for all u ∈ S . (50) Proof. By Lemma 3.10 (c), we have κ ( G ) r ( G ) ≤ 1 \kappa(G)r(G) \le1 κ ( G ) r ( G ) ≤ 1 κ ( G ) = 0 \kappa(G) = 0 κ ( G ) = 0 Lemma 3.10 (a) gives r ( G ) / N ≤ A q r(G)/N \le A_q r ( G ) / N ≤ A q Lemma 3.6 (c), we obtain
Q ( G ) r ( G ) = ( 1 − 1 N ) κ ( G ) r ( G ) + r ( G ) N ≤ 1 + A q . Q(G)r(G) = \left(1 - \frac{1}{N}\right)\kappa(G)r(G) + \frac{r(G)}{N} \le1 + A_q. Q ( G ) r ( G ) = ( 1 − N 1 ) κ ( G ) r ( G ) + N r ( G ) ≤ 1 + A q . Since r ( G ) > y r(G) > y r ( G ) > y (50) , and the second follows from the bound ∣ C G ( u ) ∣ ≤ N Q ( G ) 1 / 2 |C_G(u)| \le NQ(G)^{1/2} ∣ C G ( u ) ∣ ≤ N Q ( G ) 1/2 Lemma 3.6 (c).
Poissonization and splitting
When the number of sampled pairs is Poisson, splitting the possible pairs into disjoint sets gives independent sublists. We record the facts that we use.
Definition 3.12. Let S S S f 1 , f 2 , … f_1, f_2, \ldots f 1 , f 2 , … S S S ( Π t ) t ≥ 0 (\Pi_t)_{t \ge0} ( Π t ) t ≥ 0 ( f i ) (f_i) ( f i ) t ↦ f 1 + ⋯ + f Π t t \mapsto f_1 + \cdots+ f_{\Pi_t} t ↦ f 1 + ⋯ + f Π t S S S t t t S S S t t t a ∈ S a \in S a ∈ S S ′ ⊂ S ∖ { a } S' \subset S \setminus\{a\} S ′ ⊂ S ∖ { a } t ≥ 0 t \ge0 t ≥ 0 a a a S ′ S' S ′ t t t x x x i ≤ Π t i \le\Pi_t i ≤ Π t f i = { a , x } f_i = \{a, x\} f i = { a , x } x ∈ S ′ x \in S' x ∈ S ′ i i i
A neighbor list may have repeated entries. For μ ≥ 0 \mu\ge0 μ ≥ 0 E ′ E' E ′ ν ′ \nu' ν ′ Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) E ′ E' E ′ ν ′ \nu' ν ′ Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) ν ′ \nu' ν ′ ( Π t ) t ≥ 0 (\Pi_t)_{t \ge0} ( Π t ) t ≥ 0 ( e i ) (e_i) ( e i ) G ( t ) = G Π t G(t) = G_{\Pi_t} G ( t ) = G Π t t ≥ 0 t \ge0 t ≥ 0 ( G ( t ) ) t ≥ 0 (G(t))_{t \ge0} ( G ( t ) ) t ≥ 0 [ n ] [n] [ n ] G ( t ) G(t) G ( t ) t ≥ 0 t \ge0 t ≥ 0 S S S ( G m ) m ≥ 0 (G_m)_{m \ge0} ( G m ) m ≥ 0 m m m S S S ( G S ( t ) ) t ≥ 0 (G^S(t))_{t \ge0} ( G S ( t ) ) t ≥ 0 S S S
Lemma 3.13. Let S S S h ≥ 0 h \ge0 h ≥ 0 S S S θ > 0 \theta> 0 θ > 0
∫ 0 ∞ E h ( G S ( θ t ) ) d t = 1 θ ∑ m ≥ 0 E h ( G m ) , (51) \int_0^\infty\mathbb{E}h(G^S(\theta t))\,\mathrm{d}t = \frac{1}{\theta}\sum_{m \ge0}\mathbb{E}h(G_m), \tag*{(51)} ∫ 0 ∞ E h ( G S ( θ t )) d t = θ 1 m ≥ 0 ∑ E h ( G m ) , (51) where both sides may be infinite.
Proof. For s ≥ 0 s \ge0 s ≥ 0 Π s \Pi_s Π s Pois ( s ) \operatorname{Pois}(s) Pois ( s )
E h ( G S ( s ) ) = ∑ m ≥ 0 e − s s m m ! E h ( G m S ) . \mathbb{E}h(G^S(s))=\sum_{m\ge0}e^{-s}\frac{s^m}{m!}\mathbb{E}h(G^S_m). E h ( G S ( s )) = m ≥ 0 ∑ e − s m ! s m E h ( G m S ) . We insert this identity with s = θ t s=\theta t s = θ t (51) , and we exchange sum and integral, which is allowed by Tonelli’s theorem since all terms are nonnegative. Using that
∫ 0 ∞ e − θ t ( θ t ) m m ! d t = 1 θ \int_0^\infty e^{-\theta t}\frac{(\theta t)^m}{m!}\,dt=\frac{1}{\theta} ∫ 0 ∞ e − θ t m ! ( θ t ) m d t = θ 1 for every integer m ≥ 0 m \ge0 m ≥ 0 (51) .
Lemma 3.14. (a) Let E E E ν \nu ν E E E X 1 , X 2 , … X_1,X_2,\ldots X 1 , X 2 , … E E E ν \nu ν I I I Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) μ ≥ 0 \mu\ge0 μ ≥ 0 E 1 , … , E J E_1,\ldots,E_J E 1 , … , E J E E E ν ( E j ) > 0 \nu(E_j)>0 ν ( E j ) > 0 ( X 1 , … , X I ) (X_1,\ldots,X_I) ( X 1 , … , X I ) E 1 , … , E J E_1,\ldots,E_J E 1 , … , E J j j j E j E_j E j Pois ( μ ν ( E j ) ) \operatorname{Pois}(\mu\nu(E_j)) Pois ( μν ( E j )) E j E_j E j ν ( ⋅ ∣ E j ) \nu(\mathord{\cdot}\mid E_j) ν ( ⋅ ∣ E j )
(b) Let S S S N ≥ 3 N \ge3 N ≥ 3 S 0 ⊂ S S_0 \subset S S 0 ⊂ S S ′ = S ∖ S 0 S'=S\setminus S_0 S ′ = S ∖ S 0 S ′ S' S ′ t ≥ 0 t \ge0 t ≥ 0 G S ( t ) [ S ′ ] G^S(t)[S'] G S ( t ) [ S ′ ] G S ( t ) [ S 0 ] G^S(t)[S_0] G S ( t ) [ S 0 ] a ∈ S 0 a\in S_0 a ∈ S 0 S ′ S' S ′ t t t G S ( t ) [ S ′ ] G^S(t)[S'] G S ( t ) [ S ′ ] S ′ S' S ′ t ( ∣ S ′ ∣ 2 ) / ( N 2 ) t\binom{|S'|}{2}/\binom{N}{2} t ( 2 ∣ S ′ ∣ ) / ( 2 N ) Pois ( t ∣ S ′ ∣ / ( N 2 ) ) \operatorname{Pois}(t|S'|/\binom{N}{2}) Pois ( t ∣ S ′ ∣/ ( 2 N ) ) S ′ S' S ′
(c) Let S S S N ≥ 3 N \ge3 N ≥ 3 a ∈ S a\in S a ∈ S t ≥ 0 t \ge0 t ≥ 0 deg G S ( t ) ( a ) \deg_{G^S(t)}(a) deg G S ( t ) ( a ) a a a S ∖ { a } S\setminus\{a\} S ∖ { a } t t t G S ( t ) G^S(t) G S ( t ) G S ( t ) [ S ∖ { a } ] G^S(t)[S\setminus\{a\}] G S ( t ) [ S ∖ { a }] { a , x } \{a,x\} { a , x } x x x G S ( t ) [ S ∖ { a } ] G^S(t)[S\setminus\{a\}] G S ( t ) [ S ∖ { a }] Pois ( 2 t / N ) \operatorname{Pois}(2t/N) Pois ( 2 t / N ) S ∖ { a } S\setminus\{a\} S ∖ { a } G S ( t ) [ S ∖ { a } ] G^S(t)[S\setminus\{a\}] G S ( t ) [ S ∖ { a }] S ∖ { a } S\setminus\{a\} S ∖ { a } ( N − 2 ) t / N (N-2)t/N ( N − 2 ) t / N
Proof. We begin with (a). Fix integers k 1 , … , k J ≥ 0 k_1,\ldots,k_J \ge0 k 1 , … , k J ≥ 0 k = k 1 + ⋯ + k J k=k_1+\cdots+k_J k = k 1 + ⋯ + k J
y ( j ) = ( y 1 ( j ) , … , y k j ( j ) ) ∈ E j k j , 1 ≤ j ≤ J . y^{(j)}=(y^{(j)}_1,\ldots,y^{(j)}_{k_j})\in E_j^{k_j},\qquad1\le j\le J. y ( j ) = ( y 1 ( j ) , … , y k j ( j ) ) ∈ E j k j , 1 ≤ j ≤ J . The subsequences of ( X 1 , … , X I ) (X_1,\ldots,X_I) ( X 1 , … , X I ) E 1 , … , E J E_1,\ldots,E_J E 1 , … , E J y ( 1 ) , … , y ( J ) y^{(1)},\ldots,y^{(J)} y ( 1 ) , … , y ( J ) I = k I=k I = k ( X 1 , … , X k ) (X_1,\ldots,X_k) ( X 1 , … , X k ) k ! / ( k 1 ! ⋯ k J ! ) k!/(k_1!\cdots k_J!) k ! / ( k 1 ! ⋯ k J !) y ( 1 ) , … , y ( J ) y^{(1)},\ldots,y^{(J)} y ( 1 ) , … , y ( J ) ∏ j = 1 J ∏ i = 1 k j ν ( y i ( j ) ) \prod_{j=1}^J\prod_{i=1}^{k_j}\nu(y_i^{(j)}) ∏ j = 1 J ∏ i = 1 k j ν ( y i ( j ) )
e − μ μ k k ! ⋅ k ! k 1 ! ⋯ k J ! ∏ j = 1 J ∏ i = 1 k j ν ( y i ( j ) ) = ∏ j = 1 J ( e − μ ν ( E j ) ( μ ν ( E j ) ) k j k j ! ∏ i = 1 k j ν ( y i ( j ) ) ν ( E j ) ) , e^{-\mu}\frac{\mu^k}{k!}\cdot\frac{k!}{k_1!\cdots k_J!}\prod_{j=1}^J\prod_{i=1}^{k_j}\nu(y_i^{(j)})
=\prod_{j=1}^J\left(e^{-\mu\nu(E_j)}\frac{(\mu\nu(E_j))^{k_j}}{k_j!}\prod_{i=1}^{k_j}\frac{\nu(y_i^{(j)})}{\nu(E_j)}\right), e − μ k ! μ k ⋅ k 1 ! ⋯ k J ! k ! j = 1 ∏ J i = 1 ∏ k j ν ( y i ( j ) ) = j = 1 ∏ J e − μν ( E j ) k j ! ( μν ( E j ) ) k j i = 1 ∏ k j ν ( E j ) ν ( y i ( j ) ) , where we used that ν ( E 1 ) + ⋯ + ν ( E J ) = 1 \nu(E_1) + \cdots+ \nu(E_J) = 1 ν ( E 1 ) + ⋯ + ν ( E J ) = 1 j j j Pois ( μ ν ( E j ) ) \operatorname{Pois}(\mu\nu(E_j)) Pois ( μν ( E j )) E j E_j E j ν ( ⋅ ∣ E j ) \nu(\cdot\mid E_j) ν ( ⋅ ∣ E j ) y ( j ) y^{(j)} y ( j )
For (b), let ( f i ) i ≥ 1 (f_i)_{i \ge1} ( f i ) i ≥ 1 G S ( t ) G^S(t) G S ( t ) E E E S S S ν \nu ν X i = f i X_i = f_i X i = f i I = Π t I = \Pi_t I = Π t μ = t \mu= t μ = t E E E S ′ S' S ′ S 0 S_0 S 0 ∣ S 0 ∣ ≥ 2 \lvert S_0\rvert\ge2 ∣ S 0 ∣ ≥ 2 a ∈ S 0 a \in S_0 a ∈ S 0 ∣ S ′ ∣ \lvert S'\rvert ∣ S ′ ∣ { a , x } \{a,x\} { a , x } x ∈ S ′ x \in S' x ∈ S ′ G S ( t ) [ S ′ ] G^S(t)[S'] G S ( t ) [ S ′ ] S ′ S' S ′ Pois ( t ( ∣ S ′ ∣ 2 ) / ( N 2 ) ) \operatorname{Pois}\left(t\binom{\lvert S'\rvert}{2}/\binom{N}{2}\right) Pois ( t ( 2 ∣ S ′ ∣ ) / ( 2 N ) ) S ′ S' S ′ Π ′ \Pi' Π ′ s = t ( ∣ S ′ ∣ 2 ) / ( N 2 ) s = t\binom{\lvert S'\rvert}{2}/\binom{N}{2} s = t ( 2 ∣ S ′ ∣ ) / ( 2 N ) Π s ′ \Pi'_s Π s ′ S ′ S' S ′
The multigraph G S ( t ) [ S 0 ] G^S(t)[S_0] G S ( t ) [ S 0 ] S 0 S_0 S 0 a ∈ S 0 a \in S_0 a ∈ S 0 S ′ S' S ′ { a , x } \{a,x\} { a , x } x ∈ S ′ x \in S' x ∈ S ′ { a , x } \{a,x\} { a , x } x x x S ′ S' S ′
To prove (c), we take S 0 = { a } S_0 = \{a\} S 0 = { a } ∣ S ′ ∣ = N − 1 \lvert S'\rvert= N - 1 ∣ S ′ ∣ = N − 1 ( N − 1 2 ) / ( N 2 ) = ( N − 2 ) / N \binom{N-1}{2}/\binom{N}{2} = (N - 2)/N ( 2 N − 1 ) / ( 2 N ) = ( N − 2 ) / N ( N − 1 ) / ( N 2 ) = 2 / N (N - 1)/\binom{N}{2} = 2/N ( N − 1 ) / ( 2 N ) = 2/ N G S ( t ) G^S(t) G S ( t ) a a a { a , x } \{a,x\} { a , x } x x x a a a S ∖ { a } S \setminus\{a\} S ∖ { a }
Lemma 3.14 is a special case of the restriction and marking theorems for Poisson processes; see [44 Theorems 5.2, 5.6, and 5.8, Corollary 5.9] . We use it only at fixed times, and we have included the short computation for this case.
Tail bounds
We use the convention 0 0 = 1 0^0 = 1 0 0 = 1 z ≥ 0 z \ge0 z ≥ 0 k ≥ 0 k \ge0 k ≥ 0 π z ( k ) = e − z z k / k ! \pi_z(k) = e^{-z}z^k/k! π z ( k ) = e − z z k / k ! π z ( − 1 ) = 0 \pi_z(-1) = 0 π z ( − 1 ) = 0 k ≥ 0 k \ge0 k ≥ 0 π z ( k ) \pi_z(k) π z ( k ) Pois ( z ) \operatorname{Pois}(z) Pois ( z ) k k k Pois ( z ) \operatorname{Pois}(z) Pois ( z ) z z z Bin ( m , p ) \operatorname{Bin}(m,p) Bin ( m , p )
Lemma 3.15. (a) Let X X X Bin ( m , p ) \operatorname{Bin}(m,p) Bin ( m , p ) m ≥ 1 m \ge1 m ≥ 1 p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] k ≥ 1 k \ge1 k ≥ 1
P ( X ≥ k ) ≤ ( m k ) p k ≤ ( e m p k ) k . \mathbb{P}(X \ge k) \le\binom{m}{k}p^k \le\left(\frac{emp}{k}\right)^k. P ( X ≥ k ) ≤ ( k m ) p k ≤ ( k e m p ) k . (b) Let X X X Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) μ ≥ 0 \mu\ge0 μ ≥ 0 k > μ k > \mu k > μ P ( X ≥ k ) ≤ ( e μ k ) k \mathbb{P}(X \ge k) \le\left(\frac{e\mu}{k}\right)^k P ( X ≥ k ) ≤ ( k e μ ) k
(c) Let X X X μ \mu μ μ > 0 \mu> 0 μ > 0 t > 0 t > 0 t > 0
P ( X ≥ μ + t ) ≤ exp ( − t 2 2 ( μ + t ) ) , P ( X ≤ μ − t ) ≤ exp ( − t 2 2 μ ) . \mathbb{P}(X \ge\mu+ t) \le\exp\left(-\frac{t^2}{2(\mu+t)}\right), \qquad\mathbb{P}(X \le\mu- t) \le\exp\left(-\frac{t^2}{2\mu}\right). P ( X ≥ μ + t ) ≤ exp ( − 2 ( μ + t ) t 2 ) , P ( X ≤ μ − t ) ≤ exp ( − 2 μ t 2 ) . (d) For all real z ≥ 0 z \ge0 z ≥ 0 i ≥ 1 i \ge1 i ≥ 1
P ( Pois ( z ) ≥ i ) ≤ ( e z − 1 ) π z ( i − 1 ) . \mathbb{P}(\operatorname{Pois}(z) \ge i) \le(e^z-1)\pi_z(i-1). P ( Pois ( z ) ≥ i ) ≤ ( e z − 1 ) π z ( i − 1 ) . Proof. We begin with (a), where we may assume k ≤ m k \le m k ≤ m X ≥ k X \ge k X ≥ k k k k m m m ( m k ) ≤ m k / k ! \binom{m}{k} \le m^k/k! ( k m ) ≤ m k / k ! k ! ≥ ( k / e ) k k! \ge(k/e)^k k ! ≥ ( k / e ) k
To prove (b) and (c), we use the Cramér–Chernoff method [13 Section 2.2] . If μ = 0 \mu= 0 μ = 0 X = 0 X = 0 X = 0 μ > 0 \mu> 0 μ > 0 E e φ X = exp ( μ ( e φ − 1 ) ) \mathbb{E}e^{\varphi X} = \exp(\mu(e^\varphi-1)) E e φX = exp ( μ ( e φ − 1 )) φ \varphi φ φ ≥ 0 \varphi\ge0 φ ≥ 0 x x x
P ( X ≥ x ) ≤ exp ( − φ x + μ ( e φ − 1 ) ) . (52) \mathbb{P}(X \ge x) \le\exp\left(-\varphi x+\mu(e^\varphi-1)\right). \tag*{(52)} P ( X ≥ x ) ≤ exp ( − φ x + μ ( e φ − 1 ) ) . (52) For (b), choosing e φ = k / μ e^\varphi= k/\mu e φ = k / μ
P ( X ≥ k ) ≤ e − μ ( e μ k ) k ≤ ( e μ k ) k . \mathbb{P}(X \ge k) \le e^{-\mu}\left(\frac{e\mu}{k}\right)^k \le\left(\frac{e\mu}{k}\right)^k. P ( X ≥ k ) ≤ e − μ ( k e μ ) k ≤ ( k e μ ) k . For the first bound in (c), we set x = t / μ x = t/\mu x = t / μ
ψ ( v ) = ( 1 + v ) log ( 1 + v ) − v , v > − 1. \psi(v)=(1+v)\log(1+v)-v,\qquad v>-1. ψ ( v ) = ( 1 + v ) log ( 1 + v ) − v , v > − 1. Choosing e φ = 1 + x e^\varphi= 1+x e φ = 1 + x
P ( X ≥ μ + t ) ≤ exp ( − φ ( μ + t ) + μ ( e φ − 1 ) ) = exp ( − μ ψ ( x ) ) . \mathbb{P}(X \ge\mu+t) \le\exp\left(-\varphi(\mu+t)+\mu(e^\varphi-1)\right)=\exp\left(-\mu\psi(x)\right). P ( X ≥ μ + t ) ≤ exp ( − φ ( μ + t ) + μ ( e φ − 1 ) ) = exp ( − μ ψ ( x ) ) . Since ψ ( 0 ) = 0 \psi(0)=0 ψ ( 0 ) = 0 ψ ′ ( 0 ) = 0 \psi'(0)=0 ψ ′ ( 0 ) = 0 ψ ′ ′ ( v ) = 1 / ( 1 + v ) \psi''(v)=1/(1+v) ψ ′′ ( v ) = 1/ ( 1 + v )
ψ ( x ) = ∫ 0 x x − v 1 + v d v ≥ 1 1 + x ∫ 0 x ( x − v ) d v = x 2 2 ( 1 + x ) , \psi(x)=\int_0^x \frac{x-v}{1+v}\,\mathrm{d}v \ge\frac{1}{1+x}\int_0^x (x-v)\,\mathrm{d}v=\frac{x^2}{2(1+x)}, ψ ( x ) = ∫ 0 x 1 + v x − v d v ≥ 1 + x 1 ∫ 0 x ( x − v ) d v = 2 ( 1 + x ) x 2 , and therefore μ ψ ( x ) ≥ t 2 / ( 2 ( μ + t ) ) \mu\psi(x)\ge t^2/(2(\mu+t)) μ ψ ( x ) ≥ t 2 / ( 2 ( μ + t ))
For the second bound in (c), observe that the event is empty if t > μ t>\mu t > μ e − μ ≤ e − μ / 2 e^{-\mu}\le e^{-\mu/2} e − μ ≤ e − μ /2 t = μ t=\mu t = μ 0 < t < μ 0<t<\mu 0 < t < μ x = t / μ ∈ ( 0 , 1 ) x=t/\mu\in(0,1) x = t / μ ∈ ( 0 , 1 ) φ ≤ 0 \varphi\le0 φ ≤ 0 e φ X e^{\varphi X} e φX
P ( X ≤ μ − t ) ≤ exp ( − φ ( μ − t ) + μ ( e φ − 1 ) ) , \mathbb{P}(X\le\mu-t)\le\exp\left(-\varphi(\mu-t)+\mu(e^\varphi-1)\right), P ( X ≤ μ − t ) ≤ exp ( − φ ( μ − t ) + μ ( e φ − 1 ) ) , and the choice e φ = 1 − x e^{\varphi}=1-x e φ = 1 − x P ( X ≤ μ − t ) ≤ exp ( − μ ψ ( − x ) ) \mathbb{P}(X\le\mu-t)\le\exp(-\mu\psi(-x)) P ( X ≤ μ − t ) ≤ exp ( − μ ψ ( − x ))
ψ ( − x ) = ∫ 0 x x − v 1 − v d v ≥ ∫ 0 x ( x − v ) d v = x 2 2 . \psi(-x)=\int_0^x\frac{x-v}{1-v}\,\mathrm{d}v\ge\int_0^x(x-v)\,\mathrm{d}v=\frac{x^2}{2}. ψ ( − x ) = ∫ 0 x 1 − v x − v d v ≥ ∫ 0 x ( x − v ) d v = 2 x 2 . so that μ ψ ( − x ) ≥ t 2 / ( 2 μ ) \mu\psi(-x)\ge t^2/(2\mu) μ ψ ( − x ) ≥ t 2 / ( 2 μ )
It remains to prove (d). Both sides vanish if z = 0 z=0 z = 0 z > 0 z>0 z > 0
P ( Pois ( z ) ≥ i ) π z ( i − 1 ) = ∑ j ≥ 1 z j ( i − 1 ) ! ( i − 1 + j ) ! ≤ ∑ j ≥ 1 z j j ! = e z − 1 , \frac{\mathbb{P}(\operatorname{Pois}(z)\ge i)}{\pi_z(i-1)}
=\sum_{j\ge1}\frac{z^j(i-1)!}{(i-1+j)!}
\le\sum_{j\ge1}\frac{z^j}{j!}=e^z-1, π z ( i − 1 ) P ( Pois ( z ) ≥ i ) = j ≥ 1 ∑ ( i − 1 + j )! z j ( i − 1 )! ≤ j ≥ 1 ∑ j ! z j = e z − 1 , where we used that
( i − 1 + j ) ! ( i − 1 ) ! = ∏ l = 1 j ( i − 1 + l ) ≥ j ! . \frac{(i-1+j)!}{(i-1)!}=\prod_{l=1}^{j}(i-1+l)\ge j!. ( i − 1 )! ( i − 1 + j )! = l = 1 ∏ j ( i − 1 + l ) ≥ j ! . Bounds on the quantiles
We now bound the quantiles L n ( η ) L_n(\eta) L n ( η ) U n ( η ) U_n(\eta) U n ( η ) c 0 ∈ ( 0 , 1 / 2 ) c_0\in(0,1/2) c 0 ∈ ( 0 , 1/2 ) G ( n , ⌊ c 0 n ⌋ ) G(n,\lfloor c_0n\rfloor) G ( n , ⌊ c 0 n ⌋) [30 Theorem 5e] . A quantitative form, proved by counting witness subgraphs, appears in [40 Theorem 5.5 and its proof, pp. 112–113] . We use the same witness types below, with a count adapted to our process, which samples pairs with replacement.
We will need the following lemma on cycles in multigraphs. For an integer j ≥ 0 j\ge0 j ≥ 0 path of length j j j to consist of distinct vertices v 0 , … , v j v_0,\ldots,v_j v 0 , … , v j p 1 , … , p j p_1,\ldots,p_j p 1 , … , p j p i = { v i − 1 , v i } p_i=\{v_{i-1},v_i\} p i = { v i − 1 , v i } j ≥ 2 j\ge2 j ≥ 2 cycle of length j j j to consist of distinct vertices v 1 , … , v j v_1,\ldots,v_j v 1 , … , v j p 1 , … , p j p_1,\ldots,p_j p 1 , … , p j p i = { v i , v i + 1 } p_i=\{v_i,v_{i+1}\} p i = { v i , v i + 1 } i < j i<j i < j p j = { v j , v 1 } p_j=\{v_j,v_1\} p j = { v j , v 1 } theta if it consists of three paths of lengths at least 1 between the same two distinct end vertices, with no other vertices or occurrences in common. We call those two end vertices its branch vertices . We call a multigraph a dumbbell if it consists of two cycles without common occurrences, which either have exactly one common vertex, or have no common vertex and are joined by a path of length at least 1 whose only vertices on the two cycles are its end vertices, one on each cycle. A theta or a dumbbell with k k k k + 1 k+1 k + 1 F F F v ( F ) v(F) v ( F ) F F F ∣ F ∣ |F| ∣ F ∣ F F F
Lemma 3.16. Let F F F connected multigraph with at least two vertices .
(a) If ∣ F ∣ ≤ v ( F ) |F|\le v(F) ∣ F ∣ ≤ v ( F ) F F F
(b) If ∣ F ∣ ≥ v ( F ) + 1 |F| \ge v(F) + 1 ∣ F ∣ ≥ v ( F ) + 1 F F F
Proof. Let T \mathcal{T} T F F F v ( F ) − 1 v(F) - 1 v ( F ) − 1 F F F p = { x , y } p = \{x,y\} p = { x , y } F F F T \mathcal{T} T p p p T \mathcal{T} T x x x y y y p p p T \mathcal{T} T
For (a), we use the assumption q ≥ 3 q \ge3 q ≥ 3 ∣ F ∣ = v ( F ) − 1 |F| = v(F) - 1 ∣ F ∣ = v ( F ) − 1 F = T F = \mathcal{T} F = T F F F p p p T \mathcal{T} T C C C p p p C C C
We now extend this coloring to F F F T \mathcal{T} T p p p F F F T \mathcal{T} T F F F C C C C C C T \mathcal{T} T C C C y y y C C C F F F T \mathcal{T} T y y y C C C T \mathcal{T} T C C C y y y F F F
We now prove (b). Let p p p p ′ p' p ′ F F F T \mathcal{T} T C C C C ′ C' C ′ C C C T \mathcal{T} T p p p p ′ p' p ′ C C C
Suppose first that C C C C ′ C' C ′ P P P F F F C C C C ′ C' C ′ P P P C C C C ′ C' C ′ C C C C ′ C' C ′ P P P
Suppose now that C C C C ′ C' C ′ C ′ C' C ′ v 1 , … , v j v_1,\ldots,v_j v 1 , … , v j p ′ = { v j , v 1 } p' = \{v_j,v_1\} p ′ = { v j , v 1 } i 1 i_1 i 1 i 2 i_2 i 2 i i i v i v_i v i C C C i 1 < i 2 i_1 < i_2 i 1 < i 2 v i 2 , v i 2 + 1 , … , v j , v 1 , … , v i 1 v_{i_2},v_{i_2+1},\ldots,v_j,v_1,\ldots,v_{i_1} v i 2 , v i 2 + 1 , … , v j , v 1 , … , v i 1 C ′ C' C ′ p ′ p' p ′ v i 2 v_{i_2} v i 2 v i 1 v_{i_1} v i 1 C C C C C C C C C
Indeed, p ′ p' p ′ C C C C C C v i 1 v_{i_1} v i 1 v i 2 v_{i_2} v i 2
Lemma 3.17. Let η ∈ ( 0 , 1 / 2 ] \eta\in(0,1/2] η ∈ ( 0 , 1/2 ]
(a) Let c 1 > γ q c_1 > \gamma_q c 1 > γ q n ≥ 2 q n \ge2q n ≥ 2 q
C c 1 = q q − 1 e 2 c 1 . C_{c_1} = \frac{q}{q-1}e^{2c_1}. C c 1 = q − 1 q e 2 c 1 . Then
P ( H > ⌊ c 1 n ⌋ ) ≤ C c 1 exp ( − n ( c 1 − γ q ) log q q − 1 ) . (53) \mathbb{P}(H > \lfloor c_1n \rfloor) \le C_{c_1}\exp\left(-n(c_1-\gamma_q)\log\frac{q}{q-1}\right). \tag*{(53)} P ( H > ⌊ c 1 n ⌋) ≤ C c 1 exp ( − n ( c 1 − γ q ) log q − 1 q ) . (53) For every c 1 > γ q c_1 > \gamma_q c 1 > γ q L n ( η ) ≤ U n ( η ) ≤ c 1 n L_n(\eta) \le U_n(\eta) \le c_1n L n ( η ) ≤ U n ( η ) ≤ c 1 n n n n U n ( η ) < M U_n(\eta) < M U n ( η ) < M n n n
(b) For every c 0 ∈ ( 0 , 1 / 2 ) c_0 \in(0,1/2) c 0 ∈ ( 0 , 1/2 ) C ( c 0 ) C(c_0) C ( c 0 ) P ( H ≤ ⌊ c 0 n ⌋ ) ≤ C ( c 0 ) / n \mathbb{P}(H \le\lfloor c_0n\rfloor) \le C(c_0)/n P ( H ≤ ⌊ c 0 n ⌋) ≤ C ( c 0 ) / n n ≥ 2 q n \ge2q n ≥ 2 q L n ( η ) > c 0 n L_n(\eta) > c_0n L n ( η ) > c 0 n n n n
Proof. We begin with (a). Let σ : [ n ] → [ q ] \sigma:[n]\to[q] σ : [ n ] → [ q ] Lemma 3.10 (a), the fraction of pairs of [ n ] [n] [ n ] σ \sigma σ ( n / q − 1 ) / ( n − 1 ) (n/q-1)/(n-1) ( n / q − 1 ) / ( n − 1 ) σ \sigma σ e 1 e_1 e 1
1 − n / q − 1 n − 1 = q − 1 q ⋅ n n − 1 . 1-\frac{n/q-1}{n-1}=\frac{q-1}{q}\cdot\frac{n}{n-1}. 1 − n − 1 n / q − 1 = q q − 1 ⋅ n − 1 n . Since the pairs e i e_i e i q n q^n q n σ \sigma σ m ≥ 0 m\ge0 m ≥ 0
P ( H > m ) = P ( G m is colorable ) ≤ q n ( q − 1 q ) m ( n n − 1 ) m . (54) \mathbb{P}(H>m)=\mathbb{P}(G_m\text{ is colorable})\le q^n\left(\frac{q-1}{q}\right)^m\left(\frac{n}{n-1}\right)^m. \tag*{(54)} P ( H > m ) = P ( G m is colorable ) ≤ q n ( q q − 1 ) m ( n − 1 n ) m . (54) We set a = log ( q / ( q − 1 ) ) a=\log(q/(q-1)) a = log ( q / ( q − 1 )) m = ⌊ c 1 n ⌋ m=\lfloor c_1n\rfloor m = ⌊ c 1 n ⌋ log q = γ q a \log q=\gamma_qa log q = γ q a c 1 n − 1 < m ≤ c 1 n c_1n-1<m\le c_1n c 1 n − 1 < m ≤ c 1 n (54) , we find
log P ( H > m ) ≤ n γ q a − m a + m n − 1 ≤ n γ q a − ( c 1 n − 1 ) a + 2 c 1 = − n ( c 1 − γ q ) a + a + 2 c 1 . \begin{aligned}
\log\mathbb{P}(H>m)&\le n\gamma_qa-ma+\frac{m}{n-1}\\
&\le n\gamma_qa-(c_1n-1)a+2c_1\\
&=-n(c_1-\gamma_q)a+a+2c_1.
\end{aligned} log P ( H > m ) ≤ n γ q a − ma + n − 1 m ≤ n γ q a − ( c 1 n − 1 ) a + 2 c 1 = − n ( c 1 − γ q ) a + a + 2 c 1 . where we used log ( n / ( n − 1 ) ) ≤ 1 / ( n − 1 ) \log(n/(n-1))\le1/(n-1) log ( n / ( n − 1 )) ≤ 1/ ( n − 1 ) m / ( n − 1 ) ≤ 2 c 1 m/(n-1)\le2c_1 m / ( n − 1 ) ≤ 2 c 1 n ≥ 2 n\ge2 n ≥ 2 e a = q / ( q − 1 ) e^a=q/(q-1) e a = q / ( q − 1 ) (53) .
To deduce the bounds on the quantiles, fix c 1 > γ q c_1>\gamma_q c 1 > γ q m m m ⌊ c 1 n ⌋ < m ≤ M \lfloor c_1n\rfloor<m\le M ⌊ c 1 n ⌋ < m ≤ M (37) that
P ( V ≥ m ) = P ( H > m ) ≤ P ( H > ⌊ c 1 n ⌋ ) , \mathbb{P}(V\ge m)=\mathbb{P}(H>m)\le\mathbb{P}(H>\lfloor c_1n\rfloor), P ( V ≥ m ) = P ( H > m ) ≤ P ( H > ⌊ c 1 n ⌋) , and by (53) the right side is smaller than η \eta η n n n n n n m > ⌊ c 1 n ⌋ m>\lfloor c_1n\rfloor m > ⌊ c 1 n ⌋ U n ( η ) U_n(\eta) U n ( η ) U n ( η ) ≤ c 1 n U_n(\eta)\le c_1n U n ( η ) ≤ c 1 n L n ( η ) ≤ U n ( η ) L_n(\eta)\le U_n(\eta) L n ( η ) ≤ U n ( η ) Lemma 3.4 . To prove that U n ( η ) < M U_n(\eta)<M U n ( η ) < M c 1 ′ = ( γ q + Λ ) / 2 c'_1=(\gamma_q+\Lambda)/2 c 1 ′ = ( γ q + Λ ) /2 c 1 ′ > γ q c'_1>\gamma_q c 1 ′ > γ q c 1 ′ n < ⌊ Λ n ⌋ = M c'_1n<\lfloor\Lambda n\rfloor=M c 1 ′ n < ⌊ Λ n ⌋ = M n n n c 1 c_1 c 1
We now prove (b). Suppose that G m G_m G m F F F G m G_m G m Lemma 3.16 (a) implies that ∣ F ∣ ≥ v ( F ) + 1 |F|\ge v(F)+1 ∣ F ∣ ≥ v ( F ) + 1 Lemma 3.16 (b), it follows that G m G_m G m
Let k ≥ 2 k\ge2 k ≥ 2 k k k j 1 , j 2 , j 3 ≥ 1 j_1,j_2,j_3\ge1 j 1 , j 2 , j 3 ≥ 1 j 1 + j 2 + j 3 = k + 1 j_1+j_2+j_3=k+1 j 1 + j 2 + j 3 = k + 1 k k k j 1 , j 2 ≥ 2 j_1,j_2\ge2 j 1 , j 2 ≥ 2 j 3 ≥ 0 j_3\ge0 j 3 ≥ 0 j 1 + j 2 + j 3 = k + 1 j_1+j_2+j_3=k+1 j 1 + j 2 + j 3 = k + 1 j 3 = 0 j_3=0 j 3 = 0 j 2 − 1 j_2-1 j 2 − 1 j 3 ≥ 1 j_3\ge1 j 3 ≥ 1 j 2 j_2 j 2 j 3 − 1 j_3-1 j 3 − 1 j 1 + j 2 − 1 = k j_1+j_2-1=k j 1 + j 2 − 1 = k j 1 + j 2 + j 3 − 1 = k j_1+j_2+j_3-1=k j 1 + j 2 + j 3 − 1 = k
In both cases, the list consists of k k k [ n ] [n] [ n ] { 0 , … , k + 1 } \{0,\ldots,k+1\} { 0 , … , k + 1 } ( k + 2 ) 2 (k+2)^2 ( k + 2 ) 2 n k n^k n k [ n ] [n] [ n ] k k k 2 ( k + 2 ) 2 n k 2(k+2)^2n^k 2 ( k + 2 ) 2 n k
Let F F F p 1 , … , p k + 1 p_1,\ldots,p_{k+1} p 1 , … , p k + 1 F ⊂ G m F\subset G_m F ⊂ G m i 1 , … , i k + 1 ≤ m i_1,\ldots,i_{k+1}\le m i 1 , … , i k + 1 ≤ m e i j = p j e_{i_j}=p_j e i j = p j j j j m k + 1 m^{k+1} m k + 1 ( n 2 ) − ( k + 1 ) \binom{n}{2}^{-(k+1)} ( 2 n ) − ( k + 1 )
P ( F ⊂ G m ) ≤ ( m ( n 2 ) − 1 ) k + 1 . \mathbb{P}(F\subset G_m)\le\left(m\binom{n}{2}^{-1}\right)^{k+1}. P ( F ⊂ G m ) ≤ ( m ( 2 n ) − 1 ) k + 1 . Now let m ≤ c 0 n m\le c_0n m ≤ c 0 n m ( n 2 ) − 1 ≤ 2 c 0 / ( n − 1 ) m\binom{n}{2}^{-1}\le2c_0/(n-1) m ( 2 n ) − 1 ≤ 2 c 0 / ( n − 1 ) k k k F F F
P ( G m is not colorable ) ≤ ∑ k = 2 n 2 ( k + 2 ) 2 n k ( 2 c 0 n − 1 ) k + 1 ≤ 2 e 2 n − 1 ∑ k ≥ 2 ( k + 2 ) 2 ( 2 c 0 ) k + 1 , \mathbb{P}(G_m\text{ is not colorable})\le\sum_{k=2}^{n}2(k+2)^2n^k\left(\frac{2c_0}{n-1}\right)^{k+1}\le\frac{2e^2}{n-1}\sum_{k\ge2}(k+2)^2(2c_0)^{k+1}, P ( G m is not colorable ) ≤ k = 2 ∑ n 2 ( k + 2 ) 2 n k ( n − 1 2 c 0 ) k + 1 ≤ n − 1 2 e 2 k ≥ 2 ∑ ( k + 2 ) 2 ( 2 c 0 ) k + 1 , where we used that for k ≤ n k\le n k ≤ n
( n n − 1 ) k ≤ ( 1 + 1 n − 1 ) n ≤ e n / ( n − 1 ) ≤ e 2 . \left(\frac{n}{n-1}\right)^k\le\left(1+\frac{1}{n-1}\right)^n\le e^{n/(n-1)}\le e^2. ( n − 1 n ) k ≤ ( 1 + n − 1 1 ) n ≤ e n / ( n − 1 ) ≤ e 2 . The last series converges since 2 c 0 < 1 2c_0<1 2 c 0 < 1 m = ⌊ c 0 n ⌋ m=\lfloor c_0n\rfloor m = ⌊ c 0 n ⌋ 1 / ( n − 1 ) ≤ 2 / n 1/(n-1)\le2/n 1/ ( n − 1 ) ≤ 2/ n
C ( c 0 ) = 4 e 2 ∑ k ≥ 2 ( k + 2 ) 2 ( 2 c 0 ) k + 1 . C(c_0)=4e^2\sum_{k\ge2}(k+2)^2(2c_0)^{k+1}. C ( c 0 ) = 4 e 2 k ≥ 2 ∑ ( k + 2 ) 2 ( 2 c 0 ) k + 1 . Finally, we fix c 0 ′ ∈ ( c 0 , 1 / 2 ) c_0'\in(c_0,1/2) c 0 ′ ∈ ( c 0 , 1/2 ) n n n c 0 n + 1 ≤ c 0 ′ n < M c_0n+1\le c_0'n<M c 0 n + 1 ≤ c 0 ′ n < M n n n m m m 0 ≤ m ≤ c 0 n 0\le m\le c_0n 0 ≤ m ≤ c 0 n m < M m<M m < M { V ≤ m } = { H ≤ m + 1 } \{V\le m\}=\{H\le m+1\} { V ≤ m } = { H ≤ m + 1 }
P ( V ≤ m ) = P ( H ≤ m + 1 ) ≤ P ( H ≤ ⌊ c 0 ′ n ⌋ ) ≤ C ( c 0 ′ ) n . \mathbb{P}(V\le m)=\mathbb{P}(H\le m+1)\le\mathbb{P}(H\le\lfloor c_0'n\rfloor)\le\frac{C(c_0')}{n}. P ( V ≤ m ) = P ( H ≤ m + 1 ) ≤ P ( H ≤ ⌊ c 0 ′ n ⌋) ≤ n C ( c 0 ′ ) . The right side is smaller than η \eta η n n n n n n m ≤ c 0 n m\le c_0n m ≤ c 0 n L n ( η ) L_n(\eta) L n ( η ) L n ( η ) > c 0 n L_n(\eta)>c_0n L n ( η ) > c 0 n
The assumption q ≥ 3 q\ge3 q ≥ 3 q = 2 q=2 q = 2
Reduction to two estimates
In this section, we reduce the existence of the threshold to two estimates on the final edge e H e_H e H § 5.1 , we state our convergence criterion, Theorem 4.1. It asserts that α q \alpha_q α q η \eta η w n ( η ) = O ( n δ n ) w_n(\eta)=O(n\delta_n) w n ( η ) = O ( n δ n ) δ n \delta_n δ n n = ⌊ e k ⌋ n=\lfloor e^k\rfloor n = ⌊ e k ⌋ § 5.2 , we bound the variance of a capped version of V V V Γ \Gamma Γ w n ( η ) = O ( n ε n ) w_n(\eta)=O(n\varepsilon_n) w n ( η ) = O ( n ε n ) n ε n n\varepsilon_n n ε n O ( n ε n 2 ) O(n\varepsilon_n^2) O ( n ε n 2 ) § 5.3 , we split the last expectation according to the degrees in Γ \Gamma Γ e H e_H e H § 3 .
From § 4 , we use the quantiles of Definition 3.3 and Lemma 3.4, the frozen pairs of Definition 3.5 and Lemma 3.6, the identity (43) , and the expected lifetime of Definition 3.9. We now fix the sequences that are used in the rest of the paper. We set ε n = ( log n ) − 3 / 2 \varepsilon_n=(\log n)^{-3/2} ε n = ( log n ) − 3/2 n ≥ 3 n\ge3 n ≥ 3 ε 1 = ε 2 = 1 \varepsilon_1=\varepsilon_2=1 ε 1 = ε 2 = 1
x n = n ε n 2 , D n = ⌊ ( log n ) 1 / 4 ⌋ . (55) x_n=n\varepsilon_n^2,\qquad D_n=\left\lfloor(\log n)^{1/4}\right\rfloor. \tag*{(55)} x n = n ε n 2 , D n = ⌊ ( log n ) 1/4 ⌋ . (55) We call x n x_n x n lifetime level and D n D_n D n degree cut .
The convergence criterion
Recall the level η 0 = e − 2 Λ / 8 \eta_0=e^{-2\Lambda}/8 η 0 = e − 2Λ /8 w n ( η ) = U n ( η ) − L n ( η ) w_n(\eta)=U_n(\eta)-L_n(\eta) w n ( η ) = U n ( η ) − L n ( η ) [10 Theorem 5 and Proposition 2] and adapts the comparisons between system sizes and the balanced-tree argument of the k k k [51 Section 5] to quantiles and geometrically summable errors. The earlier additive form of near-superadditivity appears in [1, arXiv:1006.3786v1, Section 4.3, Lemma 8 and Remark 3.
Theorem 4.1. Let ( δ n ) n ≥ 1 (\delta_n)_{n\ge1} ( δ n ) n ≥ 1 ( 0 , 1 ] (0,1] ( 0 , 1 ] ∑ k ≥ 1 δ ⌊ e k ⌋ < ∞ \sum_{k\ge1}\delta_{\lfloor e^k\rfloor}<\infty ∑ k ≥ 1 δ ⌊ e k ⌋ < ∞ η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] K ′ ( η ) > 0 K'(\eta)>0 K ′ ( η ) > 0 n 1 ( η ) n_1(\eta) n 1 ( η )
w n ( η ) ≤ K ′ ( η ) n δ n for all n ≥ n 1 ( η ) . (56) w_n(\eta)\le K'(\eta)n\delta_n\qquad\text{for all }n\ge n_1(\eta). \tag*{(56)} w n ( η ) ≤ K ′ ( η ) n δ n for all n ≥ n 1 ( η ) . (56) Then there exists α q ∈ [ 1 / 2 , γ q ] \alpha_q\in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ]
(1) For every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L n ( η ) / n → α q L_n(\eta)/n \to\alpha_q L n ( η ) / n → α q U n ( η ) / n → α q U_n(\eta)/n \to\alpha_q U n ( η ) / n → α q n → ∞ n \to\infty n → ∞
(2) For every c ∈ [ 0 , α q ) c \in[0,\alpha_q) c ∈ [ 0 , α q ) P ( H > ⌊ c n ⌋ ) → 1 \mathbb{P}(H > \lfloor cn\rfloor) \to1 P ( H > ⌊ c n ⌋) → 1 P ( G ( n , ⌊ c n ⌋ ) is colorable ) → 1 \mathbb{P}(G(n,\lfloor cn\rfloor)\text{ is colorable}) \to1 P ( G ( n , ⌊ c n ⌋) is colorable ) → 1 n → ∞ n \to\infty n → ∞ c > α q c > \alpha_q c > α q
(3) For every d ∈ [ 0 , 2 α q ) d \in[0,2\alpha_q) d ∈ [ 0 , 2 α q ) P ( G ( n , d / n ) is colorable ) → 1 \mathbb{P}(G(n,d/n)\text{ is colorable}) \to1 P ( G ( n , d / n ) is colorable ) → 1 n → ∞ n \to\infty n → ∞ d > 2 α q d > 2\alpha_q d > 2 α q
Note that the statements on P ( H > ⌊ c n ⌋ ) \mathbb{P}(H > \lfloor cn\rfloor) P ( H > ⌊ c n ⌋) α q \alpha_q α q α q \alpha_q α q q q q Λ \Lambda Λ δ n \delta_n δ n Theorem 4.1 with δ n = ε n \delta_n=\varepsilon_n δ n = ε n (56) for this sequence.
Remark 4.2. The window bound at every level η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] [2 Theorem 1.1] unnecessary for the proof of Theorem 4.1 . In Appendix § 9 , we first prove convergence of the normalized quantiles at each level, use their nesting to identify a common limit, and then let η \eta η ( δ ⌊ e k ⌋ ) (\delta_{\lfloor e^k\rfloor}) ( δ ⌊ e k ⌋ )
Deletion and the expected lifetime of the last colorable graph
Throughout Sections 4.2 and 4.3, we fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n ≥ 2 q n \ge2q n ≥ 2 q L L L U U U w w w L n ( η ) L_n(\eta) L n ( η ) U n ( η ) U_n(\eta) U n ( η ) w n ( η ) w_n(\eta) w n ( η ) A A A X X X A A A E [ X ; A ] \mathbb{E}[X;A] E [ X ; A ] X X X A A A E [ X 1 A ] \mathbb{E}[X\mathbf{1}_A] E [ X 1 A ] X X X A A A
Y = min ( U , max ( L , V ) ) . (57) Y=\min(U,\max(L,V)). \tag*{(57)} Y = min ( U , max ( L , V )) . (57) Since U ≤ M U \le M U ≤ M V = min ( H − 1 , M ) V=\min(H-1,M) V = min ( H − 1 , M ) Y = min ( U , max ( L , H − 1 ) ) Y=\min(U,\max(L,H-1)) Y = min ( U , max ( L , H − 1 )) H ≤ L + 1 H \le L+1 H ≤ L + 1 L + 1 < H ≤ U L+1 < H \le U L + 1 < H ≤ U H > U H > U H > U
# { m ∈ { L + 1 , … , U } : m < H } = min ( U , max ( L , H − 1 ) ) − L . \#\{m \in\{L+1,\ldots,U\}:m<H\}=\min(U,\max(L,H-1))-L. # { m ∈ { L + 1 , … , U } : m < H } = min ( U , max ( L , H − 1 )) − L . Since G m G_m G m m < H m<H m < H
Y = L + ∑ m = L + 1 U 1 { G m is colorable } . (58) Y=L+\sum_{m=L+1}^{U}\mathbf{1}\{G_m\text{ is colorable}\}. \tag*{(58)} Y = L + m = L + 1 ∑ U 1 { G m is colorable } . (58) For 1 ≤ i ≤ U 1 \le i \le U 1 ≤ i ≤ U m ≥ 0 m \ge0 m ≥ 0 G m ( − i ) G_m^{(-i)} G m ( − i ) { e j : 1 ≤ j ≤ m , j ≠ i } \{e_j:1 \le j \le m,\ j \ne i\} { e j : 1 ≤ j ≤ m , j = i } G m ( − i ) = G m G_m^{(-i)}=G_m G m ( − i ) = G m m < i m<i m < i m ≥ i m \ge i m ≥ i G m ( − i ) G_m^{(-i)} G m ( − i ) G m G_m G m e i e_i e i (58) , we define
Y ( − i ) = L + ∑ m = L + 1 U 1 { G m ( − i ) is colorable } . (59) Y^{(-i)}=L+\sum_{m=L+1}^{U}\mathbf{1}\{G_m^{(-i)}\text{ is colorable}\}. \tag*{(59)} Y ( − i ) = L + m = L + 1 ∑ U 1 { G m ( − i ) is colorable } . (59) The variable Y ( − i ) Y^{(-i)} Y ( − i ) e j e_j e j j ≤ U j \le U j ≤ U j ≠ i j \ne i j = i Y ( − i ) − Y Y^{(-i)}-Y Y ( − i ) − Y
Bounds of this kind have antecedents. In [7 Theorem 1(b)] , Aldous considered the first time T T T T T T E T \mathbb{E}T E T T − 1 T-1 T − 1 H H H { u , v } \{u,v\} { u , v } σ : [ n ] → [ q ] \sigma:[n]\to[q] σ : [ n ] → [ q ] σ ( u ) = σ ( v ) \sigma(u)=\sigma(v) σ ( u ) = σ ( v ) n > q n>q n > q H H H e 1 , … , e H e_1,\ldots,e_H e 1 , … , e H e 1 , … , e H − 1 e_1,\ldots,e_{H-1} e 1 , … , e H − 1 Col ( Γ ) \operatorname{Col}(\Gamma) Col ( Γ ) r ( Γ ) r(\Gamma) r ( Γ ) Var ( H ) \operatorname{Var}(H) Var ( H ) E H E r ( Γ ) \mathbb{E}H\,\mathbb{E}r(\Gamma) E H E r ( Γ )
For random k k k [16 Theorem 4.1 and equation (4.1)] . The limiting-threshold preprint [51 Lemmas 3.1 and 3.2] applies the deletion form of the Efron–Stein inequality to a capped hitting time, retains the original clause indices, and expands the squares of deletion delays over pairs of times. The variance preprint [53 Section 3.3, Lemma 3.4] also combines these delays with expected lifetimes. Lemma 4.3 is the analogous computation for colorability. Instead of the bound for Var ( H ) \operatorname{Var}(H) Var ( H ) (60) , in which only final edges e H e_H e H L < H ≤ U L<H\le U L < H ≤ U
Lemma 4.3. Let η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n ≥ 2 q n\ge2q n ≥ 2 q
∑ i = 1 U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] (60) \sum_{i=1}^{U}\mathbb{E}\left[(Y^{(-i)}-Y)^2\right]=\mathbb{E}\left[H\left(2^{r_{U-H+1}(\Gamma)}-1\right);L<H\le U\right] \tag*{(60)} i = 1 ∑ U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] (60) and
Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ ) ) ; L < H ≤ U ] . (61) \operatorname{Var}(Y)\le2U\mathbb{E}\left[\min(w,r(\Gamma));L<H\le U\right]. \tag*{(61)} Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ )) ; L < H ≤ U ] . (61) Proof. We begin by computing the differences Y ( − i ) − Y Y^{(-i)}-Y Y ( − i ) − Y 1 ≤ i ≤ U 1\le i\le U 1 ≤ i ≤ U m ≥ i m\ge i m ≥ i
X m , i = { G m ( − i ) is colorable and G m is not colorable } . \mathcal{X}_{m,i}=\left\{G_m^{(-i)}\text{ is colorable and }G_m\text{ is not colorable}\right\}. X m , i = { G m ( − i ) is colorable and G m is not colorable } . For m < i m<i m < i G m ( − i ) = G m G_m^{(-i)}=G_m G m ( − i ) = G m m ≥ i m\ge i m ≥ i G m = G m ( − i ) + e i G_m=G_m^{(-i)}+e_i G m = G m ( − i ) + e i G m G_m G m G m ( − i ) G_m^{(-i)} G m ( − i )
1 { G m ( − i ) is colorable } − 1 { G m is colorable } = 1 X m , i . \mathbf{1}\{G_m^{(-i)}\text{ is colorable}\}-\mathbf{1}\{G_m\text{ is colorable}\}=\mathbf{1}_{\mathcal{X}_{m,i}}. 1 { G m ( − i ) is colorable } − 1 { G m is colorable } = 1 X m , i . Subtracting (4.4) from (4.5), we have
Y ( − i ) − Y = ∑ m = max ( i , L + 1 ) U 1 X m , i . (62) Y^{(-i)}-Y=\sum_{m=\max(i,L+1)}^{U}\mathbf{1}_{\mathcal{X}_{m,i}}. \tag*{(62)} Y ( − i ) − Y = m = m a x ( i , L + 1 ) ∑ U 1 X m , i . (62) Next, we square (62) . Let m < l m<l m < l G l G_l G l G m G_m G m X m , i \mathcal{X}_{m,i} X m , i
X m , i ∩ X l , i = X m , i ∩ { G l ( − i ) is colorable } . \mathcal{X}_{m,i}\cap\mathcal{X}_{l,i}=\mathcal{X}_{m,i}\cap\{G_l^{(-i)}\text{ is colorable}\}. X m , i ∩ X l , i = X m , i ∩ { G l ( − i ) is colorable } . Expanding the square gives
( γ ( − i ) − γ ) 2 = ∑ m = max ( i , L + 1 ) U ( 1 X m , i + 2 ∑ l = m + 1 U 1 X m , i ∩ { G l ( − i ) is colorable } ) . (63) (\gamma^{(-i)}-\gamma)^2=\sum_{m=\max(i,L+1)}^U\left(1_{\mathcal{X}_{m,i}}+2\sum_{l=m+1}^U1_{\mathcal{X}_{m,i}\cap\{G_l^{(-i)}\text{ is colorable}\}}\right). \tag*{(63)} ( γ ( − i ) − γ ) 2 = m = m a x ( i , L + 1 ) ∑ U ( 1 X m , i + 2 l = m + 1 ∑ U 1 X m , i ∩ { G l ( − i ) is colorable } ) . (63) The main step is to compute the conditional probabilities of these events given G m ( − i ) G_m^{(-i)} G m ( − i ) m ≥ i m\ge i m ≥ i l > m l>m l > m G m ( − i ) G_m^{(-i)} G m ( − i ) e j e_j e j j ≤ m j\le m j ≤ m j ≠ i j\ne i j = i
G m = G m ( − i ) + e i , G l = G m ( − i ) + e m + 1 + ⋯ + e l . G_m=G_m^{(-i)}+e_i,\qquad G_l=G_m^{(-i)}+e_{m+1}+\cdots+e_l. G m = G m ( − i ) + e i , G l = G m ( − i ) + e m + 1 + ⋯ + e l . The pairs e i , e m + 1 , … , e l e_i,e_{m+1},\ldots,e_l e i , e m + 1 , … , e l G m ( − i ) G_m^{(-i)} G m ( − i )
Let G G G [ n ] [n] [ n ] P ( G m ( − i ) = G ) > 0 \mathbb{P}(G_m^{(-i)}=G)>0 P ( G m ( − i ) = G ) > 0 Lemma 3.6 (a), the multigraph G + e i G+e_i G + e i e i e_i e i G G G κ ( G ) \kappa(G) κ ( G )
The definition of T G T_G T G Definition 3.9 shows that G + e m + 1 + ⋯ + e l G+e_{m+1}+\cdots+e_l G + e m + 1 + ⋯ + e l P ( T G > l − m ) \mathbb{P}(T_G>l-m) P ( T G > l − m ) e i e_i e i e m + 1 , … , e l e_{m+1},\ldots,e_l e m + 1 , … , e l
P ( X m , i ∣ G m ( − i ) = G ) = κ ( G ) , \mathbb{P}\left(\mathcal{X}_{m,i}\mid G_m^{(-i)}=G\right)=\kappa(G), P ( X m , i ∣ G m ( − i ) = G ) = κ ( G ) , P ( X m , i ∩ { G l ( − i ) is colorable } ∣ G m ( − i ) = G ) = κ ( G ) P ( T G > l − m ) . (64) \mathbb{P}\left(\mathcal{X}_{m,i}\cap\{G_l^{(-i)}\text{ is colorable}\}\mid G_m^{(-i)}=G\right)=\kappa(G)\mathbb{P}(T_G>l-m). \tag*{(64)} P ( X m , i ∩ { G l ( − i ) is colorable } ∣ G m ( − i ) = G ) = κ ( G ) P ( T G > l − m ) . (64) The two probabilities in (64) vanish if G G G κ ( G ) \kappa(G) κ ( G )
For m ∈ { L + 1 , … , U } m\in\{L+1,\ldots,U\} m ∈ { L + 1 , … , U } G G G [ n ] [n] [ n ]
h m ′ ( G ) = 1 { G is colorable } ( 2 r U − m + 1 ( G ) − 1 ) . h'_m(G)=1_{\{G\text{ is colorable}\}}\left(2r_{U-m+1}(G)-1\right). h m ′ ( G ) = 1 { G is colorable } ( 2 r U − m + 1 ( G ) − 1 ) . For colorable G G G P ( T G > 0 ) = 1 \mathbb{P}(T_G>0)=1 P ( T G > 0 ) = 1 (47) and Lemma 3.10 (a) give
1 + 2 ∑ l = m + 1 U P ( T G > l − m ) = 2 r U − m + 1 ( G ) − 1 = h m ′ ( G ) ≥ 1. (65) 1+2\sum_{l=m+1}^U\mathbb{P}(T_G>l-m)=2r_{U-m+1}(G)-1=h'_m(G)\ge1. \tag*{(65)} 1 + 2 l = m + 1 ∑ U P ( T G > l − m ) = 2 r U − m + 1 ( G ) − 1 = h m ′ ( G ) ≥ 1. (65) The indicator makes h m ′ h'_m h m ′ Lemma 3.8 requires. Taking expectations in (63) and inserting (64) and (65) , we obtain
E [ ( γ ( − i ) − γ ) 2 ] = ∑ m = max ( i , L + 1 ) U E [ κ ( G m ( − i ) ) h m ′ ( G m ( − i ) ) ] . (66) \mathbb{E}\left[(\gamma^{(-i)}-\gamma)^2\right]=\sum_{m=\max(i,L+1)}^U\mathbb{E}\left[\kappa(G_m^{(-i)})h'_m(G_m^{(-i)})\right]. \tag*{(66)} E [ ( γ ( − i ) − γ ) 2 ] = m = m a x ( i , L + 1 ) ∑ U E [ κ ( G m ( − i ) ) h m ′ ( G m ( − i ) ) ] . (66) We now sum over i i i m ∈ { L + 1 , … , U } m \in\{L+1,\ldots,U\} m ∈ { L + 1 , … , U } (66) with index m m m 1 ≤ i ≤ m 1 \le i \le m 1 ≤ i ≤ m i i i G m ( − i ) G_m^{(-i)} G m ( − i ) m − 1 m-1 m − 1 G m − 1 G_{m-1} G m − 1
∑ i = 1 U E [ ( Y ( − i ) − Y ) 2 ] = ∑ m = L + 1 U m E [ κ ( G m − 1 ) h m ′ ( G m − 1 ) ] . (67) \sum_{i=1}^{U} \mathbb{E}\left[(Y^{(-i)}-Y)^2\right] = \sum_{m=L+1}^{U} m\mathbb{E}\left[\kappa(G_{m-1})h'_m(G_{m-1})\right]. \tag*{(67)} i = 1 ∑ U E [ ( Y ( − i ) − Y ) 2 ] = m = L + 1 ∑ U m E [ κ ( G m − 1 ) h m ′ ( G m − 1 ) ] . (67) To evaluate the right side of (67) , let E \mathcal{E} E ( G , e ) (G,e) ( G , e ) G G G [ n ] [n] [ n ] e e e G G G κ E = κ \kappa_{\mathcal{E}}=\kappa κ E = κ Lemma 3.6 (a), the final edge e H e_H e H Γ \Gamma Γ H < ∞ H<\infty H < ∞ ( Γ , e H ) ∈ E (\Gamma,e_H)\in\mathcal{E} ( Γ , e H ) ∈ E Lemma 3.8 (a) with this E \mathcal{E} E L ′ = m − 1 L'=m-1 L ′ = m − 1 U ′ = m U'=m U ′ = m h m − 1 = h m ′ h_{m-1}=h'_m h m − 1 = h m ′
E [ κ ( G m − 1 ) h m ′ ( G m − 1 ) ] = E [ h m ′ ( Γ ) ; H = m ] . \mathbb{E}\left[\kappa(G_{m-1})h'_m(G_{m-1})\right] = \mathbb{E}\left[h'_m(\Gamma);H=m\right]. E [ κ ( G m − 1 ) h m ′ ( G m − 1 ) ] = E [ h m ′ ( Γ ) ; H = m ] . Inserting this into (67) , and using that Γ \Gamma Γ (60) . This is an identity of expectations. An individual deletion delay need not vanish on a trajectory with H ≤ L H\le L H ≤ L
To prove (61) , we use the Efron–Stein inequality [29 , 61 ] in the following form, stated in [12 (3.6)] and [13 Theorem 3.1] . Let X 1 , … , X k X_1,\ldots,X_k X 1 , … , X k Z Z Z ( X 1 , … , X k ) (X_1,\ldots,X_k) ( X 1 , … , X k ) i i i Z i Z_i Z i σ ( X j : j ≠ i ) \sigma(X_j:j\ne i) σ ( X j : j = i )
Var ( Z ) ≤ ∑ i = 1 k E [ ( Z − Z i ) 2 ] . \operatorname{Var}(Z) \le\sum_{i=1}^{k}\mathbb{E}\left[(Z-Z_i)^2\right]. Var ( Z ) ≤ i = 1 ∑ k E [ ( Z − Z i ) 2 ] . By (58) , the variable Y Y Y e 1 , … , e U e_1,\ldots,e_U e 1 , … , e U Y ( − i ) Y^{(-i)} Y ( − i ) e j e_j e j j ≤ U j\le U j ≤ U j ≠ i j\ne i j = i Z = Y Z=Y Z = Y Z i = Y ( − i ) Z_i=Y^{(-i)} Z i = Y ( − i ) (60) , we find
Var ( Y ) ≤ E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . \operatorname{Var}(Y) \le\mathbb{E}\left[H(2r_{U-H+1}(\Gamma)-1);L<H\le U\right]. Var ( Y ) ≤ E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . We have H ≤ U H\le U H ≤ U 1 ≤ U − H + 1 ≤ w 1\le U-H+1\le w 1 ≤ U − H + 1 ≤ w { L < H ≤ U } \{L<H\le U\} { L < H ≤ U }
r U − H + 1 ( Γ ) ≤ r w ( Γ ) ≤ min ( w , r ( Γ ) ) , r_{U-H+1}(\Gamma) \le r_w(\Gamma) \le\min(w,r(\Gamma)), r U − H + 1 ( Γ ) ≤ r w ( Γ ) ≤ min ( w , r ( Γ )) , where we used that Γ \Gamma Γ Lemma 3.10 (a), r b ( G ) r_b(G) r b ( G ) b b b min ( b , r ( G ) ) \min(b,r(G)) min ( b , r ( G )) G G G b ≥ 1 b\ge1 b ≥ 1 − 1 -1 − 1 (61) . □
We will also need the following elementary lower bound on the variance of Y Y Y
Lemma 4.4. Let η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n ≥ 2 q n\ge2q n ≥ 2 q Var ( Y ) ≥ η 2 w 2 \operatorname{Var}(Y)\ge\eta^2w^2 Var ( Y ) ≥ η 2 w 2
Proof. If w = 0 w=0 w = 0 L < U L<U L < U Y ′ Y' Y ′ Y Y Y ∣ Y − Y ′ ∣ = w |Y-Y'|=w ∣ Y − Y ′ ∣ = w { Y = L , Y ′ = U } \{Y=L,Y'=U\} { Y = L , Y ′ = U } { Y = U , Y ′ = L } \{Y=U,Y'=L\} { Y = U , Y ′ = L }
Var ( Y ) = 1 2 E [ ( Y − Y ′ ) 2 ] ≥ P ( Y = L ) P ( Y = U ) w 2 . \operatorname{Var}(Y)=\frac{1}{2}\mathbb{E}\left[(Y-Y')^2\right]\geq\mathbb{P}(Y=L)\mathbb{P}(Y=U)w^2. Var ( Y ) = 2 1 E [ ( Y − Y ′ ) 2 ] ≥ P ( Y = L ) P ( Y = U ) w 2 . Because L < U L<U L < U (57) gives
{ Y = L } = { V ≤ L } , { Y = U } = { V ≥ U } . \{Y=L\}=\{V\leq L\},\qquad\{Y=U\}=\{V\geq U\}. { Y = L } = { V ≤ L } , { Y = U } = { V ≥ U } . Recall from the definitions of L L L U U U Definition 3.3 that P ( V ≤ L ) ≥ η \mathbb{P}(V\leq L)\geq\eta P ( V ≤ L ) ≥ η P ( V ≥ U ) ≥ η \mathbb{P}(V\geq U)\geq\eta P ( V ≥ U ) ≥ η
Combining the two lemmas, we bound the window by the expected lifetime of Γ \Gamma Γ n ε n n\varepsilon_n n ε n
Proposition 4.5. Let η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n ≥ 2 q n\geq2q n ≥ 2 q K 0 ≥ 0 K_0\geq0 K 0 ≥ 0
E [ min ( n ε n , r ( Γ ) ) ; L < H ≤ U ] ≤ K 0 n ε n 2 . (68) \mathbb{E}\left[\min(n\varepsilon_n,r(\Gamma));L<H\leq U\right]\leq K_0n\varepsilon_n^2. \tag*{(68)} E [ min ( n ε n , r ( Γ )) ; L < H ≤ U ] ≤ K 0 n ε n 2 . (68) Then w ≤ max ( 1 , 2 Λ K 0 / η 2 ) n ε n w\leq\max(1,2\Lambda K_0/\eta^2)n\varepsilon_n w ≤ max ( 1 , 2Λ K 0 / η 2 ) n ε n
Proof. If w < n ε n w<n\varepsilon_n w < n ε n w ≥ n ε n w\geq n\varepsilon_n w ≥ n ε n w > 0 w>0 w > 0 y ≥ 0 y\geq0 y ≥ 0
min ( w , y ) ≤ w n ε n min ( n ε n , y ) . (69) \min(w,y)\leq\frac{w}{n\varepsilon_n}\min(n\varepsilon_n,y). \tag*{(69)} min ( w , y ) ≤ n ε n w min ( n ε n , y ) . (69) Indeed, if y ≤ n ε n y\leq n\varepsilon_n y ≤ n ε n y y y y > n ε n y>n\varepsilon_n y > n ε n w w w Lemma 4.4 , (61) , (69) , and (68) , we find
η 2 w 2 ≤ Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ ) ) ; L < H ≤ U ] \eta^2w^2\leq\operatorname{Var}(Y)\leq2U\mathbb{E}\left[\min(w,r(\Gamma));L<H\leq U\right] η 2 w 2 ≤ Var ( Y ) ≤ 2 U E [ min ( w , r ( Γ )) ; L < H ≤ U ] ≤ 2 U w n ε n E [ min ( n ε n , r ( Γ ) ) ; L < H ≤ U ] ≤ 2 Λ K 0 n ε n w , \leq\frac{2Uw}{n\varepsilon_n}\mathbb{E}\left[\min(n\varepsilon_n,r(\Gamma));L<H\leq U\right]\leq2\Lambda K_0n\varepsilon_nw, ≤ n ε n 2 U w E [ min ( n ε n , r ( Γ )) ; L < H ≤ U ] ≤ 2Λ K 0 n ε n w , where in the last step we also used U ≤ M ≤ Λ n U\leq M\leq\Lambda n U ≤ M ≤ Λ n η 2 w \eta^2w η 2 w w ≤ ( 2 Λ K 0 / η 2 ) n ε n w\leq(2\Lambda K_0/\eta^2)n\varepsilon_n w ≤ ( 2Λ K 0 / η 2 ) n ε n
Splitting by the degrees of the endpoints
We now split the expectation in (68) according to the degrees in Γ \Gamma Γ e H e_H e H D n D_n D n r ( Γ ) > x n r(\Gamma)>x_n r ( Γ ) > x n Lemma 3.11 , to bound the size of the frozen class of that endpoint. When both endpoints have degree larger than D n D_n D n § 7 holds without a restriction on the lifetime.
Proposition 4.6. Let η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n ≥ 2 q n \ge2q n ≥ 2 q
P ( L < H ≤ U , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n (70) \mathbb{P}\left(L < H \le U,\ \min_{v\in e_H}\deg_{\Gamma}(v) \le D_n,\ r(\Gamma) > x_n\right) \le\varepsilon_n \tag*{(70)} P ( L < H ≤ U , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n (70) and
P ( L < H ≤ U , min v ∈ e H deg Γ ( v ) > D n , r ( Γ ) > x n ) ≤ ε n . (71) \mathbb{P}\left(L < H \le U,\ \min_{v\in e_H}\deg_{\Gamma}(v) > D_n,\ r(\Gamma) > x_n\right) \le\varepsilon_n. \tag*{(71)} P ( L < H ≤ U , v ∈ e H min deg Γ ( v ) > D n , r ( Γ ) > x n ) ≤ ε n . (71) Then
E [ min ( n ε n , r ( Γ ) ) ; L < H ≤ U ] ≤ 3 n ε n 2 and w ≤ 6 Λ η 2 n ε n . \mathbb{E}\left[\min(n\varepsilon_n,r(\Gamma));L < H \le U\right] \le3n\varepsilon_n^2
\qquad\text{and}\qquad
w \le\frac{6\Lambda}{\eta^2}n\varepsilon_n. E [ min ( n ε n , r ( Γ )) ; L < H ≤ U ] ≤ 3 n ε n 2 and w ≤ η 2 6Λ n ε n . Proof. For every real y ≥ 0 y \ge0 y ≥ 0
min ( n ε n , y ) ≤ x n + n ε n 1 { y > x n } , \min(n\varepsilon_n,y) \le x_n+n\varepsilon_n\mathbf{1}\{y>x_n\}, min ( n ε n , y ) ≤ x n + n ε n 1 { y > x n } , since the left side is bounded by y ≤ x n y \le x_n y ≤ x n y ≤ x n y \le x_n y ≤ x n n ε n n\varepsilon_n n ε n (70) and (71) holds on the event { L < H ≤ U , r ( Γ ) > x n } \{L < H \le U,\ r(\Gamma) > x_n\} { L < H ≤ U , r ( Γ ) > x n } (70) , (71) , and x n = n ε n 2 x_n=n\varepsilon_n^2 x n = n ε n 2
E [ min ( n ε n , r ( Γ ) ) ; L < H ≤ U ] ≤ x n P ( L < H ≤ U ) + n ε n P ( L < H ≤ U , r ( Γ ) > x n ) ≤ n ε n 2 + 2 n ε n 2 . \begin{aligned}
\mathbb{E}\left[\min(n\varepsilon_n,r(\Gamma));L < H \le U\right]
\le x_n\mathbb{P}(L < H \le U)+n\varepsilon_n\mathbb{P}(L < H \le U,\ r(\Gamma)>x_n) \\
&\le n\varepsilon_n^2+2n\varepsilon_n^2.
\end{aligned} E [ min ( n ε n , r ( Γ )) ; L < H ≤ U ] ≤ x n P ( L < H ≤ U ) + n ε n P ( L < H ≤ U , r ( Γ ) > x n ) ≤ n ε n 2 + 2 n ε n 2 . This is the first claim. The second claim follows from Proposition 4.5 with K 0 = 3 K_0=3 K 0 = 3 6 Λ / η 2 ≥ 1 6\Lambda/\eta^2 \ge1 6Λ/ η 2 ≥ 1
The hypotheses of Proposition 4.6 are verified in the next two sections, by estimates that hold for all large n n n n n n
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n . (72) \mathbb{P}\left(H<\infty,\ \min_{v\in e_H}\deg_{\Gamma}(v)\le D_n,\ r(\Gamma)>x_n\right)\le\varepsilon_n. \tag*{(72)} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n . (72) This bound concerns final edges at all times. It implies (70) for every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] (70) is contained in the event in (72) . In Section 6, we prove Theorem 6.13, which gives, for all large n n n 0 ≤ L ′ ≤ U ′ ≤ M 0\le L'\le U'\le M 0 ≤ L ′ ≤ U ′ ≤ M
P ( L ′ < H ≤ U ′ , min v ∈ e H deg Γ ( v ) > D n ) ≤ ε n . (73) \mathbb{P}\left(L'<H\le U',\ \min_{v\in e_H}\deg_{\Gamma}(v)>D_n\right)\le\varepsilon_n. \tag*{(73)} P ( L ′ < H ≤ U ′ , v ∈ e H min deg Γ ( v ) > D n ) ≤ ε n . (73) Taking L ′ = L L'=L L ′ = L U ′ = U U'=U U ′ = U (73) implies (71) . The two estimates can then be combined with Proposition 4.6 and Theorem 4.1.
Final edges at a vertex of low degree
In this section, we prove Theorem 5.7 . It bounds the probability that the final edge e H e_H e H D n D_n D n Γ \Gamma Γ r ( Γ ) r(\Gamma) r ( Γ ) x n x_n x n Proposition 4.6 ; this is Corollary 5.8. The heuristics are explained in § 3 .
The proof has a structural part and a probabilistic part. For a multigraph G G G a a a a a a neighbor list of a a a G G G Lemma 5.3 (iii), shows that if a a a G G G a a a q ≥ 3 q \ge3 q ≥ 3 q = 2 q = 2 q = 2 q ≥ 3 q \ge3 q ≥ 3 Lemma 3.17 (b), which is used in Appendix § 9 .)
Let B B B x \mathbf{x} x B B B x \mathbf{x} x [58 Section 3, Theorem 1] to split the chromatic polynomial. Further, Poisson splitting at a vertex of degree at most D n D_n D n Lemma 5.5 . In § 6.3 , we combine these results with the bound on frozen classes of Lemma 3.11 to prove Theorem 5.7 .
Frozen pairs and neighbor lists
We first fix notation for lists and identifications. Let S S S § 4.5 that a list of elements of S S S S S S k k k x = ( x ( 1 ) , … , x ( k ) ) \mathbf{x} = (\mathbf{x}(1), \ldots, \mathbf{x}(k)) x = ( x ( 1 ) , … , x ( k )) 0 ≤ i ≤ k 0 \le i \le k 0 ≤ i ≤ k x [ i ] = ( x ( 1 ) , … , x ( i ) ) \mathbf{x}_{[i]} = (\mathbf{x}(1), \ldots, \mathbf{x}(i)) x [ i ] = ( x ( 1 ) , … , x ( i )) i i i x [ 0 ] \mathbf{x}_{[0]} x [ 0 ]
For a map σ : S → [ q ] \sigma:S\to[q] σ : S → [ q ] σ ( x ) = { σ ( x ( i ) ) : 1 ≤ i ≤ k } \sigma(\mathbf{x})=\{\sigma(x(i)):1\le i\le k\} σ ( x ) = { σ ( x ( i )) : 1 ≤ i ≤ k } x \mathbf{x} x σ \sigma σ x \mathbf{x} x ∣ σ ( x ) ∣ ≤ 1 |\sigma(\mathbf{x})|\le1 ∣ σ ( x ) ∣ ≤ 1 σ \sigma σ Z ⊂ S Z\subset S Z ⊂ S Z Z Z
For a vertex o ∉ S o\notin S o ∈ / S star ( o , x ) \operatorname{star}(o,\mathbf{x}) star ( o , x ) S ∪ { o } S\cup\{o\} S ∪ { o } k k k { o , x ( 1 ) } , … , { o , x ( k ) } \{o,x(1)\},\ldots,\{o,x(k)\} { o , x ( 1 )} , … , { o , x ( k )} B B B S S S B + star ( o , x ) B+\operatorname{star}(o,\mathbf{x}) B + star ( o , x ) S ∪ { o } S\cup\{o\} S ∪ { o } B B B star ( o , x ) \operatorname{star}(o,\mathbf{x}) star ( o , x )
Definition 5.1. Let B B B S S S x \mathbf{x} x S S S never monochromatic in B B B B B B x \mathbf{x} x ∣ σ ( x ) ∣ ≥ 2 |\sigma(\mathbf{x})|\ge2 ∣ σ ( x ) ∣ ≥ 2 σ ∈ Col ( B ) \sigma\in\operatorname{Col}(B) σ ∈ Col ( B )
Identifying a set of vertices that contains a pair occurring in a multigraph produces a loop, so we allow loops in the following definition.
Definition 5.2. Let B B B S S S Z ⊂ S Z\subset S Z ⊂ S o ∉ S o\notin S o ∈ / S ι Z : S → ( S ∖ Z ) ∪ { o } \iota_Z:S\to(S\setminus Z)\cup\{o\} ι Z : S → ( S ∖ Z ) ∪ { o } ι Z ( v ) = o \iota_Z(v)=o ι Z ( v ) = o v ∈ Z v\in Z v ∈ Z ι Z ( v ) = v \iota_Z(v)=v ι Z ( v ) = v v ∉ Z v\notin Z v ∈ / Z identification B / Z B/Z B / Z { u , v } \{u,v\} { u , v } B B B { ι Z ( u ) , ι Z ( v ) } \{\iota_Z(u),\iota_Z(v)\} { ι Z ( u ) , ι Z ( v )} T T T multigraph with loops on T T T to be a finite multiset of sets { u , v } \{u,v\} { u , v } u , v ∈ T u,v\in T u , v ∈ T u = v u=v u = v { u , u } \{u,u\} { u , u } σ ′ : T → [ q ] \sigma':T\to[q] σ ′ : T → [ q ] σ ′ ( u ) ≠ σ ′ ( v ) \sigma'(u)\ne\sigma'(v) σ ′ ( u ) = σ ′ ( v ) { u , v } \{u,v\} { u , v } Col \operatorname{Col} Col colorable if that set is nonempty. We view B / Z B/Z B / Z ( S ∖ Z ) ∪ { o } (S\setminus Z)\cup\{o\} ( S ∖ Z ) ∪ { o }
The identification B / Z B/Z B / Z B B B Z Z Z o o o Z Z Z B / Z B/Z B / Z ( S ∖ Z ) ∪ { o } (S\setminus Z)\cup\{o\} ( S ∖ Z ) ∪ { o } Definition 3.1 . (The same convention for loops is used in Appendix § 9 .)
The following lemma collects the facts about frozen pairs and identifications that we need. Part (iv) is the correspondence mentioned at the start of this section. The argument for part (iii) recolors a single vertex and is elementary, but its conclusion requires q ≥ 3 q\ge3 q ≥ 3
Lemma 5.3. Let S S S G G G S S S a ∈ S a\in S a ∈ S B = G [ S ∖ { a } ] B=G[S\setminus\{a\}] B = G [ S ∖ { a }] G G G a a a x \mathbf{x} x G G G a a a k k k x \mathbf{x} x k = deg G ( a ) k=\deg_G(a) k = deg G ( a ) G = B + star ( a , x ) G=B+\operatorname{star}(a,\mathbf{x}) G = B + star ( a , x )
(i) Let σ ′ : S → [ q ] \sigma':S\to[q] σ ′ : S → [ q ] σ \sigma σ S ∖ { a } S\setminus\{a\} S ∖ { a } σ ′ \sigma' σ ′ G G G σ \sigma σ B B B σ ′ ( a ) ∉ σ ( x ) \sigma'(a)\notin\sigma(\mathbf{x}) σ ′ ( a ) ∈ / σ ( x )
(ii) Suppose that G G G b ∈ S ∖ { a } b\in S\setminus\{a\} b ∈ S ∖ { a } a ∼ G b a\sim_G b a ∼ G b B + star ( a , ( x ( 1 ) , … , x ( k ) , b ) ) B+\operatorname{star}(a,(x(1),\ldots,x(k),b)) B + star ( a , ( x ( 1 ) , … , x ( k ) , b ))
(iii) Suppose that a ∼ G b a \sim_G b a ∼ G b b ∈ S ∖ { a } b \in S \setminus\{a\} b ∈ S ∖ { a } B B B ∣ σ ( x ) ∣ ≥ q − 1 |\sigma(x)| \ge q - 1 ∣ σ ( x ) ∣ ≥ q − 1 σ \sigma σ B B B q ≥ 3 q \ge3 q ≥ 3 x x x B B B
(iv) Let F F F S S S Z ⊂ S Z \subset S Z ⊂ S F / Z F/Z F / Z F F F Z Z Z S S S Z Z Z F F F F / Z F/Z F / Z
Proof. For (i), note that an occurrence of G G G a a a B B B { a , x ( i ) } \{a,x(i)\} { a , x ( i )} i i i σ ′ \sigma' σ ′ G G G σ ∈ Col ( B ) \sigma\in\operatorname{Col}(B) σ ∈ Col ( B ) σ ′ ( a ) ≠ σ ( x ( i ) ) \sigma'(a) \ne\sigma(x(i)) σ ′ ( a ) = σ ( x ( i )) i i i
To prove (ii), observe that
B + star ( a , ( x ( 1 ) , … , x ( k ) , b ) ) = G + { a , b } . B+\operatorname{star}(a,(x(1),\ldots,x(k),b))=G+\{a,b\}. B + star ( a , ( x ( 1 ) , … , x ( k ) , b )) = G + { a , b } . Since G G G a ≠ b a \ne b a = b G + { a , b } G+\{a,b\} G + { a , b } a ∼ G b a \sim_G b a ∼ G b
For (iii), recall that a ∼ G b a \sim_G b a ∼ G b G G G B B B σ \sigma σ B B B ∣ σ ( x ) ∣ ≤ q − 2 |\sigma(x)| \le q-2 ∣ σ ( x ) ∣ ≤ q − 2 a ′ ≠ b ′ a' \ne b' a ′ = b ′ σ ( x ) \sigma(x) σ ( x ) σ ′ \sigma' σ ′ σ ′ ′ \sigma'' σ ′′ σ \sigma σ S ∖ { a } S \setminus\{a\} S ∖ { a } σ ′ ( a ) = a ′ \sigma'(a)=a' σ ′ ( a ) = a ′ σ ′ ′ ( a ) = b ′ \sigma''(a)=b' σ ′′ ( a ) = b ′ G G G
Using b ≠ a b \ne a b = a a ∼ G b a \sim_G b a ∼ G b
a ′ = σ ′ ( a ) = σ ′ ( b ) = σ ( b ) = σ ′ ′ ( b ) = σ ′ ′ ( a ) = b ′ , a'=\sigma'(a)=\sigma'(b)=\sigma(b)=\sigma''(b)=\sigma''(a)=b', a ′ = σ ′ ( a ) = σ ′ ( b ) = σ ( b ) = σ ′′ ( b ) = σ ′′ ( a ) = b ′ , a contradiction. We conclude that ∣ σ ( x ) ∣ ≥ q − 1 |\sigma(x)| \ge q-1 ∣ σ ( x ) ∣ ≥ q − 1 σ ∈ Col ( B ) \sigma\in\operatorname{Col}(B) σ ∈ Col ( B ) q ≥ 3 q \ge3 q ≥ 3 ∣ σ ( x ) ∣ ≥ 2 |\sigma(x)| \ge2 ∣ σ ( x ) ∣ ≥ 2 x x x B B B
For (iv), suppose first that σ ′ \sigma' σ ′ F / Z F/Z F / Z F / Z F/Z F / Z { u , v } \{u,v\} { u , v } F F F { ι Z ( u ) , ι Z ( v ) } \{\iota_Z(u),\iota_Z(v)\} { ι Z ( u ) , ι Z ( v )} F / Z F/Z F / Z σ ′ ∘ ι Z \sigma' \circ\iota_Z σ ′ ∘ ι Z F F F Z Z Z
Conversely, let σ \sigma σ F F F a ′ a' a ′ Z Z Z F F F Z Z Z F / Z F/Z F / Z σ ′ ( o ) = a ′ \sigma'(o)=a' σ ′ ( o ) = a ′ σ ′ ( v ) = σ ( v ) \sigma'(v)=\sigma(v) σ ′ ( v ) = σ ( v ) v ∈ S ∖ Z v \in S \setminus Z v ∈ S ∖ Z σ = σ ′ ∘ ι Z \sigma=\sigma'\circ\iota_Z σ = σ ′ ∘ ι Z { u , v } \{u,v\} { u , v } F F F
σ ′ ( ι Z ( u ) ) = σ ( u ) ≠ σ ( v ) = σ ′ ( ι Z ( v ) ) , \sigma'(\iota_Z(u))=\sigma(u)\ne\sigma(v)=\sigma'(\iota_Z(v)), σ ′ ( ι Z ( u )) = σ ( u ) = σ ( v ) = σ ′ ( ι Z ( v )) , so σ ′ \sigma' σ ′ F / Z F/Z F / Z
The proof of (iv) shows that σ ′ ↦ σ ′ ∘ ι Z \sigma' \mapsto\sigma' \circ\iota_Z σ ′ ↦ σ ′ ∘ ι Z Col ( F / Z ) \operatorname{Col}(F/Z) Col ( F / Z ) F F F Z Z Z F F F S S S F F F F / Z F/Z F / Z
Remark 5.4. The assumption q ≥ 3 q \ge3 q ≥ 3 Lemma 5.3 (iii) cannot be removed. Suppose that q = 2 q = 2 q = 2 S S S a , v , b a,v,b a , v , b G G G S S S { a , v } \{a,v\} { a , v } { v , b } \{v,b\} { v , b } x = ( v ) \mathbf{x} = (v) x = ( v ) B = G [ S ∖ { a } ] B = G[S \setminus\{a\}] B = G [ S ∖ { a }] G G G a a a b b b v v v a ∼ G b a \sim_G b a ∼ G b x \mathbf{x} x B B B q = 2 q = 2 q = 2 G G G S ∖ { a } S \setminus\{a\} S ∖ { a } B B B a a a q = 2 q = 2 q = 2 a a a G G G q = 2 q = 2 q = 2 x \mathbf{x} x Theorem 5.7 therefore fails for q = 2 q = 2 q = 2 Lemma 5.3 (iii).
Identification of random vertices
Let N ≥ 2 q N \ge2q N ≥ 2 q ( B m ) m ≥ 0 (B_m)_{m \ge0} ( B m ) m ≥ 0 m m m [ N ] [N] [ N ] Definition 3.2 . We let H ( N ) = min { m ≥ 0 : B m is not colorable } H^{(N)} = \min\{m \ge0 : B_m \text{ is not colorable}\} H ( N ) = min { m ≥ 0 : B m is not colorable } Definition 3.9 , the variable H ( N ) H^{(N)} H ( N ) T G T_G T G G G G [ N ] [N] [ N ] Lemma 3.10 (a) implies that E H ( N ) ≤ A q N \mathbb{E}H^{(N)} \le A_qN E H ( N ) ≤ A q N H ( N ) H^{(N)} H ( N )
For D ∈ { 1 , … , N } D \in\{1,\ldots,N\} D ∈ { 1 , … , N } Z Z Z D D D [ N ] [N] [ N ] ( B m ) (B_m) ( B m ) o ∉ [ N ] o \notin[N] o ∈ / [ N ]
H Z = min { m ≥ 0 : B m / Z is not colorable } . H_Z = \min\{m \ge0 : B_m/Z \text{ is not colorable}\}. H Z = min { m ≥ 0 : B m / Z is not colorable } . Since B m ⊂ B m + 1 B_m \subset B_{m+1} B m ⊂ B m + 1 B m / Z ⊂ B m + 1 / Z B_m/Z \subset B_{m+1}/Z B m / Z ⊂ B m + 1 / Z B m / Z B_m/Z B m / Z m ≥ H Z m \ge H_Z m ≥ H Z Lemma 5.3 implies that the multigraph B m / Z B_m/Z B m / Z B m B_m B m H Z ≤ H ( N ) H_Z \le H^{(N)} H Z ≤ H ( N )
H ( N ) − H Z = ∑ m ≥ 0 1 { B m is colorable and B m / Z is not colorable } . (74) H^{(N)} - H_Z = \sum_{m \ge0} \mathbf{1}\{B_m \text{ is colorable and } B_m/Z \text{ is not colorable}\}. \tag*{(74)} H ( N ) − H Z = m ≥ 0 ∑ 1 { B m is colorable and B m / Z is not colorable } . (74) We set
z 0 = 4 q log q + 1 , C q = 2 e z 0 , N q = ⌈ 4 q log q ⌉ + 1. (75) z_0 = 4q\log q + 1,\qquad C_q = 2e^{z_0},\qquad N_q = \lceil4q\log q\rceil+ 1. \tag*{(75)} z 0 = 4 q log q + 1 , C q = 2 e z 0 , N q = ⌈ 4 q log q ⌉ + 1. (75) Then N q > 2 q N_q > 2q N q > 2 q
We first treat a vertex that is added to an edge process in the same way as the other vertices. In the proof of Proposition 5.6, the vertex obtained by identification will be compared with such a vertex.
Lemma 5.5. Let V ′ V' V ′ N ′ ≥ 2 N' \ge2 N ′ ≥ 2 o ∉ V ′ o \notin V' o ∈ / V ′ λ > 0 \lambda> 0 λ > 0 t ≥ 0 t \ge0 t ≥ 0 F ( t ) F(t) F ( t ) V ′ ∪ { o } V' \cup\{o\} V ′ ∪ { o } t t t λ ′ = λ N ′ + 1 2 \lambda' = \lambda\frac{N'+1}{2} λ ′ = λ 2 N ′ + 1 s ≥ 0 s \ge0 s ≥ 0 B ′ B' B ′ V ′ V' V ′ V ′ V' V ′ Pois ( λ s ) \operatorname{Pois}(\lambda s) Pois ( λ s ) x = ( x ( 1 ) , x ( 2 ) , … ) \mathbf{x} = (\mathbf{x}(1),\mathbf{x}(2),\ldots) x = ( x ( 1 ) , x ( 2 ) , … )
uniform elements of V ′ V' V ′ J J J Pois ( λ N ′ s ) \operatorname{Pois}(\lambda N's) Pois ( λ N ′ s ) B ′ B' B ′ x \mathbf{x} x J J J
I ⋆ = min { i ≥ 0 : B ′ + star ( o , x [ i ] ) is not colorable } , B ′ ′ = B ′ + star ( o , x [ J ] ) , I_{\star}=\min\{i\geq0:B'+\operatorname{star}(o,x_{[i]})\text{ is not colorable}\},\qquad B''=B'+\operatorname{star}(o,x_{[J]}), I ⋆ = min { i ≥ 0 : B ′ + star ( o , x [ i ] ) is not colorable } , B ′′ = B ′ + star ( o , x [ J ] ) , with I ⋆ = ∞ I_{\star}=\infty I ⋆ = ∞ i i i B ′ ′ B'' B ′′ F ( λ ′ s ) F(\lambda's) F ( λ ′ s )
P ( B ′ is colorable , I ⋆ ≤ J ) ≤ ( e λ N ′ s − 1 ) E κ ( F ( λ ′ s ) ) . (76) \mathbb{P}(B'\text{ is colorable},\ I_{\star}\leq J)\leq(e^{\lambda N's}-1)\mathbb{E}\kappa(F(\lambda's)). \tag*{(76)} P ( B ′ is colorable , I ⋆ ≤ J ) ≤ ( e λ N ′ s − 1 ) E κ ( F ( λ ′ s )) . (76) Moreover,
∫ 0 ∞ E κ ( F ( λ ′ s ) ) d s ≤ 1 λ ′ . (77) \int_0^\infty\mathbb{E}\kappa(F(\lambda's))\,\mathrm{d}s\leq\frac{1}{\lambda'}. \tag*{(77)} ∫ 0 ∞ E κ ( F ( λ ′ s )) d s ≤ λ ′ 1 . (77) Proof. We begin with the law of B ′ ′ B'' B ′′ B ′ ′ B'' B ′′ { o , v } \{o,v\} { o , v } v ∈ V ′ v\in V' v ∈ V ′ x [ J ] x_{[J]} x [ J ] v v v V ′ V' V ′ Pois ( λ s ) \operatorname{Pois}(\lambda s) Pois ( λ s ) B ′ B' B ′
The numbers of occurrences of B ′ ′ B'' B ′′ ( N ′ + 1 2 ) \binom{N'+1}{2} ( 2 N ′ + 1 ) V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } Pois ( λ s ) \operatorname{Pois}(\lambda s) Pois ( λ s ) V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } F ( λ ′ s ) F(\lambda's) F ( λ ′ s ) B ′ ′ B'' B ′′ F ( λ ′ s ) F(\lambda's) F ( λ ′ s )
Next, we prove (76) . We set z ^ = λ N ′ s \widehat z=\lambda N's z = λ N ′ s I ⋆ ≥ 1 I_{\star}\geq1 I ⋆ ≥ 1 B ′ B' B ′ J J J ( B ′ , x ) (B',\mathbf{x}) ( B ′ , x ) I ⋆ I_{\star} I ⋆ ( B ′ , x ) (B',\mathbf{x}) ( B ′ , x ) Lemma 3.15 (d) gives
P ( B ′ is colorable , I ⋆ ≤ J ) = ∑ i ≥ 1 P ( B ′ is colorable , I ⋆ = i ) P ( J ≥ i ) ≤ ( e z ^ − 1 ) ∑ i ≥ 1 P ( B ′ is colorable , I ⋆ = i ) π z ^ ( i − 1 ) . (78) \begin{aligned}
\mathbb{P}(B'\text{ is colorable},\ I_{\star}\leq J)
=\sum_{i\geq1}\mathbb{P}(B'\text{ is colorable},\ I_{\star}=i)\mathbb{P}(J\geq i) \\
&\leq(e^{\widehat z}-1)\sum_{i\geq1}\mathbb{P}(B'\text{ is colorable},\ I_{\star}=i)\pi_{\widehat z}(i-1). \tag*{(78)}
\end{aligned} P ( B ′ is colorable , I ⋆ ≤ J ) = i ≥ 1 ∑ P ( B ′ is colorable , I ⋆ = i ) P ( J ≥ i ) ≤ ( e z − 1 ) i ≥ 1 ∑ P ( B ′ is colorable , I ⋆ = i ) π z ( i − 1 ) . (78) We now identify the last sum in (78) . Using once more that J J J ( B ′ , x ) (B',\mathbf{x}) ( B ′ , x ) P ( B ′ is colorable , I ⋆ = J + 1 ) \mathbb{P}(B'\text{ is colorable},\ I_{\star}=J+1) P ( B ′ is colorable , I ⋆ = J + 1 ) I ⋆ I_{\star} I ⋆ B ′ ′ B'' B ′′ B ′ + star ( o , x [ J + 1 ] ) B'+\operatorname{star}(o,x_{[J+1]}) B ′ + star ( o , x [ J + 1 ] ) Lemma 5.3 (ii) to the multigraph B ′ ′ B'' B ′′ o o o b = x ( J + 1 ) b=x(J+1) b = x ( J + 1 ) B ′ ′ B'' B ′′ o ∼ B ′ ′ x ( J + 1 ) o\sim_{B''}x(J+1) o ∼ B ′′ x ( J + 1 )
Conditional on J J J x ( J + 1 ) x(J+1) x ( J + 1 ) V ′ V' V ′ B ′ ′ B'' B ′′
P ( B ′ is colorable , I ⋆ = J + 1 ) = 1 N ′ E [ 1 { B ′ ′ is colorable } ( ∣ C B ′ ′ ( o ) ∣ − 1 ) ] . (79) \mathbb{P}(B'\text{ is colorable},\ I_{\star}=J+1)
=\frac{1}{N'}\mathbb{E}\left[\mathbf{1}\{B''\text{ is colorable}\}\left(|C_{B''}(o)|-1\right)\right]. \tag*{(79)} P ( B ′ is colorable , I ⋆ = J + 1 ) = N ′ 1 E [ 1 { B ′′ is colorable } ( ∣ C B ′′ ( o ) ∣ − 1 ) ] . (79) The law of F ( λ ′ s ) F(\lambda's) F ( λ ′ s ) V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } (79) does not change if o o o V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } F ′ F' F ′ V ′ ∪ { o } V' \cup\{o\} V ′ ∪ { o } κ ( F ′ ) \kappa(F') κ ( F ′ ) F ′ F' F ′ ( N ′ + 1 2 ) \binom{N'+1}{2} ( 2 N ′ + 1 )
∑ v ∈ V ′ ∪ { o } ( ∣ C F ′ ( v ) ∣ − 1 ) = 2 ( N ′ + 1 2 ) κ ( F ′ ) = ( N ′ + 1 ) N ′ κ ( F ′ ) . \sum_{v\in V'\cup\{o\}} \left(\lvert C_{F'}(v)\rvert-1\right)=2\binom{N'+1}{2}\kappa(F')=(N'+1)N'\kappa(F'). v ∈ V ′ ∪ { o } ∑ ( ∣ C F ′ ( v )∣ − 1 ) = 2 ( 2 N ′ + 1 ) κ ( F ′ ) = ( N ′ + 1 ) N ′ κ ( F ′ ) . Averaging the right side of (79) over the N ′ + 1 N'+1 N ′ + 1 o o o
P ( B ′ is colorable , I ⋆ = J + 1 ) = E κ ( F ( λ ′ s ) ) . \mathbb{P}(B' \text{ is colorable},\ I_\star=J+1)=\mathbb{E}\kappa(F(\lambda's)). P ( B ′ is colorable , I ⋆ = J + 1 ) = E κ ( F ( λ ′ s )) . Inserting this into (78) , we arrive at (76) .
It remains to prove (77) . Let ( F m ) m ≥ 0 (F_m)_{m\geq0} ( F m ) m ≥ 0 m m m V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } F ( t ) F(t) F ( t ) ( m + 1 ) (m+1) ( m + 1 ) F m F_m F m E κ ( F m ) \mathbb{E}\kappa(F_m) E κ ( F m ) F m F_m F m F m + 1 F_{m+1} F m + 1 m m m
∑ m ≥ 0 E κ ( F m ) ≤ 1. \sum_{m\geq0}\mathbb{E}\kappa(F_m)\leq1. m ≥ 0 ∑ E κ ( F m ) ≤ 1. This bound and Lemma 3.13 with θ = λ ′ \theta=\lambda' θ = λ ′ (77) .
Proposition 5.6. Let N ≥ N q N\geq N_q N ≥ N q D D D 1 ≤ D ≤ N / 4 1\leq D\leq N/4 1 ≤ D ≤ N /4 k k k k > e D z 0 k>eDz_0 k > eD z 0
E [ H ( N ) − H Z ] ≤ C q ( D k + N ( e D z 0 k ) k + D 2 + 1 ) . (80) \mathbb{E}\left[H^{(N)}-H_Z\right]\leq C_q\left(D^k+N\left(\frac{eDz_0}{k}\right)^k+D^2+1\right). \tag*{(80)} E [ H ( N ) − H Z ] ≤ C q ( D k + N ( k eD z 0 ) k + D 2 + 1 ) . (80) For the following figure and the proof, let Z 0 Z_0 Z 0 D D D [ N ] [N] [ N ] ( Π s ) s ≥ 0 (\Pi_s)_{s\geq0} ( Π s ) s ≥ 0 ( B m ) (B_m) ( B m ) Z Z Z N ′ = N − D N'=N-D N ′ = N − D V ′ = [ N ] ∖ Z 0 V'=[N]\setminus Z_0 V ′ = [ N ] ∖ Z 0 λ = ( N 2 ) − 1 \lambda=\binom{N}{2}^{-1} λ = ( 2 N ) − 1 s ≥ 0 s\geq0 s ≥ 0 B s ′ = B Π s [ V ′ ] B'_s=B_{\Pi_s}[V'] B s ′ = B Π s [ V ′ ] z ^ ( s ) = λ N ′ s \widehat z(s)=\lambda N's z ( s ) = λ N ′ s
Proof. We first outline the argument, which Figure 5.1 illustrates. After Poissonization and a bound for large times, we fix a time s s s D D D Z 0 Z_0 Z 0 Z 0 Z_0 Z 0 Z 0 Z_0 Z 0 B Π s / Z 0 B_{\Pi_s}/Z_0 B Π s / Z 0 B s ′ B'_s B s ′ o o o Pois ( D z ^ ( s ) ) \operatorname{Pois}(D\widehat z(s)) Pois ( D z ( s )) V ′ V' V ′ B s ′ B'_s B s ′ Pois ( z ^ ( s ) ) \operatorname{Pois}(\widehat z(s)) Pois ( z ( s )) k k k D k D^k D k
Figure 5.1. The construction in the proof of Proposition 5.6, at a fixed time s s s D D D Z 0 Z_0 Z 0 D = 3 D=3 D = 3 B s ′ B'_s B s ′ V ′ = [ N ] ∖ Z 0 V'=[N]\setminus Z_0 V ′ = [ N ] ∖ Z 0 V ′ V' V ′ Z 0 Z_0 Z 0 (86) ; if present, it becomes a loop under identification. Middle: the complementary case in which no occurrence has both endpoints in Z 0 Z_0 Z 0 B Π s / Z 0 B_{\Pi_s}/Z_0 B Π s / Z 0 B s ′ B'_s B s ′ o o o Pois ( D z ^ ( s ) ) \operatorname{Pois}(D\widehat{z}(s)) Pois ( D z ( s )) B ′ ′ B'' B ′′ B ′ = B s ′ B'=B'_s B ′ = B s ′ o o o Pois ( z ^ ( s ) ) \operatorname{Pois}(\widehat{z}(s)) Pois ( z ( s )) N ′ + 1 N'+1 N ′ + 1 V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } (88) .
We set
s 0 = 2 q ( log q ) N . (81) s_0=2q(\log q)N. \tag*{(81)} s 0 = 2 q ( log q ) N . (81) By Definition 3.12 , s ↦ B Π s s\mapsto B_{\Pi_s} s ↦ B Π s [ N ] [N] [ N ] B B B [ N ] [N] [ N ]
h ( B ) = P ( B is colorable and B / Z is not colorable ) , h(B)=\mathbb{P}(B\text{ is colorable and }B/Z\text{ is not colorable}), h ( B ) = P ( B is colorable and B / Z is not colorable ) , where the probability is over Z Z Z Z Z Z ( B m ) (B_m) ( B m ) Π \Pi Π (74) , Tonelli’s theorem, and Lemma 3.13 with θ = 1 \theta=1 θ = 1
E [ H ( N ) − H Z ] = ∑ m ≥ 0 E h ( B m ) = ∫ 0 ∞ E h ( B Π s ) d s . (82) \mathbb{E}[H^{(N)}-H_Z]=\sum_{m\geq0}\mathbb{E}h(B_m)=\int_0^\infty\mathbb{E}h(B_{\Pi_s})\,\mathrm{d}s. \tag*{(82)} E [ H ( N ) − H Z ] = m ≥ 0 ∑ E h ( B m ) = ∫ 0 ∞ E h ( B Π s ) d s . (82) We begin with large times. Since N ≥ 2 q N\geq2q N ≥ 2 q σ : [ N ] → [ q ] \sigma:[N]\to[q] σ : [ N ] → [ q ] 1 / ( 2 q ) 1/(2q) 1/ ( 2 q ) [ N ] [N] [ N ] σ \sigma σ B m B_m B m ( 1 − 1 / ( 2 q ) ) m (1-1/(2q))^m ( 1 − 1/ ( 2 q ) ) m q N q^N q N
P ( B m is colorable ) ≤ q N ( 1 − 1 2 q ) m . \mathbb{P}(B_m\text{ is colorable})\leq q^N\left(1-\frac{1}{2q}\right)^m. P ( B m is colorable ) ≤ q N ( 1 − 2 q 1 ) m . Using h ( B ) ≤ 1 { B is colorable } h(B)\leq\mathbf{1}\{B\text{ is colorable}\} h ( B ) ≤ 1 { B is colorable } Pois ( s ) \operatorname{Pois}(s) Pois ( s ) Π s \Pi_s Π s
∫ s 0 ∞ E h ( B Π s ) d s ≤ q N ∫ s 0 ∞ e − s / ( 2 q ) d s = 2 q q N e − s 0 / ( 2 q ) = 2 q , (83) \int_{s_0}^{\infty}\mathbb{E}h(B_{\Pi_s})\,\mathrm{d}s\leq q^N\int_{s_0}^{\infty}e^{-s/(2q)}\,\mathrm{d}s=2q\,q^N e^{-s_0/(2q)}=2q, \tag*{(83)} ∫ s 0 ∞ E h ( B Π s ) d s ≤ q N ∫ s 0 ∞ e − s / ( 2 q ) d s = 2 q q N e − s 0 / ( 2 q ) = 2 q , (83) where in the last step we used s 0 = 2 q ( log q ) N s_0=2q(\log q)N s 0 = 2 q ( log q ) N
Next, we fix s ∈ [ 0 , s 0 ] s \in[0,s_0] s ∈ [ 0 , s 0 ] D D D Z 0 Z_0 Z 0 [ N ] [N] [ N ] B Π s B_{\Pi_s} B Π s B Π s / Z 0 B_{\Pi_s}/Z_0 B Π s / Z 0 Z Z Z ( B m ) (B_m) ( B m ) Π \Pi Π Z 0 Z_0 Z 0 E h ( B Π s ) \mathbb{E}h(B_{\Pi_s}) E h ( B Π s ) S = [ N ] S=[N] S = [ N ] S 0 = Z 0 S_0=Z_0 S 0 = Z 0 B s ′ B'_s B s ′ V ′ V' V ′ λ ( N ′ 2 ) s \lambda\binom{N'}{2}s λ ( 2 N ′ ) s v ∈ Z 0 v\in Z_0 v ∈ Z 0 x v x_v x v v v v V ′ V' V ′ s s s Pois ( z ^ ( s ) ) \operatorname{Pois}(\widehat z(s)) Pois ( z ( s )) V ′ V' V ′ B Π s [ Z 0 ] B_{\Pi_s}[Z_0] B Π s [ Z 0 ]
Let x Z 0 x_{Z_0} x Z 0 x v x_v x v v ∈ Z 0 v\in Z_0 v ∈ Z 0 x Z 0 x_{Z_0} x Z 0 V ′ V' V ′ x Z 0 x_{Z_0} x Z 0 Pois ( D z ^ ( s ) ) \operatorname{Pois}(D\widehat z(s)) Pois ( D z ( s )) V ′ V' V ′ B s ′ B'_s B s ′
Suppose that B Π s [ Z 0 ] B_{\Pi_s}[Z_0] B Π s [ Z 0 ] B Π s B_{\Pi_s} B Π s B s ′ B'_s B s ′ { v , x } \{v,x\} { v , x } v ∈ Z 0 v\in Z_0 v ∈ Z 0 x ∈ V ′ x\in V' x ∈ V ′ { o , x } \{o,x\} { o , x }
B Π s / Z 0 = B s ′ + star ( o , x Z 0 ) (84) B_{\Pi_s}/Z_0=B'_s+\operatorname{star}(o,x_{Z_0}) \tag*{(84)} B Π s / Z 0 = B s ′ + star ( o , x Z 0 ) (84) as multigraphs on V ′ ∪ { o } V'\cup\{o\} V ′ ∪ { o } B s ′ B'_s B s ′ B Π s B_{\Pi_s} B Π s B Π s [ Z 0 ] B_{\Pi_s}[Z_0] B Π s [ Z 0 ] λ ( D 2 ) s \lambda\binom{D}{2}s λ ( 2 D ) s (84) and Markov’s inequality imply that
P ( B Π s is colorable and B Π s / Z 0 is not colorable ) ≤ λ ( D 2 ) s \begin{aligned}
\mathbb{P}(B_{\Pi_s}\text{ is colorable and }B_{\Pi_s}/Z_0\text{ is not colorable}) \\
&\leq\lambda\binom{D}{2}s
\end{aligned} P ( B Π s is colorable and B Π s / Z 0 is not colorable ) ≤ λ ( 2 D ) s ≤ + P ( B s ′ is colorable and B s ′ + star ( o , x Z 0 ) is not colorable ) . (85) \mathrel{\phantom{\leq{}}}+\mathbb{P}(B'_s\text{ is colorable and }B'_s+\operatorname{star}(o,x_{Z_0})\text{ is not colorable}). \tag*{(85)} ≤ + P ( B s ′ is colorable and B s ′ + star ( o , x Z 0 ) is not colorable ) . (85) Integrating the first term over [ 0 , s 0 ] [0,s_0] [ 0 , s 0 ]
∫ 0 s 0 λ ( D 2 ) s d s = D ( D − 1 ) s 0 2 2 N ( N − 1 ) = 2 q 2 ( log q ) 2 D ( D − 1 ) N N − 1 ≤ 2 q 2 ( log q ) 2 D 2 , (86) \int_0^{s_0}\lambda\binom{D}{2}s\,\mathrm{d}s
=\frac{D(D-1)s_0^2}{2N(N-1)}
=2q^2(\log q)^2\frac{D(D-1)N}{N-1}
\leq2q^2(\log q)^2D^2, \tag*{(86)} ∫ 0 s 0 λ ( 2 D ) s d s = 2 N ( N − 1 ) D ( D − 1 ) s 0 2 = 2 q 2 ( log q ) 2 N − 1 D ( D − 1 ) N ≤ 2 q 2 ( log q ) 2 D 2 , (86) where in the last step we used D ≤ N D\leq N D ≤ N
We now compare the list x Z 0 x_{Z_0} x Z 0 D z ^ ( s ) D\widehat z(s) D z ( s ) z ^ ( s ) \widehat z(s) z ( s ) z ^ = z ^ ( s ) \widehat z=\widehat z(s) z = z ( s ) λ N ′ ≤ 2 / ( N − 1 ) \lambda N'\leq2/(N-1) λ N ′ ≤ 2/ ( N − 1 ) N − 1 ≥ 4 q log q N-1\geq4q\log q N − 1 ≥ 4 q log q
z ^ ≤ 2 s 0 N − 1 = 4 q log q + 4 q log q N − 1 ≤ z 0 . (87) \widehat z\leq\frac{2s_0}{N-1}
=4q\log q+\frac{4q\log q}{N-1}
\leq z_0. \tag*{(87)} z ≤ N − 1 2 s 0 = 4 q log q + N − 1 4 q log q ≤ z 0 . (87) Let x = ( x ( 1 ) , x ( 2 ) , … ) \mathbf{x}=(x(1),x(2),\ldots) x = ( x ( 1 ) , x ( 2 ) , … ) V ′ V' V ′ J J J J ′ J' J ′ Pois ( z ^ ) \operatorname{Pois}(\widehat z) Pois ( z ) Pois ( D z ^ ) \operatorname{Pois}(D\widehat z) Pois ( D z ) B s ′ B'_s B s ′ x \mathbf{x} x J J J J ′ J' J ′ ( B s ′ , x Z 0 ) (B'_s,x_{Z_0}) ( B s ′ , x Z 0 ) ( B s ′ , x [ J ′ ] ) (B'_s,\mathbf{x}_{[J']}) ( B s ′ , x [ J ′ ] ) I ⋆ I_\star I ⋆ Lemma 5.5 with B ′ = B s ′ B'=B'_s B ′ = B s ′
Non-colorability persists as j j j B s ′ B'_s B s ′ B s ′ + star ( o , x [ j ] ) B'_s + \operatorname{star}(o,x_{[j]}) B s ′ + star ( o , x [ j ] ) I ⋆ ≤ j I_\star\le j I ⋆ ≤ j J ′ J' J ′ ( B s ′ , x ) (B'_s,x) ( B s ′ , x )
P ( B s ′ is colorable and B s ′ + star ( o , x Z 0 ) is not colorable ) = ∑ j ≥ 0 π D z ^ ( j ) P ( B s ′ is colorable , I ⋆ ≤ j ) . \mathbb{P}(B'_s\text{ is colorable and }B'_s+\operatorname{star}(o,x_{Z_0})\text{ is not colorable})
=\sum_{j\ge0}\pi_{D\hat{z}}(j)\mathbb{P}(B'_s\text{ is colorable},I_\star\le j). P ( B s ′ is colorable and B s ′ + star ( o , x Z 0 ) is not colorable ) = j ≥ 0 ∑ π D z ^ ( j ) P ( B s ′ is colorable , I ⋆ ≤ j ) . For 0 ≤ j ≤ k 0\le j\le k 0 ≤ j ≤ k
π D z ^ ( j ) = D j e − ( D − 1 ) z ^ π z ^ ( j ) ≤ D k π z ^ ( j ) . \pi_{D\hat{z}}(j)=D^j e^{-(D-1)\hat{z}}\pi_{\hat{z}}(j)\le D^k\pi_{\hat{z}}(j). π D z ^ ( j ) = D j e − ( D − 1 ) z ^ π z ^ ( j ) ≤ D k π z ^ ( j ) . We use this bound for j ≤ k j\le k j ≤ k j > k j>k j > k J J J ( B s ′ , x ) (B'_s,x) ( B s ′ , x )
∑ j ≥ 0 π D z ^ ( j ) P ( B s ′ is colorable , I ⋆ ≤ j ) ≤ D k P ( B s ′ is colorable , I ⋆ ≤ J ) + P ( J ′ > k ) . (88) \sum_{j\ge0}\pi_{D\hat{z}}(j)\mathbb{P}(B'_s\text{ is colorable},I_\star\le j)
\le D^k\mathbb{P}(B'_s\text{ is colorable},I_\star\le J)+\mathbb{P}(J'>k). \tag*{(88)} j ≥ 0 ∑ π D z ^ ( j ) P ( B s ′ is colorable , I ⋆ ≤ j ) ≤ D k P ( B s ′ is colorable , I ⋆ ≤ J ) + P ( J ′ > k ) . (88) Further, k > e D z 0 ≥ D z ^ k>eDz_0\ge D\hat{z} k > eD z 0 ≥ D z ^ Lemma 3.15 (b) implies that
P ( J ′ > k ) ≤ ( e D z ^ k ) k ≤ ( e D z 0 k ) k . \mathbb{P}(J'>k)\le\left(\frac{eD\hat{z}}{k}\right)^k\le\left(\frac{eDz_0}{k}\right)^k. P ( J ′ > k ) ≤ ( k eD z ^ ) k ≤ ( k eD z 0 ) k . Integrating over [ 0 , s 0 ] [0,s_0] [ 0 , s 0 ]
∫ 0 s 0 P ( J ′ > k ) d s ≤ 2 q ( log q ) N ( e D z 0 k ) k . (89) \int_0^{s_0}\mathbb{P}(J'>k)\,\mathrm{d}s\le2q(\log q)N\left(\frac{eDz_0}{k}\right)^k. \tag*{(89)} ∫ 0 s 0 P ( J ′ > k ) d s ≤ 2 q ( log q ) N ( k eD z 0 ) k . (89) Next, we bound the first term on the right side of (88) with Lemma 5.5 . Its hypothesis on B s ′ B'_s B s ′ V ′ V' V ′ λ ( N 2 ′ ) s \lambda(N'_2)s λ ( N 2 ′ ) s V ′ V' V ′ λ ′ = λ ( N ′ + 1 2 ) \lambda'=\lambda\left(\frac{N'+1}{2}\right) λ ′ = λ ( 2 N ′ + 1 ) F ( t ) F(t) F ( t ) Lemma 5.5 . Using (76) , (87) , and (77) , we obtain
∫ 0 s 0 D k P ( B s ′ is colorable , I ⋆ ≤ J ) d s ≤ D k ( e z 0 − 1 ) ∫ 0 ∞ E K ( F ( λ ′ s ) ) d s ≤ D k ( e z 0 − 1 ) λ ′ ≤ 2 ( e z 0 − 1 ) D k . (90) \begin{aligned}
\int_0^{s_0}D^k\mathbb{P}(B'_s\text{ is colorable},I_\star\le J)\,\mathrm{d}s
&\le D^k(e^{z_0}-1)\int_0^\infty\mathbb{E}K(F(\lambda's))\,\mathrm{d}s\\
&\le\frac{D^k(e^{z_0}-1)}{\lambda'}\\
&\le2(e^{z_0}-1)D^k. \tag*{(90)}
\end{aligned} ∫ 0 s 0 D k P ( B s ′ is colorable , I ⋆ ≤ J ) d s ≤ D k ( e z 0 − 1 ) ∫ 0 ∞ E K ( F ( λ ′ s )) d s ≤ λ ′ D k ( e z 0 − 1 ) ≤ 2 ( e z 0 − 1 ) D k . (90) In the last step, we used N − D ≥ 3 N / 4 N-D\ge3N/4 N − D ≥ 3 N /4
1 λ ′ = N ( N − 1 ) ( N − D + 1 ) ( N − D ) ≤ N 2 ( 3 N / 4 ) 2 = 16 9 . \frac{1}{\lambda'}=\frac{N(N-1)}{(N-D+1)(N-D)}\le\frac{N^2}{(3N/4)^2}=\frac{16}{9}. λ ′ 1 = ( N − D + 1 ) ( N − D ) N ( N − 1 ) ≤ ( 3 N /4 ) 2 N 2 = 9 16 . Finally, we collect the bounds. We split the integral in (82) at s 0 s_0 s 0 (83) for the integral over [ s 0 , ∞ ) [s_0,\infty) [ s 0 , ∞ ) [ 0 , s 0 ] [0,s_0] [ 0 , s 0 ] (85) and (88) , and we integrate with (86) , (89) , and (90) . We obtain
E [ H ( N ) − H Z ] ≤ 2 q + 2 q 2 ( log q ) 2 D 2 + 2 q ( log q ) N ( e D z 0 k ) k + 2 ( e z 0 − 1 ) D k . \mathbb{E}[H^{(N)}-H_Z]\leq2q+2q^2(\log q)^2D^2+2q(\log q)N\left(\frac{e^D z_0}{k}\right)^k+2(e^{z_0}-1)D^k. E [ H ( N ) − H Z ] ≤ 2 q + 2 q 2 ( log q ) 2 D 2 + 2 q ( log q ) N ( k e D z 0 ) k + 2 ( e z 0 − 1 ) D k . Each of the four coefficients is bounded by C q = 2 e z 0 = 2 e q 4 q C_q=2e^{z_0}=2eq^{4q} C q = 2 e z 0 = 2 e q 4 q (80) .
Proof of the low-degree estimate
We now prove the main result of this section. Recall that D n = ⌊ ( log n ) 1 / 4 ⌋ D_n=\left\lfloor(\log n)^{1/4}\right\rfloor D n = ⌊ ( log n ) 1/4 ⌋ r ( G ) r(G) r ( G ) G G G [ n ] [n] [ n ] Definition 3.9 .
Theorem 5.7. There exist C l o w C_{\mathrm{low}} C low n q n_q n q q q q n ≥ n q n\geq n_q n ≥ n q y > 0 y>0 y > 0
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ C l o w y − 1 / 2 n 3 / 8 . (91) \mathbb{P}\left(H<\infty,\ \min_{v\in e_H}\deg_{\Gamma}(v)\leq D_n,\ r(\Gamma)>y\right)\leq C_{\mathrm{low}}y^{-1/2}n^{3/8}. \tag*{(91)} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ C low y − 1/2 n 3/8 . (91) Proof. We first fix the parameters. We set
k n = ⌊ log n log log n ⌋ , C l o w = 8 C q ( 1 + A q ) 1 / 2 , k_n=\left\lfloor\frac{\log n}{\log\log n}\right\rfloor,\qquad C_{\mathrm{low}}=8C_q(1+A_q)^{1/2}, k n = ⌊ log log n log n ⌋ , C low = 8 C q ( 1 + A q ) 1/2 , and we let n q n_q n q n ≥ n q n\geq n_q n ≥ n q n − 1 ≥ N q n-1\geq N_q n − 1 ≥ N q
log ( e z 0 log log n ) ≤ 1 8 log log n , n 1 / 4 ( log n ) 1 / 4 + ( log n ) 1 / 2 + 1 ≤ n 3 / 8 . (92) \log(ez_0\log\log n)\leq\frac{1}{8}\log\log n,\qquad n^{1/4}(\log n)^{1/4}+(\log n)^{1/2}+1\leq n^{3/8}. \tag*{(92)} log ( e z 0 log log n ) ≤ 8 1 log log n , n 1/4 ( log n ) 1/4 + ( log n ) 1/2 + 1 ≤ n 3/8 . (92) There exists such an integer, depending only on q q q log ( e z 0 log log n ) / log log n → 0 \log(ez_0\log\log n)/\log\log n\to0 log ( e z 0 log log n ) / log log n → 0 n 1 / 4 ( log n ) 1 / 4 + ( log n ) 1 / 2 + 1 = o ( n 3 / 8 ) n^{1/4}(\log n)^{1/4}+(\log n)^{1/2}+1=o(n^{3/8}) n 1/4 ( log n ) 1/4 + ( log n ) 1/2 + 1 = o ( n 3/8 ) n → ∞ n\to\infty n → ∞ n ≥ n q n\geq n_q n ≥ n q y > 0 y>0 y > 0 n − 1 ≥ N q > 2 q n-1\geq N_q>2q n − 1 ≥ N q > 2 q N q ≥ 15 N_q\geq15 N q ≥ 15 n ≥ 2 q n\geq2q n ≥ 2 q log log n > 0 \log\log n>0 log log n > 0 1 ≤ D n ≤ ( n − 1 ) / 4 1\leq D_n\leq(n-1)/4 1 ≤ D n ≤ ( n − 1 ) /4
We begin by writing the probability in (91) as a time integral. We define the set E l o w \mathcal{E}_{\mathrm{low}} E low h h h [ n ] [n] [ n ]
E l o w = { ( G , e ) : G is colorable , e is a frozen pair of G , min v ∈ e deg G ( v ) ≤ D n } , \mathcal{E}_{\mathrm{low}}=\left\{(G,e):G\text{ is colorable},\ e\text{ is a frozen pair of }G,\ \min_{v\in e}\deg_G(v)\leq D_n\right\}, E low = { ( G , e ) : G is colorable , e is a frozen pair of G , v ∈ e min deg G ( v ) ≤ D n } , h ( G ) = 1 { r ( G ) > y } . h(G)=\mathbf{1}\{r(G)>y\}. h ( G ) = 1 { r ( G ) > y } . Recall from Lemma 3.6 (a) that e H e_H e H Γ \Gamma Γ H < ∞ H<\infty H < ∞ (91) is the event that H < ∞ H<\infty H < ∞ ( Γ , e H ) ∈ E l o w (\Gamma,e_H)\in\mathcal{E}_{\mathrm{low}} ( Γ , e H ) ∈ E low r ( Γ ) > y r(\Gamma)>y r ( Γ ) > y Lemma 3.8 (a) with L ′ = 0 L'=0 L ′ = 0 U ′ = ∞ U'=\infty U ′ = ∞ Lemma 3.13 with θ = 1 \theta=1 θ = 1
∑ m ≥ 0 E [ κ E l o w ( G m ) h ( G m ) ] = ∫ 0 ∞ E [ κ E l o w ( G ( t ) ) h ( G ( t ) ) ] d t . (93) \sum_{m\geq0}\mathbb{E}\left[\kappa_{\mathcal{E}_{\mathrm{low}}}(G_m)h(G_m)\right]=\int_0^\infty\mathbb{E}\left[\kappa_{\mathcal{E}_{\mathrm{low}}}(G(t))h(G(t))\right]\,\mathrm{d}t. \tag*{(93)} m ≥ 0 ∑ E [ κ E low ( G m ) h ( G m ) ] = ∫ 0 ∞ E [ κ E low ( G ( t )) h ( G ( t )) ] d t . (93) Next, we reduce to a single vertex. Let G G G [ n ] [n] [ n ] e e e ( G , e ) ∈ E l o w (G,e) \in\mathcal{E}_{\mathrm{low}} ( G , e ) ∈ E low a a a deg G ( a ) ≤ D n \deg_G(a) \le D_n deg G ( a ) ≤ D n C G ( a ) C_G(a) C G ( a )
κ E l o w ( G ) ≤ ( n 2 ) − 1 ∑ a ∈ [ n ] 1 { G is colorable , deg G ( a ) ≤ D n } ( ∣ C G ( a ) ∣ − 1 ) . \kappa_{\mathcal{E}_{\mathrm{low}}}(G) \le\binom{n}{2}^{-1} \sum_{a\in[n]} \boldsymbol{1}\{G\text{ is colorable},\ \deg_G(a)\le D_n\}\bigl(\lvert C_G(a)\rvert-1\bigr). κ E low ( G ) ≤ ( 2 n ) − 1 a ∈ [ n ] ∑ 1 { G is colorable , deg G ( a ) ≤ D n } ( ∣ C G ( a )∣ − 1 ) . and the two sides vanish if G G G
The law of G ( t ) G(t) G ( t ) G m G_m G m [ n ] [n] [ n ] a a a a ′ a' a ′ C G ( a ) C_G(a) C G ( a ) a ′ a' a ′ Lemma 3.10 (b) shows that it preserves h h h n n n G = G ( t ) G=G(t) G = G ( t ) a 0 = n a_0=n a 0 = n
E [ κ E l o w ( G ( t ) ) h ( G ( t ) ) ] ≤ 2 n − 1 E [ 1 { G ( t ) is colorable , deg G ( t ) ( a 0 ) ≤ D n } ( ∣ C G ( t ) ( a 0 ) ∣ − 1 ) h ( G ( t ) ) ] . (94) \mathbb{E}\left[\kappa_{\mathcal{E}_{\mathrm{low}}}(G(t))h(G(t))\right]
\le\frac{2}{n-1}\mathbb{E}\left[\boldsymbol{1}\{G(t)\text{ is colorable},\ \deg_{G(t)}(a_0)\le D_n\}\bigl(\lvert C_{G(t)}(a_0)\rvert-1\bigr)h(G(t))\right]. \tag*{(94)} E [ κ E low ( G ( t )) h ( G ( t )) ] ≤ n − 1 2 E [ 1 { G ( t ) is colorable , deg G ( t ) ( a 0 ) ≤ D n } ( ∣ C G ( t ) ( a 0 )∣ − 1 ) h ( G ( t )) ] . (94) We now use the restriction on the lifetime. By Lemma 3.11 , every colorable multigraph G G G [ n ] [n] [ n ] r ( G ) > y r(G)>y r ( G ) > y ∣ C G ( a 0 ) ∣ ≤ n ( ( 1 + A q ) / y ) 1 / 2 \lvert C_G(a_0)\rvert\le n\bigl((1+A_q)/y\bigr)^{1/2} ∣ C G ( a 0 )∣ ≤ n ( ( 1 + A q ) / y ) 1/2 ∣ C G ( a 0 ) ∣ − 1 > 0 \lvert C_G(a_0)\rvert-1>0 ∣ C G ( a 0 )∣ − 1 > 0 a 0 a_0 a 0 G G G (94) , and using h ≤ 1 h\le1 h ≤ 1
E [ κ E l o w ( G ( t ) ) h ( G ( t ) ) ] ≤ 2 n n − 1 ( 1 + A q y ) 1 / 2 P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) . (95) \mathbb{E}\left[\kappa_{\mathcal{E}_{\mathrm{low}}}(G(t))h(G(t))\right]
\le\frac{2n}{n-1}\left(\frac{1+A_q}{y}\right)^{1/2}\mathbb{P}\left(\deg_{G(t)}(a_0)\le D_n\text{ and }a_0\text{ lies in a frozen pair of }G(t)\right). \tag*{(95)} E [ κ E low ( G ( t )) h ( G ( t )) ] ≤ n − 1 2 n ( y 1 + A q ) 1/2 P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) . (95) This is the only use of the lifetime in the proof.
Next, we use Lemma 5.3 (iii). This is the step where q ≥ 3 q\ge3 q ≥ 3 B ( t ) = G ( t ) [ [ n − 1 ] ] B(t)=G(t)[[n-1]] B ( t ) = G ( t ) [[ n − 1 ]] G ( t ) G(t) G ( t ) a 0 a_0 a 0 x \mathbf{x} x a 0 a_0 a 0 [ n − 1 ] [n-1] [ n − 1 ] t t t
G ( t ) = B ( t ) + star ( a 0 , x ) , G(t)=B(t)+\operatorname{star}(a_0,\mathbf{x}), G ( t ) = B ( t ) + star ( a 0 , x ) , and deg G ( t ) ( a 0 ) \deg_{G(t)}(a_0) deg G ( t ) ( a 0 ) x \mathbf{x} x
We set p = ( n − 2 ) / n p=(n-2)/n p = ( n − 2 ) / n z ( t ) = 2 t / n z(t)=2t/n z ( t ) = 2 t / n S = [ n ] S=[n] S = [ n ] a = a 0 a=a_0 a = a 0 B ( t ) B(t) B ( t ) x \mathbf{x} x B ( t ) B(t) B ( t ) [ n − 1 ] [n-1] [ n − 1 ] p t pt pt x \mathbf{x} x Pois ( z ( t ) ) \operatorname{Pois}(z(t)) Pois ( z ( t )) [ n − 1 ] [n-1] [ n − 1 ]
Suppose that a 0 a_0 a 0 G ( t ) G(t) G ( t ) Lemma 5.3 (iii) and q ≥ 3 q\ge3 q ≥ 3 B ( t ) B(t) B ( t ) x \mathbf{x} x B ( t ) B(t) B ( t )
For an integer D ≥ 0 D \ge0 D ≥ 0 B B B [ n − 1 ] [n-1] [ n − 1 ] nm D ( B ) \operatorname{nm}_D(B) nm D ( B ) D D D [ n − 1 ] [n-1] [ n − 1 ] B B B nm D ( B ) = 0 \operatorname{nm}_D(B)=0 nm D ( B ) = 0 B B B nm 0 = 0 \operatorname{nm}_0=0 nm 0 = 0
A coloring that is constant on a list is constant on each prefix of the list, so a list with a prefix that is never monochromatic in B B B B B B D ≤ D n D \le D_n D ≤ D n D n D_n D n nm D ( B ) ≤ nm D n ( B ) \operatorname{nm}_D(B) \le\operatorname{nm}_{D_n}(B) nm D ( B ) ≤ nm D n ( B )
Conditioning on the length of x x x B ( t ) B(t) B ( t ) ∑ D π Z ( t ) ( D ) ≤ 1 \sum_D \pi_{Z(t)}(D) \le1 ∑ D π Z ( t ) ( D ) ≤ 1
P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) ≤ ∑ D = 0 D n π Z ( t ) ( D ) E [ nm D ( B ( t ) ) ] ≤ E [ nm D n ( B ( t ) ) ] . (96) \begin{aligned}
\mathbb{P}\bigl(\deg_{G(t)}(a_0) \le D_n \text{ and } a_0 \text{ lies in a frozen pair of }G(t)\bigr)
&\le\sum_{D=0}^{D_n} \pi_{Z(t)}(D)\mathbb{E}\bigl[\operatorname{nm}_D(B(t))\bigr] \\
&\le\mathbb{E}\bigl[\operatorname{nm}_{D_n}(B(t))\bigr]. \tag*{(96)}
\end{aligned} P ( deg G ( t ) ( a 0 ) ≤ D n and a 0 lies in a frozen pair of G ( t ) ) ≤ D = 0 ∑ D n π Z ( t ) ( D ) E [ nm D ( B ( t )) ] ≤ E [ nm D n ( B ( t )) ] . (96) We now pass to identifications. Let ( B m ) m ≥ 0 (B_m)_{m\ge0} ( B m ) m ≥ 0 m m m [ n − 1 ] [n-1] [ n − 1 ] § 6.2 , we let Z Z Z D n D_n D n [ n − 1 ] [n-1] [ n − 1 ] ( B m ) (B_m) ( B m ) F F F [ n − 1 ] [n-1] [ n − 1 ]
nm D n ( F ) ≤ P ( F / Z is not colorable ) . (97) \operatorname{nm}_{D_n}(F) \le\mathbb{P}(F/Z\text{ is not colorable}). \tag*{(97)} nm D n ( F ) ≤ P ( F / Z is not colorable ) . (97) To prove (97) , let x ′ x' x ′ D n D_n D n [ n − 1 ] [n-1] [ n − 1 ] Z 1 Z_1 Z 1 x ′ x' x ′ Z ′ Z' Z ′ Z 1 Z_1 Z 1 [ n − 1 ] ∖ Z 1 [n-1]\setminus Z_1 [ n − 1 ] ∖ Z 1 D n − ∣ Z 1 ∣ D_n-|Z_1| D n − ∣ Z 1 ∣ x ′ x' x ′ Z ′ Z' Z ′ [ n − 1 ] [n-1] [ n − 1 ] Z ′ Z' Z ′ [ n − 1 ] [n-1] [ n − 1 ] D n D_n D n [ n − 1 ] [n-1] [ n − 1 ] Z ′ Z' Z ′ D n D_n D n
Now suppose that x ′ x' x ′ F F F F F F Z ′ ⊃ Z 1 Z' \supset Z_1 Z ′ ⊃ Z 1 F / Z ′ F/Z' F / Z ′ (97) .
Applying Lemma 3.13 on [ n − 1 ] [n-1] [ n − 1 ] p p p (97) and (5.1) with N = n − 1 N=n-1 N = n − 1 D = D n D=D_n D = D n
∫ 0 ∞ E [ nm D n ( B ( t ) ) ] d t = 1 p ∑ m ≥ 0 E [ nm D n ( B m ) ] ≤ 1 p ∑ m ≥ 0 P ( B m is colorable and B m / Z is not colorable ) = 1 p E [ H ( n − 1 ) − H Z ] . (98) \begin{aligned}
\int_0^\infty\mathbb{E}\bigl[\operatorname{nm}_{D_n}(B(t))\bigr]\,dt
&= \frac{1}{p}\sum_{m\ge0}\mathbb{E}\bigl[\operatorname{nm}_{D_n}(B_m)\bigr] \\
&\le\frac{1}{p}\sum_{m\ge0}\mathbb{P}\bigl(B_m\text{ is colorable and }B_m/Z\text{ is not colorable}\bigr) \\
&= \frac{1}{p}\mathbb{E}\bigl[H^{(n-1)}-H_Z\bigr]. \tag*{(98)}
\end{aligned} ∫ 0 ∞ E [ nm D n ( B ( t )) ] d t = p 1 m ≥ 0 ∑ E [ nm D n ( B m ) ] ≤ p 1 m ≥ 0 ∑ P ( B m is colorable and B m / Z is not colorable ) = p 1 E [ H ( n − 1 ) − H Z ] . (98) To bound the right side of (98) , we apply Proposition 5.6 with N = n − 1 ≥ N q N=n-1\ge N_q N = n − 1 ≥ N q D = D n D=D_n D = D n k = k n k=k_n k = k n log D n ≤ ( 1 / 4 ) log log n \log D_n\le(1/4)\log\log n log D n ≤ ( 1/4 ) log log n k n ≤ log n / log log n + 1 k_n\le\log n/\log\log n+1 k n ≤ log n / log log n + 1
D n k n ≤ exp ( ( log n log log n + 1 ) log log n 4 ) = n 1 / 4 ( log n ) 1 / 4 . D_n^{k_n}\le\exp\left(\left(\frac{\log n}{\log\log n}+1\right)\frac{\log\log n}{4}\right)=n^{1/4}(\log n)^{1/4}. D n k n ≤ exp ( ( log log n log n + 1 ) 4 log log n ) = n 1/4 ( log n ) 1/4 . Further, using D n ≤ ( log n ) 1 / 4 D_n\le(\log n)^{1/4} D n ≤ ( log n ) 1/4 k n ≥ log n / log log n k_n\ge\log n/\log\log n k n ≥ log n / log log n (92) , we find
e D n z 0 k n ≤ e z 0 ( log log n ) ( log n ) − 3 / 4 = exp ( log ( e z 0 log log n ) − 3 4 log log n ) ≤ exp ( − 5 8 log log n ) < 1. \begin{aligned}
\frac{eD_nz_0}{k_n}
\le ez_0(\log\log n)(\log n)^{-3/4} \\
&=\exp\left(\log(ez_0\log\log n)-\frac{3}{4}\log\log n\right) \\
&\le\exp\left(-\frac{5}{8}\log\log n\right)<1.
\end{aligned} k n e D n z 0 ≤ e z 0 ( log log n ) ( log n ) − 3/4 = exp ( log ( e z 0 log log n ) − 4 3 log log n ) ≤ exp ( − 8 5 log log n ) < 1. In particular, k n > e D n z 0 k_n>eD_nz_0 k n > e D n z 0
( n − 1 ) ( e D n z 0 k n ) k n ≤ n exp ( − 5 8 k n log log n ) ≤ n exp ( − 5 8 log n ) = n 3 / 8 . (n-1)\left(\frac{eD_nz_0}{k_n}\right)^{k_n}
\le n\exp\left(-\frac{5}{8}k_n\log\log n\right)
\le n\exp\left(-\frac{5}{8}\log n\right)=n^{3/8}. ( n − 1 ) ( k n e D n z 0 ) k n ≤ n exp ( − 8 5 k n log log n ) ≤ n exp ( − 8 5 log n ) = n 3/8 . Inserting these bounds and D n 2 ≤ ( log n ) 1 / 2 D_n^2\le(\log n)^{1/2} D n 2 ≤ ( log n ) 1/2 (80) , we obtain
E [ H ( n − 1 ) − H Z ] ≤ C q ( n 1 / 4 ( log n ) 1 / 4 + n 3 / 8 + ( log n ) 1 / 2 + 1 ) ≤ 2 C q n 3 / 8 , (99) \mathbb{E}\left[H^{(n-1)}-H_Z\right]\le C_q\left(n^{1/4}(\log n)^{1/4}+n^{3/8}+(\log n)^{1/2}+1\right)\le2C_qn^{3/8}, \tag*{(99)} E [ H ( n − 1 ) − H Z ] ≤ C q ( n 1/4 ( log n ) 1/4 + n 3/8 + ( log n ) 1/2 + 1 ) ≤ 2 C q n 3/8 , (99) where in the last step we used the second condition in (92) .
Finally, we combine (93) , (95) , (96) , (98) , and (99) . With p − 1 = n / ( n − 2 ) p^{-1}=n/(n-2) p − 1 = n / ( n − 2 )
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ 2 n 2 ( n − 1 ) ( n − 2 ) ( 1 + A q y ) 1 / 2 ⋅ 2 C q n 3 / 8 \begin{aligned}
\mathbb{P}\left(H<\infty,\min_{v\in e_H}\deg_{\Gamma}(v)\le D_n,\ r(\Gamma)>y\right) \\
&\le\frac{2n^2}{(n-1)(n-2)}\left(\frac{1+A_q}{y}\right)^{1/2}\cdot2C_qn^{3/8}
\end{aligned} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > y ) ≤ ( n − 1 ) ( n − 2 ) 2 n 2 ( y 1 + A q ) 1/2 ⋅ 2 C q n 3/8 ≤ 8 C q ( 1 + A q ) 1 / 2 y − 1 / 2 n 3 / 8 . \le8C_q(1+A_q)^{1/2}y^{-1/2}n^{3/8}. ≤ 8 C q ( 1 + A q ) 1/2 y − 1/2 n 3/8 . where in the last step we used 2 n 2 ≤ 4 ( n − 1 ) ( n − 2 ) 2n^2\le4(n-1)(n-2) 2 n 2 ≤ 4 ( n − 1 ) ( n − 2 ) n ≥ 6 n\ge6 n ≥ 6
With y = x n y=x_n y = x n § 5.3 .
Corollary 5.8. For all large n n n
P ( H < ∞ , min v ∈ e H deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n . (100) \mathbb{P}\left(H<\infty,\min_{v\in e_H}\deg_{\Gamma}(v)\le D_n,\ r(\Gamma)>x_n\right)\le\varepsilon_n. \tag*{(100)} P ( H < ∞ , v ∈ e H min deg Γ ( v ) ≤ D n , r ( Γ ) > x n ) ≤ ε n . (100) In particular, for every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n n n (70) of Proposition 4.6 holds.
Proof. Let C l o w C_{\mathrm{low}} C low n q n_q n q Theorem 5.7 , and recall that x n = n ( log n ) − 3 x_n=n(\log n)^{-3} x n = n ( log n ) − 3 n ≥ 3 n\ge3 n ≥ 3 n ≥ n q n\ge n_q n ≥ n q Theorem 5.7 with y = x n y=x_n y = x n (100) by
C l o w x n − 1 / 2 n 3 / 8 = C l o w ( log n ) 3 / 2 n − 1 / 8 . C_{\mathrm{low}}x_n^{-1/2}n^{3/8}=C_{\mathrm{low}}(\log n)^{3/2}n^{-1/8}. C low x n − 1/2 n 3/8 = C low ( log n ) 3/2 n − 1/8 . This is bounded by ( log n ) − 3 / 2 = ε n (\log n)^{-3/2}=\varepsilon_n ( log n ) − 3/2 = ε n C l o w ( log n ) 3 ≤ n 1 / 8 C_{\mathrm{low}}(\log n)^3\le n^{1/8} C low ( log n ) 3 ≤ n 1/8 n n n (70) is contained in the event in (100) .
Remark 5.9. The exponent 1 / 4 1/4 1/4 D n D_n D n ( log n ) 1 / 2 (\log n)^{1/2} ( log n ) 1/2 b 0 ∈ ( 0 , 1 ) b_0\in(0,1) b 0 ∈ ( 0 , 1 ) b 1 > 0 b_1>0 b 1 > 0 D = ⌊ ( log N ) b 0 ⌋ D=\lfloor(\log N)^{b_0}\rfloor D = ⌊( log N ) b 0 ⌋ k = ⌈ b 1 log N / log log N ⌉ k=\lceil b_1\log N/\log\log N\rceil k = ⌈ b 1 log N / log log N ⌉ N → ∞ N\to\infty N → ∞
D k = N b 0 b 1 + o ( 1 ) , N ( e D z 0 k ) k = N 1 − b 1 ( 1 − b 0 ) + o ( 1 ) . D^k=N^{b_0b_1+o(1)},\qquad N\left(\frac{eDz_0}{k}\right)^k=N^{1-b_1(1-b_0)+o(1)}. D k = N b 0 b 1 + o ( 1 ) , N ( k eD z 0 ) k = N 1 − b 1 ( 1 − b 0 ) + o ( 1 ) . The two exponents are equal at b 1 = 1 b_1=1 b 1 = 1 b 0 b_0 b 0 b 1 b_1 b 1 b 0 b_0 b 0 b 1 = 1 b_1=1 b 1 = 1 E [ H ( N ) − H Z ] ≤ N b 0 + o ( 1 ) \mathbb{E}[H^{(N)}-H_Z]\le N^{b_0+o(1)} E [ H ( N ) − H Z ] ≤ N b 0 + o ( 1 ) Theorem 5.7 with D n D_n D n ⌊ ( log n ) b 0 ⌋ \lfloor(\log n)^{b_0}\rfloor ⌊( log n ) b 0 ⌋ (100) by n b 0 − 1 / 2 + o ( 1 ) n^{b_0-1/2+o(1)} n b 0 − 1/2 + o ( 1 ) b 0 < 1 / 2 b_0<1/2 b 0 < 1/2 ε n \varepsilon_n ε n n n n
At b 0 = 1 / 2 b_0=1/2 b 0 = 1/2 o ( N ) o(\sqrt{N}) o ( N ) k k k D = ⌊ log N ⌋ D=\lfloor\sqrt{\log N}\rfloor D = ⌊ log N ⌋ D k ≤ N D^k\le\sqrt{N} D k ≤ N k ≤ ( log N ) / ( 2 log D ) k\le(\log N)/(2\log D) k ≤ ( log N ) / ( 2 log D ) k log ( k / ( e D z 0 ) ) k\log(k/(eDz_0)) k log ( k / ( eD z 0 )) k > e D z 0 k>eDz_0 k > eD z 0
k log ( k e D z 0 ) ≤ log N 2 log D log ( log N 2 e D z 0 log D ) = log N 2 − log N 2 log D log ( 2 e z 0 D 2 log D log N ) < log N 2 . \begin{aligned}
k\log\left(\frac{k}{eDz_0}\right)\le\frac{\log N}{2\log D}\log\left(\frac{\log N}{2eDz_0\log D}\right) \\
&=\frac{\log N}{2}-\frac{\log N}{2\log D}\log\left(\frac{2e z_0D^2\log D}{\log N}\right)<\frac{\log N}{2}.
\end{aligned} k log ( eD z 0 k ) ≤ 2 log D log N log ( 2 eD z 0 log D log N ) = 2 log N − 2 log D log N log ( log N 2 e z 0 D 2 log D ) < 2 log N . for all large N N N 2 e z 0 D 2 log D / log N ∼ e z 0 log log N → ∞ 2e z_0D^2\log D/\log N\sim e z_0\log\log N\to\infty 2 e z 0 D 2 log D / log N ∼ e z 0 log log N → ∞ N ( e D z 0 / k ) k > N N(eDz_0/k)^k>\sqrt{N} N ( eD z 0 / k ) k > N D k > N D^k>\sqrt{N} D k > N N \sqrt{N} N b 0 = 1 / 2 b_0=1/2 b 0 = 1/2 D n D_n D n
Final edges at vertices of high degree
This section proves Theorem 6.13 , the estimate for final edges at vertices of degree greater than D n D_n D n Proposition 4.6 . It bounds the probability that the process becomes non-colorable by time M M M e H e_H e H D n D_n D n Γ \Gamma Γ r ( Γ ) r(\Gamma) r ( Γ ) M M M 0 ≤ L ′ ≤ U ′ ≤ M 0 \le L' \le U' \le M 0 ≤ L ′ ≤ U ′ ≤ M L ′ < H ≤ U ′ L' < H \le U' L ′ < H ≤ U ′ e H e_H e H D n D_n D n Γ \Gamma Γ § 3 . The arguments use the number of colors only through the notion of colorability, and they hold for q = 2 q = 2 q = 2
The argument has three steps. In § 7.1 , we delete edge occurrences and record whether two specified vertices must still receive the same color in every coloring. Iterating the estimate for deletions that preserve this equality reduces the high-degree probability to the density of pairs for which many incident deletions destroy the equality (Proposition 6.5 ). We express this density through a relation, defined below, that records possible additions to the frozen class of a vertex (Lemma 6.4 ). In § 7.2 , we bound the density of this relation by the expected value of a logarithmic increment defined below (Lemma 6.9 ). This increment is also defined when the added edge makes the graph non-colorable. In § 7.3 , we compare these increments with the growth of the logarithm of the frozen-class size before the last step and bound the last increment separately. Their sum is bounded by log n \log n log n Theorem 6.10 ). Finally, § 7.4 combines this estimate with a binomial tail bound for the probability that a fixed vertex reaches the required degree by time M M M
Throughout this section, n ≥ 3 n \ge3 n ≥ 3 Λ > 0 \Lambda> 0 Λ > 0 M = ⌊ Λ n ⌋ M = \lfloor\Lambda n \rfloor M = ⌊ Λ n ⌋ § 4.1 ; the results of this section use only Λ > 0 \Lambda> 0 Λ > 0 D n = ⌊ ( log n ) 1 / 4 ⌋ D_n = \lfloor(\log n)^{1/4} \rfloor D n = ⌊( log n ) 1/4 ⌋ (55) . We set
K = ⌈ 8 Λ ⌉ , w 0 = ⌊ D n / 2 ⌋ . (101) K = \lceil8\Lambda\rceil,\qquad w_0 = \lfloor D_n/2 \rfloor. \tag*{(101)} K = ⌈ 8Λ ⌉ , w 0 = ⌊ D n /2 ⌋ . (101) Since n ≥ 3 n \ge3 n ≥ 3 D n ≥ 1 D_n \ge1 D n ≥ 1 0 ≤ w 0 < D n 0 \le w_0 < D_n 0 ≤ w 0 < D n a a a b b b a b ab ab { a , b } \{a,b\} { a , b } deg G ( u ) \deg_G(u) deg G ( u ) u ∼ G v u \sim_G v u ∼ G v G G G
Pivotal occurrences
We first define the two kinds of occurrences at an endpoint of a frozen pair.
Definition 6.1. Let G G G [ n ] [n] [ n ] ( u , v ) (u,v) ( u , v ) u ∼ G v u \sim_G v u ∼ G v ( u , v ) (u,v) ( u , v ) ordered frozen pair of G G G e e e G G G u ∈ e u \in e u ∈ e e e e non-pivotal for ( u , v ) (u,v) ( u , v ) u ∼ G − e v u \sim_{G-e} v u ∼ G − e v e e e pivotal for ( u , v ) (u,v) ( u , v ) j ≥ 0 j \ge0 j ≥ 0
κ j ( G ) = 1 n ( n − 1 ) # { ( u , v ) ∈ [ n ] 2 : u ≠ v , u ∼ G v , deg G ( u ) = j } . (102) \kappa_j(G) = \frac{1}{n(n-1)} \#\{(u,v) \in[n]^2 : u \ne v,\ u \sim_G v,\ \deg_G(u) = j\}. \tag*{(102)} κ j ( G ) = n ( n − 1 ) 1 # {( u , v ) ∈ [ n ] 2 : u = v , u ∼ G v , deg G ( u ) = j } . (102) and we define κ j n p ( G ) \kappa_j^{\mathrm{np}}(G) κ j np ( G ) K K K j j j u u u ( u , v ) (u,v) ( u , v ) κ j p i v ( G ) = κ j ( G ) − κ j n p ( G ) \kappa_j^{\mathrm{piv}}(G) = \kappa_j(G) - \kappa_j^{\mathrm{np}}(G) κ j piv ( G ) = κ j ( G ) − κ j np ( G ) G G G κ j ( G ) = κ j n p ( G ) = κ j p i v ( G ) = 0 \kappa_j(G) = \kappa_j^{\mathrm{np}}(G) = \kappa_j^{\mathrm{piv}}(G) = 0 κ j ( G ) = κ j np ( G ) = κ j piv ( G ) = 0
Occurrences are counted with multiplicity, so that exactly deg G ( u ) \deg_G(u) deg G ( u ) u u u G − e ⊂ G G-e \subset G G − e ⊂ G G − e G-e G − e e e e G − e G-e G − e G G G
A pair counted in n ( n − 1 ) κ j p i v ( G ) n(n-1)\kappa_j^{\mathrm{piv}}(G) n ( n − 1 ) κ j piv ( G ) j − K + 1 j-K+1 j − K + 1 u u u G − e G-e G − e e e e ( u , v ) (u,v) ( u , v ) ( G − e ) + u v (G-e)+uv ( G − e ) + uv e e e ( u , v ) (u,v) ( u , v ) [59 Section 2, p. 130] , for the increasing event that G + u v G+uv G + uv G G G G − e G-e G − e [37 Section 4.8, Exercise 4.9] ).
We will also need a relation that records how the frozen class of a vertex grows when an edge at that vertex is added.
Definition 6.2. Let G G G [ n ] [n] [ n ] u ∈ [ n ] u \in[n] u ∈ [ n ] W = [ n ] ∖ C G ( u ) W = [n] \setminus C_G(u) W = [ n ] ∖ C G ( u ) R G ( u ) R_G(u) R G ( u ) ( y , v ) ∈ W × W (y,v) \in W \times W ( y , v ) ∈ W × W
σ ( u ) ∈ { σ ( y ) , σ ( v ) } for all σ ∈ Col ( G ) . (103) \sigma(u) \in\{\sigma(y),\sigma(v)\} \qquad\text{for all } \sigma\in\operatorname{Col}(G). \tag*{(103)} σ ( u ) ∈ { σ ( y ) , σ ( v )} for all σ ∈ Col ( G ) . (103) For y ∈ [ n ] y \in[n] y ∈ [ n ] section R G ( u ; y ) = { v ∈ [ n ] : ( y , v ) ∈ R G ( u ) } R_G(u;y) = \{v \in[n] : (y,v) \in R_G(u)\} R G ( u ; y ) = { v ∈ [ n ] : ( y , v ) ∈ R G ( u )}
β G ( u ) = n − 2 ∣ R G ( u ) ∣ . (104) \beta_G(u) = n^{-2}|R_G(u)|. \tag*{(104)} β G ( u ) = n − 2 ∣ R G ( u ) ∣. (104) If G G G R G ( u ) = ∅ R_G(u) = \varnothing R G ( u ) = ∅ β G ( u ) = 0 \beta_G(u) = 0 β G ( u ) = 0
Equivalently, β G ( u ) \beta_G(u) β G ( u ) y y y v v v [ n ] [n] [ n ] C G ( u ) C_G(u) C G ( u ) G G G u u u y y y v v v R G ( u ; y ) R_G(u;y) R G ( u ; y ) y ∈ C G ( u ) y \in C_G(u) y ∈ C G ( u ) G G G u u u R ( y ) R(y) R ( y ) R G ( u ; y ) R_G(u;y) R G ( u ; y ) W W W [ n ] ∖ C G ( u ) [n] \setminus C_G(u) [ n ] ∖ C G ( u )
Lemma 6.3. Let G G G [ n ] [n] [ n ] u ∈ [ n ] u \in[n] u ∈ [ n ]
(a) The relation R G ( u ) R_G(u) R G ( u ) ( y , y ) ∉ R G ( u ) (y,y) \notin R_G(u) ( y , y ) ∈ / R G ( u ) y ∈ [ n ] y \in[n] y ∈ [ n ]
(b) For all y ∈ W y \in W y ∈ W G + u y G+uy G + u y C G + u y ( u ) = C G ( u ) ∪ R ( y ) C_{G+uy}(u) = C_G(u) \cup R(y) C G + u y ( u ) = C G ( u ) ∪ R ( y )
Proof. For (a), note that the condition (103) is symmetric in y y y v v v v = y v=y v = y σ ( u ) = σ ( y ) \sigma(u) = \sigma(y) σ ( u ) = σ ( y ) σ \sigma σ G G G y ∈ C G ( u ) y \in C_G(u) y ∈ C G ( u ) y ∈ W y \in W y ∈ W
For (b), let y ∈ W y \in W y ∈ W y ≠ u y \ne u y = u u ̸ ∼ G y u \not\sim_G y u ∼ G y Lemma 3.6 (a) implies that G + u y G+uy G + u y G + u y G+uy G + u y σ \sigma σ G G G σ ( y ) ≠ σ ( u ) \sigma(y) \ne\sigma(u) σ ( y ) = σ ( u ) v ∈ C G + u y ( u ) v \in C_{G+uy}(u) v ∈ C G + u y ( u ) σ \sigma σ G G G σ ( y ) ≠ σ ( u ) \sigma(y) \ne\sigma(u) σ ( y ) = σ ( u ) σ ( v ) = σ ( u ) \sigma(v) = \sigma(u) σ ( v ) = σ ( u )
v ∈ C G + u y ( u ) ⟺ σ ( u ) ∈ { σ ( y ) , σ ( v ) } for all σ ∈ Col ( G ) . v \in C_{G+uy}(u) \Longleftrightarrow\sigma(u) \in\{\sigma(y),\sigma(v)\}\quad\text{for all }\sigma\in\operatorname{Col}(G). v ∈ C G + u y ( u ) ⟺ σ ( u ) ∈ { σ ( y ) , σ ( v )} for all σ ∈ Col ( G ) . The right side holds for all v ∈ C G ( u ) v \in C_G(u) v ∈ C G ( u ) v ∈ W v \in W v ∈ W ( y , v ) ∈ R G ( u ) (y,v) \in R_G(u) ( y , v ) ∈ R G ( u ) □ \square □
By Lemma 6.3 (b), for y ∈ W y \in W y ∈ W ∣ R ( y ) ∣ |R(y)| ∣ R ( y ) ∣ u u u u y uy u y G G G ∣ R G ( u ) ∣ = ∑ y ∈ [ n ] ∣ R ( y ) ∣ |R_G(u)| = \sum_{y \in[n]} |R(y)| ∣ R G ( u ) ∣ = ∑ y ∈ [ n ] ∣ R ( y ) ∣ n β G ( u ) n\beta_G(u) n β G ( u ) ∣ R ( y ) ∣ |R(y)| ∣ R ( y ) ∣ y ∈ [ n ] y \in[n] y ∈ [ n ]
We now bound the densities κ j n p \kappa_j^{\mathrm{np}} κ j np κ j p i v \kappa_j^{\mathrm{piv}} κ j piv m m m e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m G m G_m G m i ≤ m i \le m i ≤ m G m − e i G_m-e_i G m − e i m − 1 m-1 m − 1 e i ′ e_{i'} e i ′ i ′ ≠ i i' \ne i i ′ = i e i e_i e i ( G m − e i , e i ) (G_m-e_i,e_i) ( G m − e i , e i ) ( G m − 1 , e ) (G_{m-1},e) ( G m − 1 , e ) e e e G m − 1 G_{m-1} G m − 1 h ≥ 0 h \ge0 h ≥ 0
E ∑ i = 1 m h ( G m − e i , e i ) = m E h ( G m − 1 , e ) . (105) \mathbb{E}\sum_{i=1}^{m} h(G_m-e_i,e_i)=m\mathbb{E}h(G_{m-1},e). \tag*{(105)} E i = 1 ∑ m h ( G m − e i , e i ) = m E h ( G m − 1 , e ) . (105) This is the analogue for m m m
Lemma 6.4. Let m ≥ 1 m \ge1 m ≥ 1 j ≥ 1 j \ge1 j ≥ 1
(a) We have
E κ j n p ( G m ) ≤ 2 m n K E κ j − 1 n p ( G m − 1 ) . (106) \mathbb{E}\kappa_j^{\mathrm{np}}(G_m) \le\frac{2m}{nK}\mathbb{E}\kappa_{j-1}^{\mathrm{np}}(G_{m-1}). \tag*{(106)} E κ j np ( G m ) ≤ n K 2 m E κ j − 1 np ( G m − 1 ) . (106) (b) If j ≥ K j \ge K j ≥ K u ∈ [ n ] u \in[n] u ∈ [ n ]
E κ j p i v ( G m ) ≤ 2 m n ( n − 1 ) 2 ( j − K + 1 ) E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . (107) \mathbb{E}\kappa_j^{\mathrm{piv}}(G_m) \le\frac{2mn}{(n-1)^2(j-K+1)}\mathbb{E}\left[\boldsymbol{1}\{\deg_{G_{m-1}}(u)=j-1\}\beta_{G_{m-1}}(u)\right]. \tag*{(107)} E κ j piv ( G m ) ≤ ( n − 1 ) 2 ( j − K + 1 ) 2 mn E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . (107) :::`
Proof. We begin with (a). For a multigraph G ′ G' G ′ e e e h 1 ( G ′ , e ) h_1(G',e) h 1 ( G ′ , e )
h 1 ( G ′ , e ) = ∑ a ∈ e ∑ v ∈ [ n ] ∖ { a } 1 { a ∼ G ′ + e v , a ̸ ∼ G ′ v , deg G ′ + e ( a ) = j } . h_1(G',e)=\sum_{a\in e}\sum_{v\in[n]\setminus\{a\}}\mathbf{1}\{a\sim_{G'+e}v,\ a\not\sim_{G'}v,\ \deg_{G'+e}(a)=j\}. h 1 ( G ′ , e ) = a ∈ e ∑ v ∈ [ n ] ∖ { a } ∑ 1 { a ∼ G ′ + e v , a ∼ G ′ v , deg G ′ + e ( a ) = j } . Let ( a , v ) (a,v) ( a , v ) n ( n − 1 ) κ j n p ( G m ) n(n-1)\kappa_j^{\mathrm{np}}(G_m) n ( n − 1 ) κ j np ( G m ) a ∼ G m v a\sim_{G_m}v a ∼ G m v deg G m ( a ) = j \deg_{G_m}(a)=j deg G m ( a ) = j K K K i ≤ m i\le m i ≤ m a ∈ e i a\in e_i a ∈ e i a ∼ G m − e i v a\sim_{G_m-e_i}v a ∼ G m − e i v 1 1 1 h 1 ( G m − e i , e i ) h_1(G_m-e_i,e_i) h 1 ( G m − e i , e i ) a a a v v v ( a , v ) (a,v) ( a , v )
K n ( n − 1 ) E κ j n p ( G m ) ≤ E ∑ i = 1 m h 1 ( G m − e i , e i ) = m E h 1 ( G m − 1 , e ) , (108) Kn(n-1)\mathbb{E}\kappa_j^{\mathrm{np}}(G_m)\le\mathbb{E}\sum_{i=1}^{m}h_1(G_m-e_i,e_i)=m\mathbb{E}h_1(G_{m-1},e), \tag*{(108)} K n ( n − 1 ) E κ j np ( G m ) ≤ E i = 1 ∑ m h 1 ( G m − e i , e i ) = m E h 1 ( G m − 1 , e ) , (108) where the identity is (6.5).
We now bound the right side of (6.8). For a ∈ e a\in e a ∈ e deg G m − 1 + e ( a ) = deg G m − 1 ( a ) + 1 \deg_{G_{m-1}+e}(a)=\deg_{G_{m-1}}(a)+1 deg G m − 1 + e ( a ) = deg G m − 1 ( a ) + 1 e e e 2 / n 2/n 2/ n e e e G m − 1 G_{m-1} G m − 1 h 1 h_1 h 1
E h 1 ( G m − 1 , e ) ≤ E ∑ a ∈ e ∑ v ∈ [ n ] ∖ { a } 1 { a ∼ G m − 1 v , deg G m − 1 ( a ) = j − 1 } = 2 n ∑ a ∈ [ n ] ∑ v ∈ [ n ] ∖ { a } P ( a ∼ G m − 1 v , deg G m − 1 ( a ) = j − 1 ) = 2 n n ( n − 1 ) E κ j − 1 ( G m − 1 ) , \begin{aligned}
\mathbb{E}h_1(G_{m-1},e)&\le\mathbb{E}\sum_{a\in e}\sum_{v\in[n]\setminus\{a\}}\mathbf{1}\{a\sim_{G_{m-1}}v,\ \deg_{G_{m-1}}(a)=j-1\}\\
&=\frac{2}{n}\sum_{a\in[n]}\sum_{v\in[n]\setminus\{a\}}\mathbb{P}(a\sim_{G_{m-1}}v,\ \deg_{G_{m-1}}(a)=j-1)\\
&=\frac{2}{n}n(n-1)\mathbb{E}\kappa_{j-1}(G_{m-1}),
\end{aligned} E h 1 ( G m − 1 , e ) ≤ E a ∈ e ∑ v ∈ [ n ] ∖ { a } ∑ 1 { a ∼ G m − 1 v , deg G m − 1 ( a ) = j − 1 } = n 2 a ∈ [ n ] ∑ v ∈ [ n ] ∖ { a } ∑ P ( a ∼ G m − 1 v , deg G m − 1 ( a ) = j − 1 ) = n 2 n ( n − 1 ) E κ j − 1 ( G m − 1 ) , where in the last step we used the definition (6.2). Inserting this bound into (6.8) gives (6.6).
We now prove (b). Let ( a , v ) (a,v) ( a , v ) n ( n − 1 ) κ j p i v ( G m ) n(n-1)\kappa_j^{\mathrm{piv}}(G_m) n ( n − 1 ) κ j piv ( G m ) K − 1 K-1 K − 1 j j j a a a ( a , v ) (a,v) ( a , v ) j − K + 1 j-K+1 j − K + 1 i ≤ m i\le m i ≤ m a ∈ e i a\in e_i a ∈ e i a ̸ ∼ G m − e i v a\not\sim_{G_m-e_i}v a ∼ G m − e i v deg G m − e i ( a ) = j − 1 \deg_{G_m-e_i}(a)=j-1 deg G m − e i ( a ) = j − 1
h 2 ( G ′ , e ) = ∑ a ∈ e ∑ v ∈ [ n ] ∖ { a } 1 { a ∼ G ′ + e v , a ̸ ∼ G ′ v , deg G ′ ( a ) = j − 1 } . h_2(G',e)=\sum_{a\in e}\sum_{v\in[n]\setminus\{a\}}\mathbf{1}\{a\sim_{G'+e}v,\ a\not\sim_{G'}v,\ \deg_{G'}(a)=j-1\}. h 2 ( G ′ , e ) = a ∈ e ∑ v ∈ [ n ] ∖ { a } ∑ 1 { a ∼ G ′ + e v , a ∼ G ′ v , deg G ′ ( a ) = j − 1 } . Each index i i i 1 1 1 h 2 ( G m − e i , e i ) h_2(G_m-e_i,e_i) h 2 ( G m − e i , e i )
( j − K + 1 ) n ( n − 1 ) E κ j p i v ( G m ) ≤ m E h 2 ( G m − 1 , e ) . (109) (j-K+1)n(n-1)\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m)\le m\mathbb{E}h_2(G_{m-1},e). \tag*{(109)} ( j − K + 1 ) n ( n − 1 ) E κ j piv ( G m ) ≤ m E h 2 ( G m − 1 , e ) . (109) To compute the right side of (6.9), we condition on G ′ = G m − 1 G'=G_{m-1} G ′ = G m − 1 G ′ G' G ′ h 2 ( G ′ , e ) = 0 h_2(G',e)=0 h 2 ( G ′ , e ) = 0 a a a y y y
∑ v ∈ [ n ] ∖ { a } 1 { a ∼ G ′ + a y v , a ̸ ∼ G ′ v } = ∣ R G ′ ( a ; y ) ∣ . (110) \sum_{v\in[n]\setminus\{a\}}\mathbf{1}\{a\sim_{G'+ay}v,\ a\not\sim_{G'}v\}=|R_{G'}(a;y)|. \tag*{(110)} v ∈ [ n ] ∖ { a } ∑ 1 { a ∼ G ′ + a y v , a ∼ G ′ v } = ∣ R G ′ ( a ; y ) ∣. (110) Indeed, suppose first that y ∈ C G ′ ( a ) y \in C_{G'}(a) y ∈ C G ′ ( a ) G ′ + a y G' + ay G ′ + a y Lemma 3.6 (a), the left side vanishes, and R G ′ ( a ; y ) = ∅ R_{G'}(a;y) = \varnothing R G ′ ( a ; y ) = ∅ Lemma 6.3 (b) shows that the vertices v v v R G ′ ( a ; y ) R_{G'}(a;y) R G ′ ( a ; y )
Every pair a y ay a y ( n 2 ) − 1 \binom{n}{2}^{-1} ( 2 n ) − 1 e e e ( a , y ) (a,y) ( a , y ) (110) and R G ′ ( a ; a ) = ∅ R_{G'}(a;a)=\varnothing R G ′ ( a ; a ) = ∅
E [ h 2 ( G ′ , e ) ∣ G ′ ] = ( n 2 ) − 1 ∑ a ∈ [ n ] 1 { deg G ′ ( a ) = j − 1 } ∑ y ∈ [ n ] ∖ { a } ∣ R G ′ ( a ; y ) ∣ = ( n 2 ) − 1 ∑ a ∈ [ n ] 1 { deg G ′ ( a ) = j − 1 } ∣ R G ′ ( a ) ∣ = 2 n n − 1 ∑ a ∈ [ n ] 1 { deg G ′ ( a ) = j − 1 } β G ′ ( a ) . \begin{aligned}
\mathbb{E}\left[h_2(G',e)\mid G'\right]=\binom{n}{2}^{-1}\sum_{a\in[n]}1\{\deg_{G'}(a)=j-1\}\sum_{y\in[n]\setminus\{a\}}\lvert R_{G'}(a;y)\rvert\\
&=\binom{n}{2}^{-1}\sum_{a\in[n]}1\{\deg_{G'}(a)=j-1\}\lvert R_{G'}(a)\rvert\\
&=\frac{2n}{n-1}\sum_{a\in[n]}1\{\deg_{G'}(a)=j-1\}\beta_{G'}(a).
\end{aligned} E [ h 2 ( G ′ , e ) ∣ G ′ ] = ( 2 n ) − 1 a ∈ [ n ] ∑ 1 { deg G ′ ( a ) = j − 1 } y ∈ [ n ] ∖ { a } ∑ ∣ R G ′ ( a ; y )∣ = ( 2 n ) − 1 a ∈ [ n ] ∑ 1 { deg G ′ ( a ) = j − 1 } ∣ R G ′ ( a )∣ = n − 1 2 n a ∈ [ n ] ∑ 1 { deg G ′ ( a ) = j − 1 } β G ′ ( a ) . where in the last step we used (104) . Recall that the law of G m − 1 G_{m-1} G m − 1 G G G a a a 1 { deg G ( a ) = j − 1 } β G ( a ) 1\{\deg_G(a)=j-1\}\beta_G(a) 1 { deg G ( a ) = j − 1 } β G ( a ) u ∈ [ n ] u\in[n] u ∈ [ n ]
E h 2 ( G m − 1 , e ) = 2 n 2 n − 1 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . \mathbb{E}h_2(G_{m-1},e)=\frac{2n^2}{n-1}\mathbb{E}\left[1\{\deg_{G_{m-1}}(u)=j-1\}\beta_{G_{m-1}}(u)\right]. E h 2 ( G m − 1 , e ) = n − 1 2 n 2 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . Inserting this identity into (109) and dividing by ( j − K + 1 ) n ( n − 1 ) (j-K+1)n(n-1) ( j − K + 1 ) n ( n − 1 ) (107) . □
We now iterate Lemma 6.4 (a). Recall that K = ⌈ 8 Λ ⌉ K=\lceil8\Lambda\rceil K = ⌈ 8Λ ⌉ (106) satisfies 2 m / ( n K ) ≤ 1 / 4 2m/(nK)\leq1/4 2 m / ( n K ) ≤ 1/4 m ≤ M m\leq M m ≤ M j − w 0 j-w_0 j − w 0 E κ j ( G m ) \mathbb{E}\kappa_j(G_m) E κ j ( G m ) j > D n j>D_n j > D n 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) 4^{-(j-w_0)}\mathbb{E}\kappa_{w_0}(G_{m-j+w_0}) 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) E κ j ′ p i v \mathbb{E}\kappa_{j'}^{\mathrm{piv}} E κ j ′ piv j ′ > w 0 j'>w_0 j ′ > w 0 (114) below. The terms of the first kind are controlled by Lemma 3.7 , which uses that the process becomes non-colorable at most once.
Proposition 6.5. Let n ≥ 3 n\geq3 n ≥ 3
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ 8 3 4 − ( D n + 1 − w 0 ) + 8 3 ∑ j = w 0 + 1 M − 1 ∑ m = j M − 1 E κ j p i v ( G m ) (111) \mathbb{P}\left(H\leq M,\max_{v\in e_H}\deg_{\Gamma}(v)>D_n\right)
\leq\frac{8}{3}4^{-(D_n+1-w_0)}+\frac{8}{3}\sum_{j=w_0+1}^{M-1}\sum_{m=j}^{M-1}\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m) \tag*{(111)} P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ 3 8 4 − ( D n + 1 − w 0 ) + 3 8 j = w 0 + 1 ∑ M − 1 m = j ∑ M − 1 E κ j piv ( G m ) (111) where an empty sum is zero.
Proof. We may assume that M ≥ 1 M\geq1 M ≥ 1 H ≥ 1 H\geq1 H ≥ 1 (111) vanishes if M = 0 M=0 M = 0 E \mathcal{E} E ( G , a b ) (G,ab) ( G , ab ) G G G [ n ] [n] [ n ] a b ab ab G G G max ( deg G ( a ) , deg G ( b ) ) > D n \max(\deg_G(a),\deg_G(b)) > D_n max ( deg G ( a ) , deg G ( b )) > D n κ E \kappa_E κ E
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) = ∑ m = 0 M − 1 E κ E ( G m ) ≤ 2 ∑ m = 0 M − 1 ∑ j = D n + 1 m E κ j ( G m ) . (112) \mathbb{P}\left(H \le M,\max_{v\in e_H}\deg_\Gamma(v)>D_n\right)=\sum_{m=0}^{M-1}\mathbb{E}\kappa_E(G_m)\le2\sum_{m=0}^{M-1}\sum_{j=D_n+1}^{m}\mathbb{E}\kappa_j(G_m). \tag*{(112)} P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) = m = 0 ∑ M − 1 E κ E ( G m ) ≤ 2 m = 0 ∑ M − 1 j = D n + 1 ∑ m E κ j ( G m ) . (112) Since H ≥ 1 H\ge1 H ≥ 1 h = 1 h=1 h = 1 L ′ = 0 L'=0 L ′ = 0 U ′ = M U'=M U ′ = M a b ab ab κ E ( G m ) \kappa_E(G_m) κ E ( G m ) ( a , b ) (a,b) ( a , b ) ( b , a ) (b,a) ( b , a ) j > D n j>D_n j > D n G m G_m G m m m m ( n 2 ) − 1 = 2 / ( n ( n − 1 ) ) \binom{n}{2}^{-1}=2/(n(n-1)) ( 2 n ) − 1 = 2/ ( n ( n − 1 ))
Next, let 1 ≤ j ≤ m ≤ M 1\le j\le m\le M 1 ≤ j ≤ m ≤ M κ j = κ j p i v + κ j n p \kappa_j=\kappa_j^{\mathrm{piv}}+\kappa_j^{\mathrm{np}} κ j = κ j piv + κ j np
E κ j ( G m ) ≤ E κ j p i v ( G m ) + 2 m n K E κ j − 1 ( G m − 1 ) ≤ E κ j p i v ( G m ) + 1 4 E κ j − 1 ( G m − 1 ) , (113) \mathbb{E}\kappa_j(G_m)\le\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m)+\frac{2m}{nK}\mathbb{E}\kappa_{j-1}(G_{m-1})\le\mathbb{E}\kappa_j^{\mathrm{piv}}(G_m)+\frac{1}{4}\mathbb{E}\kappa_{j-1}(G_{m-1}), \tag*{(113)} E κ j ( G m ) ≤ E κ j piv ( G m ) + n K 2 m E κ j − 1 ( G m − 1 ) ≤ E κ j piv ( G m ) + 4 1 E κ j − 1 ( G m − 1 ) , (113) where in the last step we used m ≤ M ≤ Λ n m\le M\le\Lambda n m ≤ M ≤ Λ n K ≥ 8 Λ K\ge8\Lambda K ≥ 8Λ
Now let w 0 < j ≤ m ≤ M − 1 w_0<j\le m\le M-1 w 0 < j ≤ m ≤ M − 1 j , j − 1 , … , w 0 + 1 j,j-1,\ldots,w_0+1 j , j − 1 , … , w 0 + 1 m , m − 1 , … , m − j + w 0 + 1 m,m-1,\ldots,m-j+w_0+1 m , m − 1 , … , m − j + w 0 + 1 j ′ j' j ′ m − j + j ′ m-j+j' m − j + j ′ 1 ≤ j ′ ≤ m − j + j ′ ≤ M 1\le j'\le m-j+j'\le M 1 ≤ j ′ ≤ m − j + j ′ ≤ M j − w 0 j-w_0 j − w 0
E κ j ( G m ) ≤ ∑ j ′ = w 0 + 1 j 4 − ( j − j ′ ) E κ j ′ p i v ( G m − j + j ′ ) + 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) . (114) \mathbb{E}\kappa_j(G_m)\le\sum_{j'=w_0+1}^{j}4^{-(j-j')}\mathbb{E}\kappa_{j'}^{\mathrm{piv}}(G_{m-j+j'})+4^{-(j-w_0)}\mathbb{E}\kappa_{w_0}(G_{m-j+w_0}). \tag*{(114)} E κ j ( G m ) ≤ j ′ = w 0 + 1 ∑ j 4 − ( j − j ′ ) E κ j ′ piv ( G m − j + j ′ ) + 4 − ( j − w 0 ) E κ w 0 ( G m − j + w 0 ) . (114) We insert (6.14) into (6.12), which is possible since D n > w 0 D_n>w_0 D n > w 0 j ≥ D n + 1 j\ge D_n+1 j ≥ D n + 1 m m m { j , … , M − 1 } \{j,\ldots,M-1\} { j , … , M − 1 } m − j + w 0 m-j+w_0 m − j + w 0 κ w 0 ≤ κ \kappa_{w_0}\le\kappa κ w 0 ≤ κ
∑ m = j M − 1 E κ w 0 ( G m − j + w 0 ) ≤ 1. \sum_{m=j}^{M-1}\mathbb{E}\kappa_{w_0}(G_{m-j+w_0})\le1. m = j ∑ M − 1 E κ w 0 ( G m − j + w 0 ) ≤ 1. Summing over j j j
2 ∑ j = D n + 1 ∞ 4 − ( j − w 0 ) = 8 3 4 − ( D n + 1 − w 0 ) 2\sum_{j=D_n+1}^{\infty}4^{-(j-w_0)}=\frac{8}{3}4^{-(D_n+1-w_0)} 2 j = D n + 1 ∑ ∞ 4 − ( j − w 0 ) = 3 8 4 − ( D n + 1 − w 0 ) to the right side of (6.12).
For the first sum in (6.14), fix j ′ ≥ w 0 + 1 j'\ge w_0+1 j ′ ≥ w 0 + 1 j ≥ max ( j ′ , D n + 1 ) j\ge\max(j',D_n+1) j ≥ max ( j ′ , D n + 1 ) m m m { j , … , M − 1 } \{j,\ldots,M-1\} { j , … , M − 1 } m − j + j ′ m-j+j' m − j + j ′ { j ′ , … , M − 1 } \{j',\ldots,M-1\} { j ′ , … , M − 1 } j j j
2 ∑ j ′ = w 0 + 1 M − 1 ( ∑ j ≥ j ′ 4 − ( j − j ′ ) ) ∑ m = j ′ M − 1 E κ j ′ p i v ( G m ) = 8 3 ∑ j ′ = w 0 + 1 M − 1 ∑ m = j ′ M − 1 E κ j ′ p i v ( G m ) 2\sum_{j'=w_0+1}^{M-1}\left(\sum_{j\ge j'}4^{-(j-j')}\right)\sum_{m=j'}^{M-1}\mathbb{E}\kappa_{j'}^{\mathrm{piv}}(G_m)=\frac{8}{3}\sum_{j'=w_0+1}^{M-1}\sum_{m=j'}^{M-1}\mathbb{E}\kappa_{j'}^{\mathrm{piv}}(G_m) 2 j ′ = w 0 + 1 ∑ M − 1 j ≥ j ′ ∑ 4 − ( j − j ′ ) m = j ′ ∑ M − 1 E κ j ′ piv ( G m ) = 3 8 j ′ = w 0 + 1 ∑ M − 1 m = j ′ ∑ M − 1 E κ j ′ piv ( G m ) to the right side of (112) . Adding the two contributions and renaming j ′ j' j ′ j j j (111) .
The descent in Proposition 6.5 runs over all degrees, since the degrees in G m G_m G m m m m 4 − ( j − j ′ ) 4^{-(j-j')} 4 − ( j − j ′ )
Credited sets
In this section and the next, we fix a vertex u ∈ [ n ] u \in[n] u ∈ [ n ] root . For a colorable multigraph G G G log ∣ C G ( u ) ∣ \log|C_G(u)| log ∣ C G ( u ) ∣ potential at $u . By Lemma 6.3 (b), the relation R G ( u ) R_G(u) R G ( u ) C G ( u ) C_G(u) C G ( u )
Definition 6.6. Let G G G [ n ] [n] [ n ] W = [ n ] ∖ C G ( u ) W = [n] \setminus C_G(u) W = [ n ] ∖ C G ( u ) R ( y ) R(y) R ( y ) R G ( u ; y ) R_G(u;y) R G ( u ; y ) f = a b f = ab f = ab [ n ] [n] [ n ] credited set of f f f
A G ( f ) = { R ( a ) ∩ R ( b ) if a , b ∈ W , R ( b ) if a ∈ C G ( u ) and b ∈ W , R ( a ) if a ∈ W and b ∈ C G ( u ) , ∅ if a , b ∈ C G ( u ) . (115) \mathcal{A}_G(f) =
\begin{cases}
R(a) \cap R(b) & \text{if } a,b \in W,\\
R(b) & \text{if } a \in C_G(u) \text{ and } b \in W,\\
R(a) & \text{if } a \in W \text{ and } b \in C_G(u),\\
\varnothing& \text{if } a,b \in C_G(u).
\end{cases} \tag*{(115)} A G ( f ) = ⎩ ⎨ ⎧ R ( a ) ∩ R ( b ) R ( b ) R ( a ) ∅ if a , b ∈ W , if a ∈ C G ( u ) and b ∈ W , if a ∈ W and b ∈ C G ( u ) , if a , b ∈ C G ( u ) . (115) If G G G A G ( f ) = ∅ \mathcal{A}_G(f) = \varnothing A G ( f ) = ∅
For a colorable multigraph G G G f f f
log ( 1 + ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ ) \log\left(1+\frac{|\mathcal{A}_G(f)|}{|C_G(u)|}\right) log ( 1 + ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ ) the credited logarithmic increment of f f f u u u
We call the first case of (115) the direct case and the second and third cases the cross case . Note that A G ( f ) ⊂ W \mathcal{A}_G(f) \subset W A G ( f ) ⊂ W f = u y f = uy f = u y y ∈ W y \in W y ∈ W A G ( u y ) = R ( y ) \mathcal{A}_G(uy) = R(y) A G ( u y ) = R ( y ) Lemma 6.3 (b), this is exactly the set of vertices that join the class of the root when u y uy u y G + f G+f G + f Theorem 6.10 (see Remark 6.11 ).
Lemma 6.7. Let G G G [ n ] [n] [ n ] f f f [ n ] [n] [ n ] G + f G+f G + f C G ( u ) C_G(u) C G ( u ) A G ( f ) \mathcal{A}_G(f) A G ( f ) C G + f ( u ) C_{G+f}(u) C G + f ( u ) ∣ C G + f ( u ) ∣ ≥ ∣ C G ( u ) ∣ + ∣ A G ( f ) ∣ |C_{G+f}(u)| \ge|C_G(u)| + |\mathcal{A}_G(f)| ∣ C G + f ( u ) ∣ ≥ ∣ C G ( u ) ∣ + ∣ A G ( f ) ∣
Proof. By Lemma 3.6 (b), we have C G ( u ) ⊂ C G + f ( u ) C_G(u) \subset C_{G+f}(u) C G ( u ) ⊂ C G + f ( u ) A G ( f ) ⊂ W \mathcal{A}_G(f) \subset W A G ( f ) ⊂ W C G ( u ) C_G(u) C G ( u ) f = a b f = ab f = ab y ∈ A G ( f ) y \in\mathcal{A}_G(f) y ∈ A G ( f ) y ∈ W y \in W y ∈ W Lemma 6.3 (b) shows that C G + u y ( u ) = C G ( u ) ∪ R ( y ) C_{G+uy}(u) = C_G(u) \cup R(y) C G + u y ( u ) = C G ( u ) ∪ R ( y ) a , b ∈ C G + u y ( u ) a,b \in C_{G+uy}(u) a , b ∈ C G + u y ( u )
In the direct case, we have y ∈ R ( a ) ∩ R ( b ) y \in R(a) \cap R(b) y ∈ R ( a ) ∩ R ( b ) R G ( u ) R_G(u) R G ( u ) a , b ∈ R ( y ) a,b \in R(y) a , b ∈ R ( y ) a ∈ C G ( u ) a \in C_G(u) a ∈ C G ( u ) y ∈ R ( b ) y \in R(b) y ∈ R ( b ) b ∈ R ( y ) b \in R(y) b ∈ R ( y )
The claim implies that a ∼ G + u y b a \sim_{G+uy} b a ∼ G + u y b a ≠ b a \ne b a = b G + u y + f G+uy+f G + u y + f σ \sigma σ G + f G+f G + f σ ( y ) = σ ( u ) \sigma(y)=\sigma(u) σ ( y ) = σ ( u ) σ \sigma σ G + f + u y G+f+uy G + f + u y G + f G+f G + f y ∈ C G + f ( u ) y \in C_{G+f}(u) y ∈ C G + f ( u )
Figure 6.1 illustrates the two cases of the proof.
Figure 6.1. The credited set of a pair f = a b f=ab f = ab u u u C G ( u ) C_G(u) C G ( u ) R G ( u ) R_G(u) R G ( u ) u y uy u y G G G a b ab ab a , b ∈ W a,b \in W a , b ∈ W y ∈ R ( a ) ∩ R ( b ) y \in R(a) \cap R(b) y ∈ R ( a ) ∩ R ( b ) a ∈ C G ( u ) a \in C_G(u) a ∈ C G ( u ) b ∈ W b \in W b ∈ W y ∈ R ( b ) y \in R(b) y ∈ R ( b ) u y uy u y a a a b b b u u u G + u y + a b G+uy+ab G + u y + ab G + a b G+ab G + ab y y y u u u
We next compare ∣ R G ( u ) ∣ |R_G(u)| ∣ R G ( u ) ∣ (115) to the expected credited logarithmic increment. The comparison is purely combinatorial; it uses the symmetry of R G ( u ) R_G(u) R G ( u )
For a colorable multigraph G G G [ n ] [n] [ n ] u u u
a 1 = ∑ ( a , b ) ∈ W × W min ( ∣ R ( a ) ∩ R ( b ) ∣ ∣ C G ( u ) ∣ , 1 ) , a 2 = 2 ( ∣ C G ( u ) ∣ − 1 ) ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) . (116) \begin{aligned}
a_1 &= \sum_{(a,b)\in W\times W} \min\left(\frac{|R(a)\cap R(b)|}{|C_G(u)|},1\right),\\
a_2 &= 2\bigl(|C_G(u)|-1\bigr)\sum_{a\in W}\min\left(\frac{|R(a)|}{|C_G(u)|},1\right). \tag*{(116)}
\end{aligned} a 1 a 2 = ( a , b ) ∈ W × W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∩ R ( b ) ∣ , 1 ) , = 2 ( ∣ C G ( u ) ∣ − 1 ) a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) . (116) Lemma 6.8. For every colorable multigraph G G G [ n ] [n] [ n ] ∣ R G ( u ) ∣ ≤ 2 a 1 + a 2 |R_G(u)| \le2a_1+a_2 ∣ R G ( u ) ∣ ≤ 2 a 1 + a 2
Proof. We first bound a 1 a_1 a 1 a , b ∈ W a,b \in W a , b ∈ W
min ( ∣ R ( a ) ∩ R ( b ) ∣ ∣ C G ( u ) ∣ , 1 ) = ∑ y ∈ R ( a ) ∩ R ( b ) 1 max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∩ R ( b ) ∣ ) . \min\left(\frac{|R(a)\cap R(b)|}{|C_G(u)|},1\right)
=
\sum_{y\in R(a)\cap R(b)}
\frac{1}{\max\left(|C_G(u)|,|R(a)\cap R(b)|\right)}. min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∩ R ( b ) ∣ , 1 ) = y ∈ R ( a ) ∩ R ( b ) ∑ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∩ R ( b ) ∣ ) 1 . where both sides vanish if R ( a ) ∩ R ( b ) = ∅ R(a) \cap R(b) = \varnothing R ( a ) ∩ R ( b ) = ∅ R G ( u ) R_G(u) R G ( u ) y ∈ R ( a ) ∩ R ( b ) y \in R(a) \cap R(b) y ∈ R ( a ) ∩ R ( b ) a , b ∈ R ( y ) a,b \in R(y) a , b ∈ R ( y )
a 1 = ∑ y ∈ W ∑ a ∈ R ( y ) ∑ b ∈ R ( y ) 1 max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∩ R ( b ) ∣ ) ≥ ∑ y ∈ W ∑ a ∈ R ( y ) ∣ R ( y ) ∣ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) . a_1 = \sum_{y \in W} \sum_{a \in R(y)} \sum_{b \in R(y)} \frac{1}{\max\left(|C_G(u)|, |R(a) \cap R(b)|\right)} \ge\sum_{y \in W} \sum_{a \in R(y)} \frac{|R(y)|}{\max\left(|C_G(u)|, |R(a)|\right)}. a 1 = y ∈ W ∑ a ∈ R ( y ) ∑ b ∈ R ( y ) ∑ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∩ R ( b ) ∣ ) 1 ≥ y ∈ W ∑ a ∈ R ( y ) ∑ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) ∣ R ( y ) ∣ . where in the last step we used ∣ R ( a ) ∩ R ( b ) ∣ ≤ ∣ R ( a ) ∣ |R(a) \cap R(b)| \le|R(a)| ∣ R ( a ) ∩ R ( b ) ∣ ≤ ∣ R ( a ) ∣ b ∈ R ( y ) b \in R(y) b ∈ R ( y ) ( y , a ) ∈ R G ( u ) (y,a) \in R_G(u) ( y , a ) ∈ R G ( u ) ( y , a ) ↦ ( a , y ) (y,a) \mapsto(a,y) ( y , a ) ↦ ( a , y ) R G ( u ) R_G(u) R G ( u ) y y y a a a
2 a 1 ≥ ∑ ( y , a ) ∈ R G ( u ) ( ∣ R ( y ) ∣ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) + ∣ R ( a ) ∣ max ( ∣ C G ( u ) ∣ , ∣ R ( y ) ∣ ) ) . (117) 2a_1 \ge\sum_{(y,a) \in R_G(u)} \left( \frac{|R(y)|}{\max\left(|C_G(u)|, |R(a)|\right)} + \frac{|R(a)|}{\max\left(|C_G(u)|, |R(y)|\right)} \right). \tag*{(117)} 2 a 1 ≥ ( y , a ) ∈ R G ( u ) ∑ ( max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) ∣ R ( y ) ∣ + max ( ∣ C G ( u ) ∣ , ∣ R ( y ) ∣ ) ∣ R ( a ) ∣ ) . (117) We now split R G ( u ) R_G(u) R G ( u ) ( y , a ) (y,a) ( y , a ) max ( ∣ R ( y ) ∣ , ∣ R ( a ) ∣ ) ≥ ∣ C G ( u ) ∣ \max\left(|R(y)|, |R(a)|\right) \ge|C_G(u)| max ( ∣ R ( y ) ∣ , ∣ R ( a ) ∣ ) ≥ ∣ C G ( u ) ∣ ∣ R ( y ) ∣ ≥ ∣ R ( a ) ∣ |R(y)| \ge|R(a)| ∣ R ( y ) ∣ ≥ ∣ R ( a ) ∣ ∣ R ( y ) ∣ ≥ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) |R(y)| \ge\max\left(|C_G(u)|, |R(a)|\right) ∣ R ( y ) ∣ ≥ max ( ∣ C G ( u ) ∣ , ∣ R ( a ) ∣ ) 2 a 1 2a_1 2 a 1
It remains to count the pairs ( y , a ) ∈ R G ( u ) (y,a) \in R_G(u) ( y , a ) ∈ R G ( u ) ∣ R ( y ) ∣ < ∣ C G ( u ) ∣ |R(y)| < |C_G(u)| ∣ R ( y ) ∣ < ∣ C G ( u ) ∣ ∣ R ( a ) ∣ < ∣ C G ( u ) ∣ |R(a)| < |C_G(u)| ∣ R ( a ) ∣ < ∣ C G ( u ) ∣ ∣ C G ( u ) ∣ = 1 |C_G(u)| = 1 ∣ C G ( u ) ∣ = 1 ( y , a ) ∈ R G ( u ) (y,a) \in R_G(u) ( y , a ) ∈ R G ( u ) a ∈ R ( y ) a \in R(y) a ∈ R ( y ) ∣ R ( y ) ∣ ≥ 1 |R(y)| \ge1 ∣ R ( y ) ∣ ≥ 1 ∣ C G ( u ) ∣ ≥ 2 |C_G(u)| \ge2 ∣ C G ( u ) ∣ ≥ 2
By symmetry, the vertices y y y ( y , a ) ∈ R G ( u ) (y,a) \in R_G(u) ( y , a ) ∈ R G ( u ) R ( a ) R(a) R ( a ) ( y , a ) ∈ R G ( u ) (y,a) \in R_G(u) ( y , a ) ∈ R G ( u ) ∣ R ( a ) ∣ < ∣ C G ( u ) ∣ |R(a)| < |C_G(u)| ∣ R ( a ) ∣ < ∣ C G ( u ) ∣
∑ a ∈ W ∣ R ( a ) ∣ < ∣ C G ( u ) ∣ ∣ R ( a ) ∣ ≤ ∣ C G ( u ) ∣ ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) ≤ a 2 , \sum_{\substack{a \in W \\ |R(a)| < |C_G(u)|}} |R(a)| \le|C_G(u)| \sum_{a \in W} \min\left(\frac{|R(a)|}{|C_G(u)|}, 1\right) \le a_2, a ∈ W ∣ R ( a ) ∣ < ∣ C G ( u ) ∣ ∑ ∣ R ( a ) ∣ ≤ ∣ C G ( u ) ∣ a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) ≤ a 2 , where we used ∣ C G ( u ) ∣ ≤ 2 ( ∣ C G ( u ) ∣ − 1 ) |C_G(u)| \le2\left(|C_G(u)| - 1\right) ∣ C G ( u ) ∣ ≤ 2 ( ∣ C G ( u ) ∣ − 1 ) ∣ R G ( u ) ∣ ≤ 2 a 1 + a 2 |R_G(u)| \le2a_1 + a_2 ∣ R G ( u ) ∣ ≤ 2 a 1 + a 2
Lemma 6.9. Let G G G [ n ] [n] [ n ] f f f [ n ] [n] [ n ]
β G ( u ) ≤ 2 log 2 E f log ( 1 + ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ ) , (118) \beta_G(u) \le\frac{2}{\log2} \mathbb{E}_f \log\left(1 + \frac{|\mathcal{A}_G(f)|}{|C_G(u)|}\right), \tag*{(118)} β G ( u ) ≤ log 2 2 E f log ( 1 + ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ ) , (118) where E f \mathbb{E}_f E f f f f
Proof. For z ≥ 0 z \ge0 z ≥ 0 log ( 1 + z ) ≥ ( log 2 ) min ( z , 1 ) \log(1+z) \ge(\log2)\min(z,1) log ( 1 + z ) ≥ ( log 2 ) min ( z , 1 ) z ∈ [ 0 , 1 ] z \in[0,1] z ∈ [ 0 , 1 ] z ↦ log ( 1 + z ) z \mapsto\log(1+z) z ↦ log ( 1 + z ) z > 1 z > 1 z > 1
E f min ( ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ , 1 ) ≥ ∣ R G ( u ) ∣ 2 n ( n − 1 ) , (119) \mathbb{E}_f \min\left(\frac{|\mathcal{A}_G(f)|}{|C_G(u)|},1\right) \ge\frac{|R_G(u)|}{2n(n-1)}, \tag*{(119)} E f min ( ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ , 1 ) ≥ 2 n ( n − 1 ) ∣ R G ( u ) ∣ , (119) since the right side of (119) is bounded below by ∣ R G ( u ) ∣ / ( 2 n 2 ) = β G ( u ) / 2 |R_G(u)|/(2n^2)=\beta_G(u)/2 ∣ R G ( u ) ∣/ ( 2 n 2 ) = β G ( u ) /2
Since each pair has probability ( n 2 ) − 1 = 2 / ( n ( n − 1 ) ) \binom{n}{2}^{-1}=2/(n(n-1)) ( 2 n ) − 1 = 2/ ( n ( n − 1 ))
E f min ( ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ , 1 ) = 1 n ( n − 1 ) ∑ ( a , b ) ∈ [ n ] 2 a ≠ b min ( ∣ A G ( a b ) ∣ ∣ C G ( u ) ∣ , 1 ) . (120) \mathbb{E}_f \min\left(\frac{|\mathcal{A}_G(f)|}{|C_G(u)|},1\right)=\frac{1}{n(n-1)}\sum_{\substack{(a,b)\in[n]^2\\a\ne b}}\min\left(\frac{|\mathcal{A}_G(ab)|}{|C_G(u)|},1\right). \tag*{(120)} E f min ( ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ , 1 ) = n ( n − 1 ) 1 ( a , b ) ∈ [ n ] 2 a = b ∑ min ( ∣ C G ( u ) ∣ ∣ A G ( ab ) ∣ , 1 ) . (120) We split the sum in (120) according to the cases of (6.15). The ordered pairs of distinct vertices of W W W
a 1 − ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) , a_1-\sum_{a\in W}\min\left(\frac{|R(a)|}{|C_G(u)|},1\right), a 1 − a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) , since the terms of a 1 a_1 a 1 a = b a=b a = b min ( ∣ R ( a ) ∣ / ∣ C G ( u ) ∣ , 1 ) \min(|R(a)|/|C_G(u)|,1) min ( ∣ R ( a ) ∣/∣ C G ( u ) ∣ , 1 )
The ordered pairs with one vertex in C G ( u ) C_G(u) C G ( u ) W W W
2 ∣ C G ( u ) ∣ ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) , 2|C_G(u)|\sum_{a\in W}\min\left(\frac{|R(a)|}{|C_G(u)|},1\right), 2∣ C G ( u ) ∣ a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) , since there are ∣ C G ( u ) ∣ |C_G(u)| ∣ C G ( u ) ∣ C G ( u ) C_G(u) C G ( u ) C G ( u ) C_G(u) C G ( u )
Adding these contributions, we find that the sum in (120) satisfies
∑ ( a , b ) ∈ [ n ] 2 a ≠ b min ( ∣ A G ( a b ) ∣ ∣ C G ( u ) ∣ , 1 ) = a 1 + ( 2 ∣ C G ( u ) ∣ − 1 ) ∑ a ∈ W min ( ∣ R ( a ) ∣ ∣ C G ( u ) ∣ , 1 ) ≥ a 1 + a 2 ≥ 1 2 ∣ R G ( u ) ∣ , \begin{aligned}
\sum_{\substack{(a,b)\in[n]^2\\a\ne b}}\min\left(\frac{|\mathcal{A}_G(ab)|}{|C_G(u)|},1\right)=a_1+\left(2|C_G(u)|-1\right)\sum_{a\in W}\min\left(\frac{|R(a)|}{|C_G(u)|},1\right) \\
&\ge a_1+a_2\ge\frac{1}{2}|R_G(u)|,
\end{aligned} ( a , b ) ∈ [ n ] 2 a = b ∑ min ( ∣ C G ( u ) ∣ ∣ A G ( ab ) ∣ , 1 ) = a 1 + ( 2∣ C G ( u ) ∣ − 1 ) a ∈ W ∑ min ( ∣ C G ( u ) ∣ ∣ R ( a ) ∣ , 1 ) ≥ a 1 + a 2 ≥ 2 1 ∣ R G ( u ) ∣ , where we used 2 ∣ C G ( u ) ∣ − 1 ≥ 2 ( ∣ C G ( u ) ∣ − 1 ) 2|C_G(u)|-1\ge2(|C_G(u)|-1) 2∣ C G ( u ) ∣ − 1 ≥ 2 ( ∣ C G ( u ) ∣ − 1 ) a 2 ≥ 0 a_2\ge0 a 2 ≥ 0 Lemma 6.8 in the second. Dividing by n ( n − 1 ) n(n-1) n ( n − 1 )
The pairs f f f a = b a=b a = b a 1 a_1 a 1 1 / n 1/n 1/ n M M M Λ \Lambda Λ
The potential
We now sum the one-step bound of Lemma 6.9 over time. The potential log ∣ C G m ( u ) ∣ \log|C_{G_m}(u)| log ∣ C G m ( u ) ∣ [ 0 , log n ] [0,\log n] [ 0 , log n ] Lemma 6.7 , it increases at each step at least by the increment appearing in (118) , as long as the graph remains colorable. However, Lemma 6.7 does not apply at the step at which the graph becomes non-colorable. The increment is still defined at that step, and we bound it using that C G ( u ) C_G(u) C G ( u ) A G ( f ) \mathcal{A}_G(f) A G ( f ) [ n ] [n] [ n ]
For each integer m ≥ 0 m \ge0 m ≥ 0 m < H m < H m < H
Φ m = log ∣ C G m ( u ) ∣ , Ψ m = log ( 1 + ∣ A G m ( e m + 1 ) ∣ ∣ C G m ( u ) ∣ ) . \Phi_m = \log|C_{G_m}(u)|,\qquad\Psi_m = \log\left(1+\frac{|\mathcal{A}_{G_m}(e_{m+1})|}{|C_{G_m}(u)|}\right). Φ m = log ∣ C G m ( u ) ∣ , Ψ m = log ( 1 + ∣ C G m ( u ) ∣ ∣ A G m ( e m + 1 ) ∣ ) . For m ≥ H m \ge H m ≥ H Φ m = Ψ m = 0 \Phi_m=\Psi_m=0 Φ m = Ψ m = 0 m < H m < H m < H Ψ m \Psi_m Ψ m e m + 1 e_{m+1} e m + 1 u u u
Theorem 6.10. Let u ∈ [ n ] u \in[n] u ∈ [ n ] τ = min ( H , M ) \tau=\min(H,M) τ = min ( H , M ) τ 0 \tau_0 τ 0 ( F m ) m ≥ 0 (\mathcal{F}_m)_{m\ge0} ( F m ) m ≥ 0 { 0 , 1 , 2 , … } ∪ { ∞ } \{0,1,2,\ldots\}\cup\{\infty\} { 0 , 1 , 2 , … } ∪ { ∞ }
E ∑ m : τ 0 ≤ m < τ β G m ( u ) ≤ 2 log 2 ( log n ) P ( τ 0 < τ ) . (121) \mathbb{E}\sum_{m:\,\tau_0\le m<\tau}\beta_{G_m}(u)\le\frac{2}{\log2}(\log n)\mathbb{P}(\tau_0<\tau). \tag*{(121)} E m : τ 0 ≤ m < τ ∑ β G m ( u ) ≤ log 2 2 ( log n ) P ( τ 0 < τ ) . (121) Proof. For m < H m < H m < H G m G_m G m 0 ≤ Φ m ≤ log n 0 \le\Phi_m \le\log n 0 ≤ Φ m ≤ log n Ψ m ≥ 0 \Psi_m \ge0 Ψ m ≥ 0 Ψ m \Psi_m Ψ m G m G_m G m e m + 1 e_{m+1} e m + 1 G m + 1 G_{m+1} G m + 1
We first prove a pathwise bound. Let m + 1 < H m+1 < H m + 1 < H G m + 1 = G m + e m + 1 G_{m+1}=G_m+e_{m+1} G m + 1 = G m + e m + 1 Lemma 6.7 gives
∣ C G m + 1 ( u ) ∣ ≥ ∣ C G m ( u ) ∣ + ∣ A G m ( e m + 1 ) ∣ , and hence Ψ m ≤ Φ m + 1 − Φ m . (122) |C_{G_{m+1}}(u)|\ge|C_{G_m}(u)|+|\mathcal{A}_{G_m}(e_{m+1})|,\qquad\text{and hence}\qquad\Psi_m\le\Phi_{m+1}-\Phi_m. \tag*{(122)} ∣ C G m + 1 ( u ) ∣ ≥ ∣ C G m ( u ) ∣ + ∣ A G m ( e m + 1 ) ∣ , and hence Ψ m ≤ Φ m + 1 − Φ m . (122) Suppose now that τ 0 < τ \tau_0<\tau τ 0 < τ τ − 1 < H \tau-1<H τ − 1 < H C G τ − 1 ( u ) C_{G_{\tau-1}}(u) C G τ − 1 ( u ) A G τ − 1 ( e τ ) \mathcal{A}_{G_{\tau-1}}(e_\tau) A G τ − 1 ( e τ ) [ n ] [n] [ n ]
Ψ τ − 1 = log ( ∣ C G τ − 1 ( u ) ∣ + ∣ A G τ − 1 ( e τ ) ∣ ) − Φ τ − 1 ≤ log n − Φ τ − 1 . \Psi_{\tau-1}=\log\left(|C_{G_{\tau-1}}(u)|+|\mathcal{A}_{G_{\tau-1}}(e_\tau)|\right)-\Phi_{\tau-1}\le\log n-\Phi_{\tau-1}. Ψ τ − 1 = log ( ∣ C G τ − 1 ( u ) ∣ + ∣ A G τ − 1 ( e τ ) ∣ ) − Φ τ − 1 ≤ log n − Φ τ − 1 . For τ 0 ≤ m ≤ τ − 2 \tau_0\le m\le\tau-2 τ 0 ≤ m ≤ τ − 2 m + 1 < τ ≤ H m+1<\tau\le H m + 1 < τ ≤ H (122) applies. Telescoping, we obtain
∑ m = τ 0 τ − 1 Ψ m ≤ ∑ m = τ 0 τ − 2 ( Φ m + 1 − Φ m ) + log n − Φ τ − 1 = log n − Φ τ 0 ≤ log n . \sum_{m=\tau_0}^{\tau-1}\Psi_m\le\sum_{m=\tau_0}^{\tau-2}(\Phi_{m+1}-\Phi_m)+\log n-\Phi_{\tau-1}=\log n-\Phi_{\tau_0}\le\log n. m = τ 0 ∑ τ − 1 Ψ m ≤ m = τ 0 ∑ τ − 2 ( Φ m + 1 − Φ m ) + log n − Φ τ − 1 = log n − Φ τ 0 ≤ log n . The sum is empty if τ 0 ≥ τ \tau_0\ge\tau τ 0 ≥ τ
∑ m : τ 0 ≤ m < τ Ψ m ≤ ( log n ) 1 { τ 0 < τ } . (123) \sum_{m:\,\tau_0\le m<\tau}\Psi_m\le(\log n)\mathbf{1}\{\tau_0<\tau\}. \tag*{(123)} m : τ 0 ≤ m < τ ∑ Ψ m ≤ ( log n ) 1 { τ 0 < τ } . (123) Next, we compare the expectations. For 0 ≤ m < M 0 \le m < M 0 ≤ m < M I m I_m I m
I m = 1 { τ 0 ≤ m < τ } . I_m = 1\{\tau_0 \le m < \tau\}. I m = 1 { τ 0 ≤ m < τ } . Since τ 0 \tau_0 τ 0 m < M m < M m < M { m < τ } \{m < \tau\} { m < τ } G m G_m G m I m I_m I m F m \mathcal{F}_m F m
The pair e m + 1 e_{m+1} e m + 1 F m \mathcal{F}_m F m { m < H } \{m < H\} { m < H } Ψ m \Psi_m Ψ m F m \mathcal{F}_m F m (118) with G = G m G = G_m G = G m I m ≤ 1 { m < H } I_m \le1\{m < H\} I m ≤ 1 { m < H }
E [ I m β G m ( u ) ] ≤ 2 log 2 E [ I m E [ Ψ m ∣ F m ] ] = 2 log 2 E [ I m Ψ m ] . \mathbb{E}\left[I_m\beta_{G_m}(u)\right] \le\frac{2}{\log2}\mathbb{E}\left[I_m\mathbb{E}\left[\Psi_m \mid\mathcal{F}_m\right]\right] = \frac{2}{\log2}\mathbb{E}\left[I_m\Psi_m\right]. E [ I m β G m ( u ) ] ≤ log 2 2 E [ I m E [ Ψ m ∣ F m ] ] = log 2 2 E [ I m Ψ m ] . Since τ ≤ M \tau\le M τ ≤ M 0 ≤ m < M 0 \le m < M 0 ≤ m < M (123) , we conclude that
E ∑ m : τ 0 ≤ m < τ β G m ( u ) = ∑ m = 0 M − 1 E [ I m β G m ( u ) ] ≤ 2 log 2 E ∑ m : τ 0 ≤ m < τ Ψ m ≤ 2 log 2 ( log n ) P ( τ 0 < τ ) . \mathbb{E}\sum_{m:\,\tau_0\le m<\tau}\beta_{G_m}(u)
=\sum_{m=0}^{M-1}\mathbb{E}\left[I_m\beta_{G_m}(u)\right]
\le\frac{2}{\log2}\mathbb{E}\sum_{m:\,\tau_0\le m<\tau}\Psi_m
\le\frac{2}{\log2}(\log n)\mathbb{P}(\tau_0<\tau). E m : τ 0 ≤ m < τ ∑ β G m ( u ) = m = 0 ∑ M − 1 E [ I m β G m ( u ) ] ≤ log 2 2 E m : τ 0 ≤ m < τ ∑ Ψ m ≤ log 2 2 ( log n ) P ( τ 0 < τ ) . This completes the proof.
The passage from the one-step bound to (121) is a standard compensator argument, in the spirit of the drift bounds for hitting and occupation times in [39 Sections 2–3] . A related use of a bounded monotone potential to control occupation, retaining terminal increments, appears in [52 Lemmas 2.2 and 4.1] . Here the potential is the logarithm of a frozen-class size, and the input specific to colorings is Lemma 6.9.
which tends to 2 / 9 2/9 2/9 n → ∞ n \to\infty n → ∞
On the other hand, a pair f f f A G ( f ) ≠ ∅ \mathcal{A}_G(f) \ne\varnothing A G ( f ) = ∅ C 1 C_1 C 1 C 2 C_2 C 2 G + f G+f G + f u u u C 1 ∪ C 2 C_1 \cup C_2 C 1 ∪ C 2 2 ( n − 1 ) / 3 2(n-1)/3 2 ( n − 1 ) /3 ( n − 1 ) / 3 (n-1)/3 ( n − 1 ) /3
E f [ log ( 1 + ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ ) 1 { G + f is colorable } ] = ( n 2 ) − 1 2 ( n − 1 ) 3 log n + 2 3 ≤ 4 log n 3 n . \mathbb{E}_f\left[\log\left(1+\frac{|\mathcal{A}_G(f)|}{|C_G(u)|}\right)1\{G+f\text{ is colorable}\}\right]
=\binom{n}{2}^{-1}\frac{2(n-1)}{3}\log\frac{n+2}{3}
\le\frac{4\log n}{3n}. E f [ log ( 1 + ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ ) 1 { G + f is colorable } ] = ( 2 n ) − 1 3 2 ( n − 1 ) log 3 n + 2 ≤ 3 n 4 log n . Let C , C ′ ≥ 0 C,C' \ge0 C , C ′ ≥ 0 n n n
β G ( u ) ≤ C E f [ log ( 1 + ∣ A G ( f ) ∣ ∣ C G ( u ) ∣ ) 1 { G + f is colorable } ] + C ′ log n n , \beta_G(u)\le C\mathbb{E}_f\left[\log\left(1+\frac{|\mathcal{A}_G(f)|}{|C_G(u)|}\right)1\{G+f\text{ is colorable}\}\right]+C'\frac{\log n}{n}, β G ( u ) ≤ C E f [ log ( 1 + ∣ C G ( u ) ∣ ∣ A G ( f ) ∣ ) 1 { G + f is colorable } ] + C ′ n log n , fails for large n n n Theorem 6.10 . About a third of all pairs lie inside one of the classes C 1 C_1 C 1 C 2 C_2 C 2 C 3 C_3 C 3 κ ( G ) \kappa(G) κ ( G ) 1 / 3 1/3 1/3 Lemma 3.10 (c) and (d), on the event { G m = G } \{G_m=G\} { G m = G } H − m H-m H − m F m \mathcal{F}_m F m r ( G ) ≤ 1 / κ ( G ) r(G)\le1/\kappa(G) r ( G ) ≤ 1/ κ ( G )
We apply Theorem 6.10 at the time at which the root reaches a given degree.
Corollary 6.12. For all u ∈ [ n ] u\in[n] u ∈ [ n ] j ≥ 1 j\ge1 j ≥ 1
∑ m = 0 M − 1 E [ 1 { deg G m ( u ) ≥ j } β G m ( u ) ] ≤ 2 log 2 ( log n ) ( 2 e Λ j ) j . (124) \sum_{m=0}^{M-1}\mathbb{E}\left[1\{\deg_{G_m}(u)\ge j\}\beta_{G_m}(u)\right]\le\frac{2}{\log2}(\log n)\left(\frac{2e\Lambda}{j}\right)^j. \tag*{(124)} m = 0 ∑ M − 1 E [ 1 { deg G m ( u ) ≥ j } β G m ( u ) ] ≤ log 2 2 ( log n ) ( j 2 e Λ ) j . (124) Proof. We may assume that M ≥ 1 M\ge1 M ≥ 1 τ = min ( H , M ) \tau=\min(H,M) τ = min ( H , M ) τ 0 \tau_0 τ 0 m ≥ 0 m\ge0 m ≥ 0 deg G m ( u ) ≥ j \deg_{G_m}(u)\ge j deg G m ( u ) ≥ j τ 0 = ∞ \tau_0=\infty τ 0 = ∞ τ 0 \tau_0 τ 0
Since degrees do not decrease in m m m deg G m ( u ) ≥ j \deg_{G_m}(u)\ge j deg G m ( u ) ≥ j τ 0 ≤ m \tau_0\le m τ 0 ≤ m β G m ( u ) = 0 \beta_{G_m}(u)=0 β G m ( u ) = 0 m ≥ H m\ge H m ≥ H (124) is the left side of (121) . By Theorem 6.10 , it is bounded by ( 2 / log 2 ) ( log n ) P ( τ 0 < τ ) (2/\log2)(\log n)\mathbb{P}(\tau_0<\tau) ( 2/ log 2 ) ( log n ) P ( τ 0 < τ )
On the event { τ 0 < τ } \{\tau_0<\tau\} { τ 0 < τ } τ 0 ≤ M − 1 \tau_0\le M-1 τ 0 ≤ M − 1 deg G M − 1 ( u ) ≥ j ≥ 1 \deg_{G_{M-1}}(u)\ge j\ge1 deg G M − 1 ( u ) ≥ j ≥ 1 M = 1 M=1 M = 1 M ≥ 2 M\ge2 M ≥ 2 deg G M − 1 ( u ) \deg_{G_{M-1}}(u) deg G M − 1 ( u ) Bin ( M − 1 , 2 / n ) \operatorname{Bin}(M-1,2/n) Bin ( M − 1 , 2/ n ) Lemma 3.15 (a) and M − 1 ≤ Λ n M-1\le\Lambda n M − 1 ≤ Λ n
P ( τ 0 < τ ) ≤ P ( Bin ( M − 1 , 2 / n ) ≥ j ) ≤ ( 2 e ( M − 1 ) n j ) j ≤ ( 2 e Λ j ) j , \mathbb{P}(\tau_0<\tau)\le\mathbb{P}(\operatorname{Bin}(M-1,2/n)\ge j)\le\left(\frac{2e(M-1)}{nj}\right)^j\le\left(\frac{2e\Lambda}{j}\right)^j, P ( τ 0 < τ ) ≤ P ( Bin ( M − 1 , 2/ n ) ≥ j ) ≤ ( nj 2 e ( M − 1 ) ) j ≤ ( j 2 e Λ ) j , which completes the proof.
Combining the estimates
We now combine Proposition 6.5, Lemma 6.4(b), and Corollary 6.12.
Theorem 6.13. Let n ≥ 3 n \ge3 n ≥ 3 w 0 ≥ K w_0 \ge K w 0 ≥ K K K K w 0 w_0 w 0
P ( H ≤ M , max v ∈ e H deg Γ ( v ) > D n ) ≤ 8 3 4 − ( D n + 1 − w 0 ) + 32 Λ 3 log 2 ( n n − 1 ) 2 log n w 0 − K + 2 ( 2 e Λ w 0 ) w 0 . (125) \mathbb{P}\left(H \le M,\ \max_{v\in e_H}\deg_{\Gamma}(v)>D_n\right)
\le\frac{8}{3}4^{-(D_n+1-w_0)}+\frac{32\Lambda}{3\log2}\left(\frac{n}{n-1}\right)^2\frac{\log n}{w_0-K+2}\left(\frac{2e\Lambda}{w_0}\right)^{w_0}. \tag*{(125)} P ( H ≤ M , v ∈ e H max deg Γ ( v ) > D n ) ≤ 3 8 4 − ( D n + 1 − w 0 ) + 3 log 2 32Λ ( n − 1 n ) 2 w 0 − K + 2 log n ( w 0 2 e Λ ) w 0 . (125) Further, there exists n 0 n_0 n 0 Λ \Lambda Λ n ≥ n 0 n \ge n_0 n ≥ n 0 ( log n ) − 3 / 2 (\log n)^{-3/2} ( log n ) − 3/2 n ≥ n 0 n \ge n_0 n ≥ n 0 0 ≤ L ′ ≤ U ′ ≤ M 0 \le L' \le U' \le M 0 ≤ L ′ ≤ U ′ ≤ M
P ( L ′ < H ≤ U ′ , min v ∈ e H deg Γ ( v ) > D n ) ≤ ( log n ) − 3 / 2 . (126) \mathbb{P}\left(L'<H\le U',\ \min_{v\in e_H}\deg_{\Gamma}(v)>D_n\right)\le(\log n)^{-3/2}. \tag*{(126)} P ( L ′ < H ≤ U ′ , v ∈ e H min deg Γ ( v ) > D n ) ≤ ( log n ) − 3/2 . (126) Proof. We bound the pivotal sum in (6.11). Fix u ∈ [ n ] u\in[n] u ∈ [ n ] w 0 + 1 ≤ j ≤ m ≤ M − 1 w_0+1\le j\le m\le M-1 w 0 + 1 ≤ j ≤ m ≤ M − 1 j ≥ K + 1 j\ge K+1 j ≥ K + 1 j − K + 1 ≥ w 0 − K + 2 j-K+1\ge w_0-K+2 j − K + 1 ≥ w 0 − K + 2 m ≤ M ≤ Λ n m\le M\le\Lambda n m ≤ M ≤ Λ n
E K j p i v ( G m ) ≤ 2 Λ w 0 − K + 2 ( n n − 1 ) 2 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . \mathbb{E}K_j^{\mathrm{piv}}(G_m)\le\frac{2\Lambda}{w_0-K+2}\left(\frac{n}{n-1}\right)^2\mathbb{E}\left[1\{\deg_{G_{m-1}}(u)=j-1\}\beta_{G_{m-1}}(u)\right]. E K j piv ( G m ) ≤ w 0 − K + 2 2Λ ( n − 1 n ) 2 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] . For fixed m m m { deg G m − 1 ( u ) = j − 1 } \{\deg_{G_{m-1}}(u)=j-1\} { deg G m − 1 ( u ) = j − 1 } w 0 + 1 ≤ j ≤ m w_0+1\le j\le m w 0 + 1 ≤ j ≤ m { deg G m − 1 ( u ) ≥ w 0 } \{\deg_{G_{m-1}}(u)\ge w_0\} { deg G m − 1 ( u ) ≥ w 0 }
∑ j = w 0 + 1 M − 1 ∑ m = j M − 1 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] ≤ ∑ m = 0 M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] ≤ 2 log 2 ( log n ) ( 2 e Λ w 0 ) w 0 , \begin{aligned}
\sum_{j=w_0+1}^{M-1}\sum_{m=j}^{M-1}\mathbb{E}\left[1\{\deg_{G_{m-1}}(u)=j-1\}\beta_{G_{m-1}}(u)\right]
\le\sum_{m=0}^{M-1}\mathbb{E}\left[1\{\deg_{G_m}(u)\ge w_0\}\beta_{G_m}(u)\right] \\
&\le\frac{2}{\log2}(\log n)\left(\frac{2e\Lambda}{w_0}\right)^{w_0},
\end{aligned} j = w 0 + 1 ∑ M − 1 m = j ∑ M − 1 E [ 1 { deg G m − 1 ( u ) = j − 1 } β G m − 1 ( u ) ] ≤ m = 0 ∑ M − 1 E [ 1 { deg G m ( u ) ≥ w 0 } β G m ( u ) ] ≤ log 2 2 ( log n ) ( w 0 2 e Λ ) w 0 , where the last inequality is Corollary 6.12 with j = w 0 ≥ 1 j=w_0\ge1 j = w 0 ≥ 1 ( 8 / 3 ) ⋅ 2 Λ ⋅ ( 2 / log 2 ) = 32 Λ / ( 3 log 2 ) (8/3)\cdot2\Lambda\cdot(2/\log2)=32\Lambda/(3\log2) ( 8/3 ) ⋅ 2Λ ⋅ ( 2/ log 2 ) = 32Λ/ ( 3 log 2 )
We now show that the right side of (6.25) is small for large n n n
D n + 1 − w 0 ≥ D n 2 + 1 ≥ ( log n ) 1 / 4 2 , w 0 ≥ D n − 1 2 ≥ ( log n ) 1 / 4 2 − 1 , D_n+1-w_0\ge\frac{D_n}{2}+1\ge\frac{(\log n)^{1/4}}{2},
\qquad
w_0\ge\frac{D_n-1}{2}\ge\frac{(\log n)^{1/4}}{2}-1, D n + 1 − w 0 ≥ 2 D n + 1 ≥ 2 ( log n ) 1/4 , w 0 ≥ 2 D n − 1 ≥ 2 ( log n ) 1/4 − 1 , where we used D n ≥ ( log n ) 1 / 4 − 1 D_n\ge(\log n)^{1/4}-1 D n ≥ ( log n ) 1/4 − 1 ( D n − 1 ) / 2 ≤ w 0 ≤ D n / 2 (D_n-1)/2\le w_0\le D_n/2 ( D n − 1 ) /2 ≤ w 0 ≤ D n /2 ( 8 / 3 ) 2 − ( log n ) 1 / 4 (8/3)2^{-(\log n)^{1/4}} ( 8/3 ) 2 − ( l o g n ) 1/4 w 0 → ∞ w_0\to\infty w 0 → ∞ n n n w 0 ≥ max ( K , 2 e 2 Λ ) w_0\ge\max(K,2e^2\Lambda) w 0 ≥ max ( K , 2 e 2 Λ ) log ( w 0 / ( 2 e Λ ) ) ≥ 1 \log(w_0/(2e\Lambda))\ge1 log ( w 0 / ( 2 e Λ )) ≥ 1
( 2 e Λ w 0 ) w 0 = exp ( − w 0 log w 0 2 e Λ ) ≤ e − w 0 ≤ e 1 − ( log n ) 1 / 4 / 2 . (127) \left(\frac{2e\Lambda}{w_0}\right)^{w_0}
=\exp\left(-w_0\log\frac{w_0}{2e\Lambda}\right)
\le e^{-w_0}
\le e^{1-(\log n)^{1/4}/2}. \tag*{(127)} ( w 0 2 e Λ ) w 0 = exp ( − w 0 log 2 e Λ w 0 ) ≤ e − w 0 ≤ e 1 − ( l o g n ) 1/4 /2 . (127) Using also w 0 − K + 2 ≥ 2 w_0-K+2\ge2 w 0 − K + 2 ≥ 2 ( n / ( n − 1 ) ) 2 ≤ 9 / 4 \left(n/(n-1)\right)^2\le9/4 ( n / ( n − 1 ) ) 2 ≤ 9/4 (125) by ( 12 e Λ / log 2 ) ( log n ) e − ( log n ) 1 / 4 / 2 (12e\Lambda/\log2)(\log n)e^{-(\log n)^{1/4}/2} ( 12 e Λ/ log 2 ) ( log n ) e − ( l o g n ) 1/4 /2 x 5 / 2 e − x 1 / 4 / 2 → 0 x^{5/2}e^{-x^{1/4}/2}\to0 x 5/2 e − x 1/4 /2 → 0 x 3 / 2 2 − x 1 / 4 → 0 x^{3/2}2^{-x^{1/4}}\to0 x 3/2 2 − x 1/4 → 0 x → ∞ x\to\infty x → ∞ ( log n ) − 3 / 2 / 2 (\log n)^{-3/2}/2 ( log n ) − 3/2 /2 n n n Λ \Lambda Λ
Finally, let 0 ≤ L ′ ≤ U ′ ≤ M 0\le L'\le U'\le M 0 ≤ L ′ ≤ U ′ ≤ M (126) is contained in the event in (125) , since H ≤ U ′ ≤ M H\le U'\le M H ≤ U ′ ≤ M (126) .
Remark 6.14. For q = 2 q=2 q = 2 κ j p i v = 0 \kappa_j^{\mathrm{piv}}=0 κ j piv = 0 j > K j>K j > K q = 2 q=2 q = 2 G G G u ∼ G v u\sim_G v u ∼ G v u ≠ v u\ne v u = v u u u v v v G G G e e e u u u u u u v v v G − e G-e G − e G − e G-e G − e G − e G-e G − e G G G u ∼ G − e v u\sim_{G-e}v u ∼ G − e v u u u u u u v v v u u u u u u u u u ( u , v ) (u,v) ( u , v ) deg G ( u ) = j > K \deg_G(u)=j>K deg G ( u ) = j > K j − 1 ≥ K j-1\ge K j − 1 ≥ K u u u κ j p i v ( G ) = 0 \kappa_j^{\mathrm{piv}}(G)=0 κ j piv ( G ) = 0 w 0 ≥ K w_0\ge K w 0 ≥ K Proposition 6.5 then vanish, which leaves the bound ( 8 / 3 ) 4 − ( D n + 1 − w 0 ) (8/3)4^{-(D_n+1-w_0)} ( 8/3 ) 4 − ( D n + 1 − w 0 ) Theorem 6.13 does not use q ≥ 3 q\ge3 q ≥ 3
Proof of the main theorems
In this section, we prove the window bound of Theorem 7.1 and deduce Theorems 1.1 and 1.2 from it. The window bound combines the reduction of § 5 with the estimates of Sections 5 and § 7 , and the deduction uses the convergence criterion of Theorem 4.1 , which is proved in Appendix § 9 . We end with three remarks on the scope of the argument.
Parameters and the window bound
Throughout this section, we retain the fixed integer q ≥ 3 q \ge3 q ≥ 3 Λ = γ q + 2 \Lambda= \gamma_q + 2 Λ = γ q + 2
γ q = log q log ( q / ( q − 1 ) ) , Λ = γ q + 2 , M = ⌊ Λ n ⌋ , η 0 = e − 2 Λ 8 , A q = 2 q log q + 2 q + 1 , K = ⌈ 8 Λ ⌉ , ε 1 = ε 2 = 1 , x n = n ε n 2 , ε n = ( log n ) − 3 / 2 , k n = ⌈ log n log log n ⌉ , D n = ⌊ ( log n ) 1 / 4 ⌋ , w 0 = ⌊ D n 2 ⌋ , (128) \begin{aligned}
\gamma_q &= \frac{\log q}{\log(q/(q-1))}, \qquad\Lambda= \gamma_q + 2, \qquad M = \lfloor\Lambda n \rfloor, \qquad\eta_0 = \frac{e^{-2\Lambda}}{8},\\
A_q &= 2q\log q + 2q + 1, \qquad K = \lceil8\Lambda\rceil, \qquad\varepsilon_1 = \varepsilon_2 = 1, \qquad x_n = n\varepsilon_n^2,\\
\varepsilon_n &= (\log n)^{-3/2}, \qquad k_n = \left\lceil\frac{\log n}{\log\log n}\right\rceil, \qquad D_n = \left\lfloor(\log n)^{1/4}\right\rfloor, \qquad w_0 = \left\lfloor\frac{D_n}{2}\right\rfloor, \tag*{(128)}
\end{aligned} γ q A q ε n = log ( q / ( q − 1 )) log q , Λ = γ q + 2 , M = ⌊ Λ n ⌋ , η 0 = 8 e − 2Λ , = 2 q log q + 2 q + 1 , K = ⌈ 8Λ ⌉ , ε 1 = ε 2 = 1 , x n = n ε n 2 , = ( log n ) − 3/2 , k n = ⌈ log log n log n ⌉ , D n = ⌊ ( log n ) 1/4 ⌋ , w 0 = ⌊ 2 D n ⌋ , (128) where the last line is for n ≥ 3 n \ge3 n ≥ 3 q q q η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L n ( η ) L_n(\eta) L n ( η ) U n ( η ) U_n(\eta) U n ( η ) w n ( η ) w_n(\eta) w n ( η ) Definition 3.3 for this value of Λ \Lambda Λ γ q \gamma_q γ q A q A_q A q Lemma 3.10 , ε n \varepsilon_n ε n Proposition 4.6 , x n = n ε n 2 x_n = n\varepsilon_n^2 x n = n ε n 2 k n k_n k n D n D_n D n K K K w 0 w_0 w 0 Proposition 6.5 and Theorem 6.13 .
Every estimate used below holds for all sufficiently large n n n n n n n q n_q n q Theorem 5.7 and, in Corollary 5.8, a comparison of the negative power n − 1 / 8 n^{-1/8} n − 1/8 log n \log n log n § 7 use that w 0 log ( w 0 / ( 2 e Λ ) ) w_0\log(w_0/(2e\Lambda)) w 0 log ( w 0 / ( 2 e Λ )) log log n \log\log n log log n q q q Λ \Lambda Λ Lemma 3.17 hold for each fixed η \eta η Theorem 7.1 depends on η \eta η
Theorem 7.1. Let q ≥ 3 q \ge3 q ≥ 3 (128) . For every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] n 0 n_0 n 0 q q q η \eta η n ≥ n 0 n \ge n_0 n ≥ n 0
w n ( η ) ≤ 6 Λ η 2 n ( log n ) − 3 / 2 . (129) w_n(\eta) \le\frac{6\Lambda}{\eta^2}n(\log n)^{-3/2}. \tag*{(129)} w n ( η ) ≤ η 2 6Λ n ( log n ) − 3/2 . (129) Proof. Fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L = L n ( η ) L=L_n(\eta) L = L n ( η ) U = U n ( η ) U=U_n(\eta) U = U n ( η ) (70) and (71) of Proposition 4.6 for all large n n n
For (70) , recall that Corollary 5.8 gives (72) for all large n n n q q q (70) is contained in the event in (72) , so (70) holds for these n n n
For (71) , note that $0 \le L \le U \le M$ by Lemma 3.4. We apply (126) with L ′ = L L'=L L ′ = L U ′ = U U'=U U ′ = U ε n = ( log n ) − 3 / 2 \varepsilon_n=(\log n)^{-3/2} ε n = ( log n ) − 3/2 n ≥ 3 n \ge3 n ≥ 3 n n n
P ( L < H ≤ U , min v ∈ E H deg Γ ( v ) > D n , r ( Γ ) > x n ) ≤ P ( L < H ≤ U , min v ∈ E H deg Γ ( v ) > D n ) ≤ ε n , \mathbb{P}\left(L < H \le U,\min_{v\in\mathcal{E}_H}\deg_{\Gamma}(v)>D_n,r(\Gamma)>x_n\right)
\le\mathbb{P}\left(L < H \le U,\min_{v\in\mathcal{E}_H}\deg_{\Gamma}(v)>D_n\right)
\le\varepsilon_n, P ( L < H ≤ U , v ∈ E H min deg Γ ( v ) > D n , r ( Γ ) > x n ) ≤ P ( L < H ≤ U , v ∈ E H min deg Γ ( v ) > D n ) ≤ ε n , which is (71) .
The proposition Proposition 4.6 now applies for all large n n n w n ( η ) ≤ ( 6 Λ / η 2 ) n ε n w_n(\eta)\le(6\Lambda/\eta^2)n\varepsilon_n w n ( η ) ≤ ( 6Λ/ η 2 ) n ε n (129) .
Proofs of the main theorems
Proof of Theorem 1.1. We apply Theorem 4.1 with δ n = ε n \delta_n=\varepsilon_n δ n = ε n log 3 > 1 \log3>1 log 3 > 1 ( ε n ) (\varepsilon_n) ( ε n ) ( 0 , 1 ] (0,1] ( 0 , 1 ] j ≥ 2 j\ge2 j ≥ 2 ⌊ e j ⌋ ≥ 3 \lfloor e^j\rfloor\ge3 ⌊ e j ⌋ ≥ 3
⌊ e j ⌋ ≥ e j − 1 ≥ e j − 1 , and hence ε ⌊ e j ⌋ ≤ ( log e j − 1 ) − 3 / 2 = ( j − 1 ) − 3 / 2 . \lfloor e^j\rfloor\ge e^j-1\ge e^{j-1},
\qquad\text{and hence}\qquad
\varepsilon_{\lfloor e^j\rfloor}\le(\log e^{j-1})^{-3/2}=(j-1)^{-3/2}. ⌊ e j ⌋ ≥ e j − 1 ≥ e j − 1 , and hence ε ⌊ e j ⌋ ≤ ( log e j − 1 ) − 3/2 = ( j − 1 ) − 3/2 . Since ε ⌊ e ⌋ = ε 2 = 1 \varepsilon_{\lfloor e\rfloor}=\varepsilon_2=1 ε ⌊ e ⌋ = ε 2 = 1
∑ j ≥ 1 ε ⌊ e j ⌋ ≤ 1 + ∑ j ≥ 2 ( j − 1 ) − 3 / 2 < ∞ . \sum_{j\ge1}\varepsilon_{\lfloor e^j\rfloor}\le1+\sum_{j\ge2}(j-1)^{-3/2}<\infty. j ≥ 1 ∑ ε ⌊ e j ⌋ ≤ 1 + j ≥ 2 ∑ ( j − 1 ) − 3/2 < ∞. Next, Theorem 7.1 shows that for every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] (56) holds with K ′ ( η ) = 6 Λ / η 2 K'(\eta)=6\Lambda/\eta^2 K ′ ( η ) = 6Λ/ η 2 n 1 ( η ) n_1(\eta) n 1 ( η )
The hypotheses of Theorem 4.1 hold, so there exists a constant α q ∈ [ 1 / 2 , γ q ] \alpha_q\in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ] G ( n , ⌊ c n ⌋ ) G(n,\lfloor cn\rfloor) G ( n , ⌊ c n ⌋) q q q c ∈ [ 0 , α q ) c\in[0,\alpha_q) c ∈ [ 0 , α q ) c > α q c>\alpha_q c > α q G ( n , d / n ) G(n,d/n) G ( n , d / n ) d q = 2 α q d_q=2\alpha_q d q = 2 α q
Proof of Theorem 1.2. Let ϵ ∈ ( 0 , 1 ) \epsilon\in(0,1) ϵ ∈ ( 0 , 1 ) η 1 = min ( ϵ / 2 , η 0 ) \eta_1=\min(\epsilon/2,\eta_0) η 1 = min ( ϵ /2 , η 0 ) m n = L n ( η 1 ) m_n=L_n(\eta_1) m n = L n ( η 1 ) Theorem 4.1 applies. Its property (1) gives m n / n → α q m_n/n\to\alpha_q m n / n → α q
We now bound the probability that H H H m n m_n m n Lemma 3.17 (a) that U n ( η 1 ) < M U_n(\eta_1)<M U n ( η 1 ) < M n n n n n n Lemma 3.4 at the level η 1 \eta_1 η 1 m = L n ( η 1 ) m=L_n(\eta_1) m = L n ( η 1 ) m = U n ( η 1 ) + 1 m=U_n(\eta_1)+1 m = U n ( η 1 ) + 1
P ( H ≤ L n ( η 1 ) ) < η 1 , P ( H > U n ( η 1 ) + 1 ) < η 1 . \mathbb{P}(H\le L_n(\eta_1))<\eta_1,\qquad\mathbb{P}(H>U_n(\eta_1)+1)<\eta_1. P ( H ≤ L n ( η 1 )) < η 1 , P ( H > U n ( η 1 ) + 1 ) < η 1 . Hence, with probability at least 1 − 2 η 1 ≥ 1 − ϵ 1-2\eta_1\ge1-\epsilon 1 − 2 η 1 ≥ 1 − ϵ L n ( η 1 ) < H ≤ U n ( η 1 ) + 1 L_n(\eta_1)<H\le U_n(\eta_1)+1 L n ( η 1 ) < H ≤ U n ( η 1 ) + 1
0 < H − m n ≤ w n ( η 1 ) + 1. 0<H-m_n\le w_n(\eta_1)+1. 0 < H − m n ≤ w n ( η 1 ) + 1. Finally, applying Theorem 7.1 at the level η 1 ∈ ( 0 , η 0 ] \eta_1\in(0,\eta_0] η 1 ∈ ( 0 , η 0 ] w n ( η 1 ) ≤ ( 6 Λ / η 1 2 ) n ( log n ) − 3 / 2 w_n(\eta_1)\le(6\Lambda/\eta_1^2)n(\log n)^{-3/2} w n ( η 1 ) ≤ ( 6Λ/ η 1 2 ) n ( log n ) − 3/2 n n n C = 6 Λ / η 1 2 + 1 C=6\Lambda/\eta_1^2+1 C = 6Λ/ η 1 2 + 1 n ( log n ) − 3 / 2 ≥ 1 n(\log n)^{-3/2}\ge1 n ( log n ) − 3/2 ≥ 1 n ≥ 3 n\ge3 n ≥ 3 n n n
P ( ∣ H − m n ∣ ≤ C n ( log n ) − 3 / 2 ) ≥ 1 − ϵ . \mathbb{P}\left(\lvert H-m_n\rvert\le Cn(\log n)^{-3/2}\right)\ge1-\epsilon. P ( ∣ H − m n ∣ ≤ C n ( log n ) − 3/2 ) ≥ 1 − ϵ . This proves Theorem 1.2, with a constant C C C q q q ϵ \epsilon ϵ
Remarks
Remark 7.2. We record where the proof uses the assumption q ≥ 3 q\ge3 q ≥ 3 Lemma 5.3 which shows that if a vertex lies in a frozen pair, then its neighbors use at least q − 1 q-1 q − 1 q − 1 ≥ 2 q-1\ge2 q − 1 ≥ 2 q ≥ 3 q\ge3 q ≥ 3 Theorem 5.7 , and the implication fails for q = 2 q=2 q = 2 Lemma 3.17 (b), which uses that a multigraph whose components each contain at most one cycle is q q q q ≥ 3 q\ge3 q ≥ 3 α q ≥ 1 / 2 \alpha_q\ge1/2 α q ≥ 1/2 q ≥ 2 q\ge2 q ≥ 2 § 7 , see Remark 6.14. Some use of q ≥ 3 q\ge3 q ≥ 3 q = 2 q=2 q = 2 d ∈ ( 0 , 1 ) d\in(0,1) d ∈ ( 0 , 1 ) G ( n , d / n ) G(n,d/n) G ( n , d / n ) e d / 2 ( ( 1 − d ) / ( 1 + d ) ) 1 / 4 ∈ ( 0 , 1 ) e^{d/2}\left((1-d)/(1+d)\right)^{1/4}\in(0,1) e d /2 ( ( 1 − d ) / ( 1 + d ) ) 1/4 ∈ ( 0 , 1 ) [57 Section 1, equation (1.5)] , and [2 p. 64] for a discussion of the case q = 2 q=2 q = 2
Remark 7.3. We now explain why the variance bound of § 5 cannot certify windows below the order n \sqrt{n} n η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L = L n ( η ) L=L_n(\eta) L = L n ( η ) U = U n ( η ) U=U_n(\eta) U = U n ( η ) Y Y Y (57) and the variables Y ( − i ) Y^{(-i)} Y ( − i ) (59) at this level; in the latter, the occurrence e i e_i e i Proposition 4.5 rests on the inequalities
η 2 w n ( η ) 2 ≤ Var ( Y ) ≤ ∑ i = 1 U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . \eta^2w_n(\eta)^2\le\operatorname{Var}(Y)\le\sum_{i=1}^{U}\mathbb{E}\left[(Y^{(-i)}-Y)^2\right]=\mathbb{E}\left[H(2r_{U-H+1}(\Gamma)-1);L<H\le U\right]. η 2 w n ( η ) 2 ≤ Var ( Y ) ≤ i = 1 ∑ U E [ ( Y ( − i ) − Y ) 2 ] = E [ H ( 2 r U − H + 1 ( Γ ) − 1 ) ; L < H ≤ U ] . The first inequality is Lemma 4.4, the second is the Efron–Stein inequality used in the proof of Lemma 4.3, and the identity is (60) . On the event { L < H ≤ U } \{L < H \le U\} { L < H ≤ U } Γ \Gamma Γ U − H + 1 ≥ 1 U - H + 1 \ge1 U − H + 1 ≥ 1 r U − H + 1 ( Γ ) ≥ 1 r_{U-H+1}(\Gamma) \ge1 r U − H + 1 ( Γ ) ≥ 1 E [ H 1 { L < H ≤ U } ] ≥ ( L + 1 ) P ( L < H ≤ U ) \mathbb{E}[H1\{L < H \le U\}] \ge(L + 1)\mathbb{P}(L < H \le U) E [ H 1 { L < H ≤ U }] ≥ ( L + 1 ) P ( L < H ≤ U ) c 0 = 1 / 4 c_0 = 1/4 c 0 = 1/4 L > n / 4 L > n/4 L > n /4 n n n P ( L < H ≤ U ) \mathbb{P}(L < H \le U) P ( L < H ≤ U ) (60) is at least of order n n n w n ( η ) w_n(\eta) w n ( η ) n \sqrt{n} n q q q n \sqrt{n} n [62 Corollary 6] . The window n ( log n ) − 3 / 2 n(\log n)^{-3/2} n ( log n ) − 3/2
Remark 7.4. Theorems 1.1 and 1.2 are asymptotic statements, and the proof gives no useful bounds at practical sizes of n n n n n n n n n w 0 ≥ K w_0 \ge K w 0 ≥ K ( log n ) 1 / 4 ≥ 2 K (\log n)^{1/4} \ge2K ( log n ) 1/4 ≥ 2 K log n ≥ ( 2 K ) 4 ≥ ( 16 Λ ) 4 \log n \ge(2K)^4 \ge(16\Lambda)^4 log n ≥ ( 2 K ) 4 ≥ ( 16Λ ) 4 α q \alpha_q α q L n ( η ) / n L_n(\eta)/n L n ( η ) / n α q \alpha_q α q α q \alpha_q α q 1 / 2 ≤ α q ≤ γ q 1/2 \le\alpha_q \le\gamma_q 1/2 ≤ α q ≤ γ q
Convergence of the threshold location
In this appendix, we prove Theorem 4.1 , and we first recall its statement. The parameter Λ > γ q + 1 \Lambda> \gamma_q + 1 Λ > γ q + 1 § 4.1 , and η 0 = e − 2 Λ / 8 \eta_0 = e^{-2\Lambda}/8 η 0 = e − 2Λ /8 ( δ n ) n ≥ 1 (\delta_n)_{n \ge1} ( δ n ) n ≥ 1 ( 0 , 1 ] (0,1] ( 0 , 1 ]
∑ k ≥ 1 δ ⌊ e k ⌋ < ∞ , (130) \sum_{k \ge1} \delta_{\lfloor e_k\rfloor} < \infty, \tag*{(130)} k ≥ 1 ∑ δ ⌊ e k ⌋ < ∞ , (130) and suppose that for every η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] K ′ ( η ) > 0 K'(\eta) > 0 K ′ ( η ) > 0 n 1 ( η ) n_1(\eta) n 1 ( η )
w n ( η ) ≤ K ′ ( η ) n δ n for all n ≥ n 1 ( η ) . (131) w_n(\eta) \le K'(\eta)n\delta_n \qquad\text{for all } n \ge n_1(\eta). \tag*{(131)} w n ( η ) ≤ K ′ ( η ) n δ n for all n ≥ n 1 ( η ) . (131) Theorem 4.1 asserts that, under these hypotheses, there exists α q ∈ [ 1 / 2 , γ q ] \alpha_q \in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ] η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L n ( η ) / n → α q L_n(\eta)/n \to\alpha_q L n ( η ) / n → α q U n ( η ) / n → α q U_n(\eta)/n \to\alpha_q U n ( η ) / n → α q n → ∞ n \to\infty n → ∞ c ≥ 0 c \ge0 c ≥ 0 P ( H > ⌊ c n ⌋ ) \mathbb{P}(H > \lfloor cn\rfloor) P ( H > ⌊ c n ⌋) P ( G ( n , ⌊ c n ⌋ ) is colorable ) \mathbb{P}(G(n,\lfloor cn\rfloor)\text{ is colorable}) P ( G ( n , ⌊ c n ⌋) is colorable ) c < α q c < \alpha_q c < α q c > α q c > \alpha_q c > α q d ≥ 0 d \ge0 d ≥ 0 P ( G ( n , d / n ) is colorable ) \mathbb{P}(G(n,d/n)\text{ is colorable}) P ( G ( n , d / n ) is colorable ) d < 2 α q d < 2\alpha_q d < 2 α q d > 2 α q d > 2\alpha_q d > 2 α q
The method adapts the proof of the corresponding statement for random k k k [51 Section 5] . Its main input is a product inequality for colorability probabilities at different system sizes, which is the coloring case of the interpolation of Bayati, Gamarnik, and Tetali [10 Theorem 5 and Proposition 2] , written for a Poisson number of edges as in [51 Lemma 5.2] . Translated to quantiles, it shows that the normalized lower quantile L n ( η ) / n L_n(\eta)/n L n ( η ) / n N 1 + N 2 N_1 + N_2 N 1 + N 2 N 1 N_1 N 1 N 2 N_2 N 2 [51 Lemma 5.4] . In [51 ] , the comparisons concern normalized expectations and have polynomially small errors. Our window bound concerns quantiles and gives only errors of order δ n \delta_n δ n [25 Theorem 23] and in [1 arXiv:1006.3786v1, Section 4.3, Lemma 8 and Remark 3] . This summability cannot be dropped from the tree argument (see Remark A.6). In the last step, we use the window bound at every small quantile level. All quantiles then share one limit, and the zero-one laws follow without the sharp-threshold theorem of Achlioptas and Friedgut [2 ] (see Remark 4.2).
In Section A.1, we prove the product inequality for a Poisson model with loops, and in Section A.2 we relate this model to the quantiles of V V V G ( n , m ) G(n,m) G ( n , m ) G ( n , p ) G(n,p) G ( n , p )
The product inequality
In this appendix, we also use multigraphs with loops. For a finite set S S S multigraph with loops on S S S to be a finite multiset of sets { x , y } \{x,y\} { x , y } x , y ∈ S x,y \in S x , y ∈ S x = y x=y x = y occurrences , and an occurrence { x , x } \{x,x\} { x , x } loop . We define its colorings and colorability by the same inequality as in Definition 3.1, and let Col ( G ) \operatorname{Col}(G) Col ( G ) x , y ∈ S x,y \in S x , y ∈ S G + { x , y } G+\{x,y\} G + { x , y } x = y x=y x = y S ′ ⊂ S S' \subset S S ′ ⊂ S G [ S ′ ] G[S'] G [ S ′ ] S ′ S' S ′ ∣ S ∣ ≥ 2 |S| \ge2 ∣ S ∣ ≥ 2
Definition A.1. Let n ≥ 1 n \ge1 n ≥ 1 c ≥ 0 c \ge0 c ≥ 0 ( u i , v i ) i ≥ 1 (u_i,v_i)_{i\ge1} ( u i , v i ) i ≥ 1 [ n ] × [ n ] [n] \times[n] [ n ] × [ n ] I I I Pois ( c n ) \operatorname{Pois}(cn) Pois ( c n ) A n ( c ) A_n(c) A n ( c ) [ n ] [n] [ n ] { u i , v i } \{u_i,v_i\} { u i , v i } 1 ≤ i ≤ I 1 \le i \le I 1 ≤ i ≤ I
ρ n ( c ) = P ( A n ( c ) is colorable ) . (132) \rho_n(c) = \mathbb{P}\bigl(A_n(c)\text{ is colorable}\bigr). \tag*{(132)} ρ n ( c ) = P ( A n ( c ) is colorable ) . (132) For n = 1 n=1 n = 1 A 1 ( c ) A_1(c) A 1 ( c )
ρ 1 ( c ) = P ( I = 0 ) = e − c . (133) \rho_1(c) = \mathbb{P}(I=0) = e^{-c}. \tag*{(133)} ρ 1 ( c ) = P ( I = 0 ) = e − c . (133) Lemma A.2. For all integers N 1 , N 2 ≥ 1 N_1,N_2 \ge1 N 1 , N 2 ≥ 1 c ≥ 0 c \ge0 c ≥ 0
ρ N 1 + N 2 ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) . (134) \rho_{N_1+N_2}(c) \ge\rho_{N_1}(c)\rho_{N_2}(c). \tag*{(134)} ρ N 1 + N 2 ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) . (134) Proof. Both sides of (A.5) equal 1 when c = 0 c=0 c = 0 c > 0 c>0 c > 0 N = N 1 + N 2 N=N_1+N_2 N = N 1 + N 2 S 1 = [ N 1 ] S_1=[N_1] S 1 = [ N 1 ] S 2 = [ N ] ∖ [ N 1 ] S_2=[N]\setminus[N_1] S 2 = [ N ] ∖ [ N 1 ] ( ω i ) i ≥ 1 (\omega_i)_{i\ge1} ( ω i ) i ≥ 1 ( ω i ( j ) ) i ≥ 1 (\omega_i^{(j)})_{i\ge1} ( ω i ( j ) ) i ≥ 1 j = 1 , 2 j=1,2 j = 1 , 2 [ N ] × [ N ] [N]\times[N] [ N ] × [ N ] j j j S j × S j S_j\times S_j S j × S j
For μ = ( μ 0 , μ 1 , μ 2 ) \mu=(\mu_0,\mu_1,\mu_2) μ = ( μ 0 , μ 1 , μ 2 ) μ 0 , μ 1 , μ 2 ≥ 0 \mu_0,\mu_1,\mu_2\ge0 μ 0 , μ 1 , μ 2 ≥ 0 I 0 I_0 I 0 I 1 I_1 I 1 I 2 I_2 I 2 Pois ( μ 0 ) \operatorname{Pois}(\mu_0) Pois ( μ 0 ) Pois ( μ 1 ) \operatorname{Pois}(\mu_1) Pois ( μ 1 ) Pois ( μ 2 ) \operatorname{Pois}(\mu_2) Pois ( μ 2 ) A ∗ ( μ ) A^*(\mu) A ∗ ( μ ) [ N ] [N] [ N ] { x , y } \{x,y\} { x , y } ( x , y ) (x,y) ( x , y ) ( ω i ) i ≤ I 0 (\omega_i)_{i\le I_0} ( ω i ) i ≤ I 0 ( ω i ( 1 ) ) i ≤ I 1 (\omega_i^{(1)})_{i\le I_1} ( ω i ( 1 ) ) i ≤ I 1 ( ω i ( 2 ) ) i ≤ I 2 (\omega_i^{(2)})_{i\le I_2} ( ω i ( 2 ) ) i ≤ I 2
χ ˉ ( μ ) = P ( A ∗ ( μ ) is colorable ) . \bar{\chi}(\mu)=\mathbb{P}\bigl(A^*(\mu)\text{ is colorable}\bigr). χ ˉ ( μ ) = P ( A ∗ ( μ ) is colorable ) . {#eq:A.6}
For ν ∈ [ 0 , c ] \nu\in[0,c] ν ∈ [ 0 , c ]
A ( ν ) ≔ A ∗ ( N ν , N 1 ( c − ν ) , N 2 ( c − ν ) ) , χ ( ν ) = χ ˉ ( N ν , N 1 ( c − ν ) , N 2 ( c − ν ) ) . (135) A^{(\nu)} \coloneqq A^*(N\nu,N_1(c-\nu),N_2(c-\nu)), \qquad\chi(\nu)=\bar{\chi}(N\nu,N_1(c-\nu),N_2(c-\nu)). \tag*{(135)} A ( ν ) : = A ∗ ( N ν , N 1 ( c − ν ) , N 2 ( c − ν )) , χ ( ν ) = χ ˉ ( N ν , N 1 ( c − ν ) , N 2 ( c − ν )) . (135) The occurrences contributed by ( ω i ) (\omega_i) ( ω i ) N ν N\nu N ν ( ω i ( j ) ) (\omega_i^{(j)}) ( ω i ( j ) ) N j ( c − ν ) N_j(c-\nu) N j ( c − ν ) j = 1 , 2 j=1,2 j = 1 , 2
We first identify the values of χ \chi χ A ( 0 ) A^{(0)} A ( 0 ) S 1 S_1 S 1 S 2 S_2 S 2 A ( 0 ) [ S 1 ] A^{(0)}[S_1] A ( 0 ) [ S 1 ] A ( 0 ) [ S 2 ] A^{(0)}[S_2] A ( 0 ) [ S 2 ] S 2 S_2 S 2 [ N 2 ] [N_2] [ N 2 ] A N 1 ( c ) A_{N_1}(c) A N 1 ( c ) A N 2 ( c ) A_{N_2}(c) A N 2 ( c ) A ( c ) A^{(c)} A ( c ) A N ( c ) A_N(c) A N ( c )
χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) , χ ( c ) = ρ N ( c ) , \chi(0)=\rho_{N_1}(c)\rho_{N_2}(c),\qquad\chi(c)=\rho_N(c), χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) , χ ( c ) = ρ N ( c ) , and it suffices to show that χ \chi χ [ 0 , c ] [0,c] [ 0 , c ]
Next, we compute the partial derivatives of χ ˉ \bar{\chi} χ ˉ k = ( k 0 , k 1 , k 2 ) \mathbf{k}=(k_0,k_1,k_2) k = ( k 0 , k 1 , k 2 ) a k a_{\mathbf{k}} a k ( ω i ) i ≤ k 0 (\omega_i)_{i\le k_0} ( ω i ) i ≤ k 0 ( ω i ( 1 ) ) i ≤ k 1 (\omega_i^{(1)})_{i\le k_1} ( ω i ( 1 ) ) i ≤ k 1 ( ω i ( 2 ) ) i ≤ k 2 (\omega_i^{(2)})_{i\le k_2} ( ω i ( 2 ) ) i ≤ k 2
χ ˉ ( μ ) = ∑ k π μ 0 ( k 0 ) π μ 1 ( k 1 ) π μ 2 ( k 2 ) a k = e − μ 0 − μ 1 − μ 2 ∑ k μ 0 k 0 μ 1 k 1 μ 2 k 2 k 0 ! k 1 ! k 2 ! a k . \bar{\chi}(\boldsymbol{\mu})=\sum_{\mathbf{k}}\pi_{\mu_0}(k_0)\pi_{\mu_1}(k_1)\pi_{\mu_2}(k_2)a_{\mathbf{k}}=e^{-\mu_0-\mu_1-\mu_2}\sum_{\mathbf{k}}\frac{\mu_0^{k_0}\mu_1^{k_1}\mu_2^{k_2}}{k_0!k_1!k_2!}a_{\mathbf{k}}. χ ˉ ( μ ) = k ∑ π μ 0 ( k 0 ) π μ 1 ( k 1 ) π μ 2 ( k 2 ) a k = e − μ 0 − μ 1 − μ 2 k ∑ k 0 ! k 1 ! k 2 ! μ 0 k 0 μ 1 k 1 μ 2 k 2 a k . Since ∣ a k ∣ ≤ 1 |a_{\mathbf{k}}|\le1 ∣ a k ∣ ≤ 1 R 3 \mathbb{R}^3 R 3 χ ˉ \bar{\chi} χ ˉ R 3 \mathbb{R}^3 R 3
Let ( x , y ) (x,y) ( x , y ) [ N ] × [ N ] [N]\times[N] [ N ] × [ N ] A ∗ ( μ ) A^*(\boldsymbol{\mu}) A ∗ ( μ ) μ 0 ↦ π μ 0 ( k 0 ) \mu_0\mapsto\pi_{\mu_0}(k_0) μ 0 ↦ π μ 0 ( k 0 ) π μ 0 ( k 0 − 1 ) − π μ 0 ( k 0 ) \pi_{\mu_0}(k_0-1)-\pi_{\mu_0}(k_0) π μ 0 ( k 0 − 1 ) − π μ 0 ( k 0 ) k 0 k_0 k 0 π μ 0 ( k 0 − 1 ) \pi_{\mu_0}(k_0-1) π μ 0 ( k 0 − 1 )
∂ μ 0 χ ˉ ( μ ) = ∑ k π μ 0 ( k 0 ) π μ 1 ( k 1 ) π μ 2 ( k 2 ) ( a ( k 0 + 1 , k 1 , k 2 ) − a k ) = − P ( A ∗ ( μ ) is colorable , A ∗ ( μ ) + { x , y } is not colorable ) , \begin{aligned}
\partial_{\mu_0}\bar{\chi}(\boldsymbol{\mu})=\sum_{\mathbf{k}}\pi_{\mu_0}(k_0)\pi_{\mu_1}(k_1)\pi_{\mu_2}(k_2)\left(a_{(k_0+1,k_1,k_2)}-a_{\mathbf{k}}\right) \\
&=-\mathbb{P}\left(A^*(\boldsymbol{\mu})\text{ is colorable},\ A^*(\boldsymbol{\mu})+\{x,y\}\text{ is not colorable}\right),
\end{aligned} ∂ μ 0 χ ˉ ( μ ) = k ∑ π μ 0 ( k 0 ) π μ 1 ( k 1 ) π μ 2 ( k 2 ) ( a ( k 0 + 1 , k 1 , k 2 ) − a k ) = − P ( A ∗ ( μ ) is colorable , A ∗ ( μ ) + { x , y } is not colorable ) , In the second step, we used that adding an occurrence can only destroy colorability. Repeating this computation for μ 1 \mu_1 μ 1 μ 2 \mu_2 μ 2 ( x , y ) (x,y) ( x , y ) S j × S j S_j\times S_j S j × S j [44 Theorem 19.1] in the case that we need.)
We now evaluate these probabilities. Let A A A [ N ] [N] [ N ] A A A [ N ] [N] [ N ] x , y ∈ [ N ] x,y\in[N] x , y ∈ [ N ] A + { x , y } A+\{x,y\} A + { x , y } x ∼ A y x\sim_A y x ∼ A y x = y x=y x = y A + { x , x } A+\{x,x\} A + { x , x } x ∼ A x x\sim_A x x ∼ A x A A A x ≠ y x \ne y x = y C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k A A A A + { x , y } A+\{x,y\} A + { x , y }
∑ i = 1 k ( ∣ C i ∣ N ) 2 if ( x , y ) is uniform on [ N ] × [ N ] , ∑ i = 1 k ( ∣ C i ∩ S j ∣ N j ) 2 if ( x , y ) is uniform on S j × S j , \begin{aligned}
&\sum_{i=1}^{k}\left(\frac{|C_i|}{N}\right)^2 && \text{if } (x,y) \text{ is uniform on } [N]\times[N],\\
&\sum_{i=1}^{k}\left(\frac{|C_i\cap S_j|}{N_j}\right)^2 && \text{if } (x,y) \text{ is uniform on } S_j\times S_j,
\end{aligned} i = 1 ∑ k ( N ∣ C i ∣ ) 2 i = 1 ∑ k ( N j ∣ C i ∩ S j ∣ ) 2 if ( x , y ) is uniform on [ N ] × [ N ] , if ( x , y ) is uniform on S j × S j , where the sums run over these k k k
On the event that A ( v ) A^{(v)} A ( v ) C 1 , … , C k C_1,\ldots,C_k C 1 , … , C k
χ ′ ( v ) = N ∂ μ 0 χ ˉ − N 1 ∂ μ 1 χ ˉ − N 2 ∂ μ 2 χ ˉ = E [ 1 { A ( v ) is colorable } ∑ i = 1 k ( ∣ C i ∩ S 1 ∣ 2 N 1 + ∣ C i ∩ S 2 ∣ 2 N 2 − ∣ C i ∣ 2 N ) ] , (136) \begin{aligned}
\chi'(v)&=N\partial_{\mu_0}\bar{\chi}-N_1\partial_{\mu_1}\bar{\chi}-N_2\partial_{\mu_2}\bar{\chi}\\
&=\mathbb{E}\left[\mathbf{1}\{A^{(v)}\text{ is colorable}\}\sum_{i=1}^{k}\left(\frac{|C_i\cap S_1|^2}{N_1}+\frac{|C_i\cap S_2|^2}{N_2}-\frac{|C_i|^2}{N}\right)\right], \tag*{(136)}
\end{aligned} χ ′ ( v ) = N ∂ μ 0 χ ˉ − N 1 ∂ μ 1 χ ˉ − N 2 ∂ μ 2 χ ˉ = E [ 1 { A ( v ) is colorable } i = 1 ∑ k ( N 1 ∣ C i ∩ S 1 ∣ 2 + N 2 ∣ C i ∩ S 2 ∣ 2 − N ∣ C i ∣ 2 ) ] , (136) where the partial derivatives are evaluated at ( N v , N 1 ( c − v ) , N 2 ( c − v ) ) (Nv,N_1(c-v),N_2(c-v)) ( N v , N 1 ( c − v ) , N 2 ( c − v )) a , b ≥ 0 a,b\geq0 a , b ≥ 0
a 2 N 1 + b 2 N 2 − ( a + b ) 2 N 1 + N 2 = ( N 2 a − N 1 b ) 2 N 1 N 2 ( N 1 + N 2 ) ≥ 0 , (137) \frac{a^2}{N_1}+\frac{b^2}{N_2}-\frac{(a+b)^2}{N_1+N_2}
=\frac{(N_2a-N_1b)^2}{N_1N_2(N_1+N_2)}\geq0, \tag*{(137)} N 1 a 2 + N 2 b 2 − N 1 + N 2 ( a + b ) 2 = N 1 N 2 ( N 1 + N 2 ) ( N 2 a − N 1 b ) 2 ≥ 0 , (137) as one checks by multiplying both sides by N 1 N 2 ( N 1 + N 2 ) N_1N_2(N_1+N_2) N 1 N 2 ( N 1 + N 2 ) a = ∣ C i ∩ S 1 ∣ a=|C_i\cap S_1| a = ∣ C i ∩ S 1 ∣ b = ∣ C i ∩ S 2 ∣ b=|C_i\cap S_2| b = ∣ C i ∩ S 2 ∣ χ ′ ( v ) ≥ 0 \chi'(v)\geq0 χ ′ ( v ) ≥ 0 v ∈ [ 0 , c ] v\in[0,c] v ∈ [ 0 , c ]
ρ N ( c ) = χ ( c ) ≥ χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) . \rho_N(c)=\chi(c)\geq\chi(0)=\rho_{N_1}(c)\rho_{N_2}(c). ρ N ( c ) = χ ( c ) ≥ χ ( 0 ) = ρ N 1 ( c ) ρ N 2 ( c ) . System-splitting interpolation was used by Guerra and Toninelli to prove thermodynamic limits for mean-field spin glasses [38 Section 3, Theorems 1–2] . Franz and Leone [31 Section III and appendices] and Panchenko and Talagrand [54 Sections 1–2] developed free-energy bounds for diluted models under their stated interaction and arity hypotheses. For the coloring problem, Bayati, Gamarnik, and Tetali compare the colorability probability of the global model with a given number of edges with that of the two parts [10 Theorem 5] , and their proof of [10 Proposition 2] uses, as above, the classes of vertices that receive the same color in every coloring. Averaging their comparison over a Poisson number of edges gives (A.5). The Poisson form and its proof by differentiation in the means are those of [51 Lemma 5.2] for k k k x ↦ x k x\mapsto x^k x ↦ x k
The Poisson model and the quantiles
We now relate ρ n ( c ) \rho_n(c) ρ n ( c ) H H H [ n ] [n] [ n ] n ≥ 1 n \ge1 n ≥ 1
Δ n = ⌈ n log n ⌉ . (138) \Delta_n = \lceil\sqrt{n}\log n\rceil. \tag*{(138)} Δ n = ⌈ n log n ⌉ . (138) Lemma A.3. (a) For all integers n ≥ 2 q n \ge2q n ≥ 2 q c ≥ 0 c \ge0 c ≥ 0
ρ n ( c ) = e − c ∑ j ≥ 0 π c ( n − 1 ) ( j ) P ( H > j ) . (139) \rho_n(c) = e^{-c}\sum_{j\ge0}\pi_{c(n-1)}(j)\mathbb{P}(H > j). \tag*{(139)} ρ n ( c ) = e − c j ≥ 0 ∑ π c ( n − 1 ) ( j ) P ( H > j ) . (139) (b) Let η ∈ ( 0 , 1 / 2 ] \eta\in(0,1/2] η ∈ ( 0 , 1/2 ] n n n c ∈ ( 0 , Λ ] c \in(0,\Lambda] c ∈ ( 0 , Λ ] c ( n − 1 ) ≤ L n ( η ) − Δ n c(n-1) \le L_n(\eta)-\Delta_n c ( n − 1 ) ≤ L n ( η ) − Δ n
ρ n ( c ) ≥ e − Λ ( 1 − η ) ( 1 − e − ( log n ) 2 / ( 2 Λ ) ) , (140) \rho_n(c) \ge e^{-\Lambda}(1-\eta)\left(1-e^{-(\log n)^2/(2\Lambda)}\right), \tag*{(140)} ρ n ( c ) ≥ e − Λ ( 1 − η ) ( 1 − e − ( l o g n ) 2 / ( 2Λ ) ) , (140) and if c ( n − 1 ) ≥ U n ( η ) + Δ n c(n-1) \ge U_n(\eta)+\Delta_n c ( n − 1 ) ≥ U n ( η ) + Δ n
ρ n ( c ) ≤ η + e − ( log n ) 2 / ( 2 Λ ) . (141) \rho_n(c) \le\eta+e^{-(\log n)^2/(2\Lambda)}. \tag*{(141)} ρ n ( c ) ≤ η + e − ( l o g n ) 2 / ( 2Λ ) . (141) Proof. We begin with (a). Each occurrence of A n ( c ) A_n(c) A n ( c ) 1 / n 1/n 1/ n E = [ n ] × [ n ] E=[n]\times[n] E = [ n ] × [ n ] ν \nu ν μ = c n \mu=cn μ = c n E E E { ( x , x ) : x ∈ [ n ] } \{(x,x):x\in[n]\} {( x , x ) : x ∈ [ n ]} A n ( c ) A_n(c) A n ( c ) Pois ( c ) \operatorname{Pois}(c) Pois ( c ) ( u i , v i ) (u_i,v_i) ( u i , v i ) u i ≠ v i u_i\ne v_i u i = v i Pois ( c ( n − 1 ) ) \operatorname{Pois}(c(n-1)) Pois ( c ( n − 1 )) { ( x , y ) ∈ [ n ] 2 : x ≠ y } \{(x,y)\in[n]^2:x\ne y\} {( x , y ) ∈ [ n ] 2 : x = y } ( x , y ) (x,y) ( x , y ) { x , y } \{x,y\} { x , y } [ n ] [n] [ n ]
Conditionally on the number j j j G j G_j G j A n ( c ) A_n(c) A n ( c )
ρ n ( c ) = e − c ∑ j ≥ 0 π c ( n − 1 ) ( j ) P ( G j is colorable ) , \rho_n(c)=e^{-c}\sum_{j\ge0}\pi_{c(n-1)}(j)\mathbb{P}(G_j\text{ is colorable}), ρ n ( c ) = e − c j ≥ 0 ∑ π c ( n − 1 ) ( j ) P ( G j is colorable ) , which is (A.10).
We now prove (b). We set μ = c ( n − 1 ) \mu=c(n-1) μ = c ( n − 1 ) J J J Pois ( μ ) \operatorname{Pois}(\mu) Pois ( μ ) L = L n ( η ) L=L_n(\eta) L = L n ( η ) U = U n ( η ) U=U_n(\eta) U = U n ( η ) μ ≤ L − Δ n \mu\le L-\Delta_n μ ≤ L − Δ n Lemma 3.4 that P ( H > j ) > 1 − η \mathbb{P}(H>j)>1-\eta P ( H > j ) > 1 − η j ≤ L j\le L j ≤ L
ρ n ( c ) ≥ e − c ( 1 − η ) ∑ j ≤ L π μ ( j ) = e − c ( 1 − η ) P ( J ≤ L ) ≥ e − Λ ( 1 − η ) ( 1 − P ( J ≥ μ + Δ n ) ) , \rho_n(c)\ge e^{-c}(1-\eta)\sum_{j\le L}\pi_\mu(j)=e^{-c}(1-\eta)\mathbb{P}(J\le L)\ge e^{-\Lambda}(1-\eta)\left(1-\mathbb{P}(J\ge\mu+\Delta_n)\right), ρ n ( c ) ≥ e − c ( 1 − η ) j ≤ L ∑ π μ ( j ) = e − c ( 1 − η ) P ( J ≤ L ) ≥ e − Λ ( 1 − η ) ( 1 − P ( J ≥ μ + Δ n ) ) , where we used c ≤ Λ c\le\Lambda c ≤ Λ { J > L } ⊂ { J ≥ μ + Δ n } \{J>L\}\subset\{J\ge\mu+\Delta_n\} { J > L } ⊂ { J ≥ μ + Δ n } μ + Δ n ≤ L ≤ M ≤ Λ n \mu+\Delta_n\le L\le M\le\Lambda n μ + Δ n ≤ L ≤ M ≤ Λ n Δ n 2 ≥ n ( log n ) 2 \Delta_n^2\ge n(\log n)^2 Δ n 2 ≥ n ( log n ) 2 Lemma 3.15 (c) gives
P ( J ≥ μ + Δ n ) ≤ exp ( − Δ n 2 2 ( μ + Δ n ) ) ≤ exp ( − n ( log n ) 2 2 Λ n ) = e − ( log n ) 2 / ( 2 Λ ) . \mathbb{P}(J\ge\mu+\Delta_n)\le\exp\left(-\frac{\Delta_n^2}{2(\mu+\Delta_n)}\right)\le\exp\left(-\frac{n(\log n)^2}{2\Lambda n}\right)=e^{-(\log n)^2/(2\Lambda)}. P ( J ≥ μ + Δ n ) ≤ exp ( − 2 ( μ + Δ n ) Δ n 2 ) ≤ exp ( − 2Λ n n ( log n ) 2 ) = e − ( l o g n ) 2 / ( 2Λ ) . This proves (A.11).
Suppose now that μ ≥ U + Δ n \mu\ge U + \Delta_n μ ≥ U + Δ n Lemma 3.17 (a), we have U + 1 ≤ M U + 1 \le M U + 1 ≤ M n n n n n n Lemma 3.4 and the monotonicity of P ( H > j ) \mathbb{P}(H > j) P ( H > j ) j j j P ( H > j ) < η \mathbb{P}(H > j) < \eta P ( H > j ) < η j ≥ U + 1 j \ge U + 1 j ≥ U + 1 P ( H > j ) \mathbb{P}(H > j) P ( H > j ) 1 1 1 j ≤ U j \le U j ≤ U η \eta η j > U j > U j > U e − c ≤ 1 e^{-c} \le1 e − c ≤ 1 0 < μ ≤ Λ n 0 < \mu\le\Lambda n 0 < μ ≤ Λ n Lemma 3.15 (c) then gives
ρ n ( c ) ≤ P ( J ≤ U ) + η ≤ P ( J ≤ μ − Δ n ) + η ≤ exp ( − Δ n 2 2 μ ) + η ≤ e − ( log n ) 2 / ( 2 Λ ) + η , \rho_n(c) \le\mathbb{P}(J \le U) + \eta\le\mathbb{P}(J \le\mu- \Delta_n) + \eta\le\exp\left(-\frac{\Delta_n^2}{2\mu}\right) + \eta\le e^{-(\log n)^2/(2\Lambda)} + \eta, ρ n ( c ) ≤ P ( J ≤ U ) + η ≤ P ( J ≤ μ − Δ n ) + η ≤ exp ( − 2 μ Δ n 2 ) + η ≤ e − ( l o g n ) 2 / ( 2Λ ) + η , which is (A.12). The only condition on n n n U + 1 ≤ M U + 1 \le M U + 1 ≤ M c c c
Comparison of the quantiles at different sizes
In this section, we fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] K ′ = K ′ ( η ) K' = K'(\eta) K ′ = K ′ ( η ) n 1 = n 1 ( η ) n_1 = n_1(\eta) n 1 = n 1 ( η ) n ≥ 2 q n \ge2q n ≥ 2 q ζ n = L n ( η ) / n \zeta_n = L_n(\eta)/n ζ n = L n ( η ) / n Lemma 3.4 that L n ( η ) ≤ M L_n(\eta) \le M L n ( η ) ≤ M 0 ≤ ζ n ≤ Λ 0 \le\zeta_n \le\Lambda 0 ≤ ζ n ≤ Λ ( δ n ) (\delta_n) ( δ n ) δ ( u ) = δ ⌊ u ⌋ \delta(u) = \delta_{\lfloor u\rfloor} δ ( u ) = δ ⌊ u ⌋ u ≥ 1 u \ge1 u ≥ 1
g ( u ) = K ′ δ ( u ) + ( Λ + 6 ) log u u , u ≥ 1. (142) g(u) = K'\delta(u) + (\Lambda+ 6)\frac{\log u}{\sqrt{u}}, \qquad u \ge1. \tag*{(142)} g ( u ) = K ′ δ ( u ) + ( Λ + 6 ) u log u , u ≥ 1. (142) Lemma A.4. There exists an integer n 0 ≥ 2 q n_0 \ge2q n 0 ≥ 2 q N 1 , N 2 ≥ n 0 N_1, N_2 \ge n_0 N 1 , N 2 ≥ n 0
ζ N 1 + N 2 ≥ min ( ζ N 1 , ζ N 2 ) − g ( min ( N 1 , N 2 ) ) , ζ N 1 + 1 ≥ ζ N 1 − g ( N 1 ) . (143) \zeta_{N_1+N_2} \ge\min(\zeta_{N_1},\zeta_{N_2}) - g(\min(N_1,N_2)), \qquad\zeta_{N_1+1} \ge\zeta_{N_1} - g(N_1). \tag*{(143)} ζ N 1 + N 2 ≥ min ( ζ N 1 , ζ N 2 ) − g ( min ( N 1 , N 2 )) , ζ N 1 + 1 ≥ ζ N 1 − g ( N 1 ) . (143) The integer n 0 n_0 n 0 q q q Λ \Lambda Λ η \eta η K ′ K' K ′ n 1 n_1 n 1 ( δ n ) (\delta_n) ( δ n )
Proof. We first choose n 0 n_0 n 0 v ≥ 1 v \ge1 v ≥ 1 Δ v ≤ v log v + 1 \Delta_v \le\sqrt{v}\log v + 1 Δ v ≤ v log v + 1
Δ v v ≤ log v v + 1 v , (144) \frac{\Delta_v}{v} \le\frac{\log v}{\sqrt{v}} + \frac{1}{v}, \tag*{(144)} v Δ v ≤ v log v + v 1 , (144) where the right side is nonincreasing in v ≥ e 2 v \ge e^2 v ≥ e 2 ϑ n = e − ( log n ) 2 / ( 2 Λ ) \vartheta_n = e^{-(\log n)^2/(2\Lambda)} ϑ n = e − ( l o g n ) 2 / ( 2Λ ) Lemma 3.17 (b) with c 0 = 1 / 4 c_0 = 1/4 c 0 = 1/4 ϑ n → 0 \vartheta_n \to0 ϑ n → 0 n 0 ≥ max ( 2 q , n 1 , e 2 ) n_0 \ge\max(2q,n_1,e^2) n 0 ≥ max ( 2 q , n 1 , e 2 ) Lemma A.3 (b) holds at every size n ≥ n 0 n \ge n_0 n ≥ n 0
ζ n > 1 4 , 2 Δ n n < 1 4 , ϑ n ≤ min ( η , 1 8 ) for all n ≥ n 0 . (145) \zeta_n > \frac{1}{4}, \qquad\frac{2\Delta_n}{n} < \frac{1}{4}, \qquad\vartheta_n \le\min\left(\eta,\frac{1}{8}\right) \quad\text{for all } n \ge n_0. \tag*{(145)} ζ n > 4 1 , n 2 Δ n < 4 1 , ϑ n ≤ min ( η , 8 1 ) for all n ≥ n 0 . (145) We begin with the first inequality in (A.14). Let N 1 , N 2 ≥ n 0 N_1,N_2 \ge n_0 N 1 , N 2 ≥ n 0 N = N 1 + N 2 N = N_1 + N_2 N = N 1 + N 2 N min = min ( N 1 , N 2 ) N_{\min} = \min(N_1,N_2) N m i n = min ( N 1 , N 2 )
c = min ( ζ N 1 − 2 Δ N 1 N 1 , ζ N 2 − 2 Δ N 2 N 2 ) . c = \min\left(\zeta_{N_1} - \frac{2\Delta_{N_1}}{N_1}, \zeta_{N_2} - \frac{2\Delta_{N_2}}{N_2}\right). c = min ( ζ N 1 − N 1 2 Δ N 1 , ζ N 2 − N 2 2 Δ N 2 ) . The bounds (A.16) give c > 0 c > 0 c > 0 c ≤ ξ ~ N 1 ≤ Λ c \le\widetilde{\xi}_{N_1} \le\Lambda c ≤ ξ N 1 ≤ Λ j = 1 , 2 j = 1,2 j = 1 , 2
c ( N j − 1 ) ≤ c N j ≤ L N j ( η ) − 2 Δ N j ≤ L N j ( η ) − Δ N j . c(N_j - 1) \le cN_j \le L_{N_j}(\eta) - 2\Delta_{N_j} \le L_{N_j}(\eta) - \Delta_{N_j}. c ( N j − 1 ) ≤ c N j ≤ L N j ( η ) − 2 Δ N j ≤ L N j ( η ) − Δ N j . Therefore (A.11) applies at size N j N_j N j η ≤ η 0 ≤ 1 / 8 \eta\le\eta_0 \le1/8 η ≤ η 0 ≤ 1/8
ρ N j ( c ) ≥ e − Λ ( 1 − η ) ( 1 − ϑ N j ) ≥ e − Λ ( 7 8 ) 2 . \rho_{N_j}(c) \ge e^{-\Lambda}(1-\eta)(1-\vartheta_{N_j}) \ge e^{-\Lambda}\left(\frac{7}{8}\right)^2. ρ N j ( c ) ≥ e − Λ ( 1 − η ) ( 1 − ϑ N j ) ≥ e − Λ ( 8 7 ) 2 . By Lemma A.2, we obtain
ρ N ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) ≥ e − 2 Λ ( 7 8 ) 4 > e − 2 Λ 2 . (146) \rho_N(c) \ge\rho_{N_1}(c)\rho_{N_2}(c) \ge e^{-2\Lambda}\left(\frac{7}{8}\right)^4 > \frac{e^{-2\Lambda}}{2}. \tag*{(146)} ρ N ( c ) ≥ ρ N 1 ( c ) ρ N 2 ( c ) ≥ e − 2Λ ( 8 7 ) 4 > 2 e − 2Λ . (146) However, recall that η 0 = e − 2 Λ / 8 \eta_0 = e^{-2\Lambda}/8 η 0 = e − 2Λ /8
η + ϑ N ≤ 2 η ≤ 2 η 0 = e − 2 Λ 4 . \eta+ \vartheta_N \le2\eta\le2\eta_0 = \frac{e^{-2\Lambda}}{4}. η + ϑ N ≤ 2 η ≤ 2 η 0 = 4 e − 2Λ . Suppose that c ( N − 1 ) ≥ U N ( η ) + Δ N c(N-1) \ge U_N(\eta) + \Delta_N c ( N − 1 ) ≥ U N ( η ) + Δ N N ≥ n 0 N \ge n_0 N ≥ n 0 ρ N ( c ) ≤ e − 2 Λ / 4 \rho_N(c) \le e^{-2\Lambda}/4 ρ N ( c ) ≤ e − 2Λ /4 c ( N − 1 ) < U N ( η ) + Δ N c(N-1) < U_N(\eta) + \Delta_N c ( N − 1 ) < U N ( η ) + Δ N N ≥ n 1 N \ge n_1 N ≥ n 1
U N ( η ) = L N ( η ) + w ϑ N ( η ) ≤ N ξ ~ N + K ′ N δ N , U_N(\eta) = L_N(\eta) + w\vartheta_N(\eta) \le N\widetilde{\xi}_N + K'N\delta_N, U N ( η ) = L N ( η ) + w ϑ N ( η ) ≤ N ξ N + K ′ N δ N , and dividing by N N N
ξ ~ N > c − c N − K ′ δ N − Δ N N ≥ c − Λ N − K ′ δ N − Δ N N . (147) \widetilde{\xi}_N > c - \frac{c}{N} - K'\delta_N - \frac{\Delta_N}{N} \ge c - \frac{\Lambda}{N} - K'\delta_N - \frac{\Delta_N}{N}. \tag*{(147)} ξ N > c − N c − K ′ δ N − N Δ N ≥ c − N Λ − K ′ δ N − N Δ N . (147) We now express the errors through N min N_{\min} N m i n N 1 , N 2 ≥ N min ≥ e 2 N_1,N_2 \ge N_{\min} \ge e^2 N 1 , N 2 ≥ N m i n ≥ e 2 [ e 2 , ∞ ) [e^2,\infty) [ e 2 , ∞ )
c ≥ min ( ξ ~ N 1 , ξ ~ N 2 ) − 2 log N min N min − 2 N min . c \ge\min(\widetilde{\xi}_{N_1},\widetilde{\xi}_{N_2}) - 2\frac{\log N_{\min}}{\sqrt{N_{\min}}} - \frac{2}{N_{\min}}. c ≥ min ( ξ N 1 , ξ N 2 ) − 2 N m i n log N m i n − N m i n 2 . Similarly, using N ≥ N min N \ge N_{\min} N ≥ N m i n Δ N / N ≤ ( log N min ) / N min + 1 / N min \Delta_N/N \le(\log N_{\min})/\sqrt{N_{\min}} + 1/N_{\min} Δ N / N ≤ ( log N m i n ) / N m i n + 1/ N m i n Λ / N ≤ Λ / N min \Lambda/N \le\Lambda/N_{\min} Λ/ N ≤ Λ/ N m i n δ N ≤ δ ( N min ) \delta_N \le\delta(N_{\min}) δ N ≤ δ ( N m i n ) ( δ n ) (\delta_n) ( δ n )
ξ ~ N ≥ min ( ξ ~ N 1 , ξ ~ N 2 ) − K ′ δ ( N min ) − 3 log N min N min − Λ + 3 N min ≥ min ( ξ ~ N 1 , ξ ~ N 2 ) − g ( N min ) , \widetilde{\xi}_N \ge\min(\widetilde{\xi}_{N_1},\widetilde{\xi}_{N_2}) - K'\delta(N_{\min}) - 3\frac{\log N_{\min}}{\sqrt{N_{\min}}} - \frac{\Lambda+3}{N_{\min}} \ge\min(\widetilde{\xi}_{N_1},\widetilde{\xi}_{N_2}) - g(N_{\min}), ξ N ≥ min ( ξ N 1 , ξ N 2 ) − K ′ δ ( N m i n ) − 3 N m i n log N m i n − N m i n Λ + 3 ≥ min ( ξ N 1 , ξ N 2 ) − g ( N m i n ) , where in the last step we used 1 / N min ≤ ( log N min ) / N min 1/N_{\min} \le(\log N_{\min})/\sqrt{N_{\min}} 1/ N m i n ≤ ( log N m i n ) / N m i n N min ≥ e N_{\min} \ge e N m i n ≥ e
For the second inequality, let N 1 ≥ n 0 N_1 \ge n_0 N 1 ≥ n 0 c = ζ ~ N 1 − 2 Δ N 1 / N 1 c=\widetilde{\zeta}_{N_1}-2\Delta_{N_1}/N_1 c = ζ N 1 − 2 Δ N 1 / N 1 0 < c ≤ Λ 0<c\le\Lambda 0 < c ≤ Λ N 2 = 1 N_2=1 N 2 = 1 N 1 N_1 N 1
ρ N 1 + 1 ( c ) ≥ ρ N 1 ( c ) e − c ≥ e − 2 Λ ( 7 8 ) 2 > e − 2 Λ 2 . \rho_{N_1+1}(c)\ge\rho_{N_1}(c)e^{-c}\ge e^{-2\Lambda}\left(\frac{7}{8}\right)^2>\frac{e^{-2\Lambda}}{2}. ρ N 1 + 1 ( c ) ≥ ρ N 1 ( c ) e − c ≥ e − 2Λ ( 8 7 ) 2 > 2 e − 2Λ . The argument above, at size N 1 + 1 ≥ n 0 N_1+1\ge n_0 N 1 + 1 ≥ n 0 N N N c N 1 < U N 1 + 1 ( η ) + Δ N 1 + 1 cN_1<U_{N_1+1}(\eta)+\Delta_{N_1+1} c N 1 < U N 1 + 1 ( η ) + Δ N 1 + 1
ζ ~ N 1 + 1 ≥ c − Λ N 1 + 1 − K ′ δ N 1 + 1 − Δ N 1 + 1 N 1 + 1 ≥ ζ ~ N 1 − K ′ δ ( N 1 ) − 3 log N 1 N 1 − Λ + 3 N 1 ≥ ζ ~ N 1 − g ( N 1 ) . \begin{aligned}
\widetilde{\zeta}_{N_1+1}\ge c-\frac{\Lambda}{N_1+1}-K'\delta_{N_1+1}-\frac{\Delta_{N_1+1}}{N_1+1} \\
&\ge\widetilde{\zeta}_{N_1}-K'\delta(N_1)-3\frac{\log N_1}{\sqrt{N_1}}-\frac{\Lambda+3}{N_1}\ge\widetilde{\zeta}_{N_1}-g(N_1).
\end{aligned} ζ N 1 + 1 ≥ c − N 1 + 1 Λ − K ′ δ N 1 + 1 − N 1 + 1 Δ N 1 + 1 ≥ ζ N 1 − K ′ δ ( N 1 ) − 3 N 1 log N 1 − N 1 Λ + 3 ≥ ζ N 1 − g ( N 1 ) . This completes the proof.
The use of the one-vertex model (A.4) for the comparison of adjacent sizes follows [51 Section 5.3] . This comparison cannot be omitted, since Remark A.6 gives a sequence that satisfies the first inequality in (A.14) with g = 0 g=0 g = 0
A balanced tree of blocks
We now prove a general convergence lemma from the two inequalities in (A.14). The proof applies those inequalities along a binary tree. The next statement specifies the condition on the error function.
Lemma A.5. Let u 0 ≥ 1 u_0\ge1 u 0 ≥ 1 g : [ 1 , ∞ ) → ( 0 , ∞ ) g:[1,\infty)\to(0,\infty) g : [ 1 , ∞ ) → ( 0 , ∞ ) [ u 0 , ∞ ) [u_0,\infty) [ u 0 , ∞ ) ∑ k ≥ 1 g ( e k ) < ∞ \sum_{k\ge1}g(e^k)<\infty ∑ k ≥ 1 g ( e k ) < ∞ n 0 ≥ 1 n_0\ge1 n 0 ≥ 1 ( ζ ~ n ) n ≥ n 0 (\widetilde{\zeta}_n)_{n\ge n_0} ( ζ n ) n ≥ n 0 N 1 , N 2 ≥ n 0 N_1,N_2\ge n_0 N 1 , N 2 ≥ n 0 ( ζ ~ n ) (\widetilde{\zeta}_n) ( ζ n )
Proof. We first record the consequences of the summability of g g g u 1 = e ⌈ log u 0 ⌉ u_1=e^{\lceil\log u_0\rceil} u 1 = e ⌈ l o g u 0 ⌉ k k k e k ≥ u 1 e^k\ge u_1 e k ≥ u 1 g g g
∫ e k e k + 1 g ( v ) d v v ≤ g ( e k ) , \int_{e^k}^{e^{k+1}}g(v)\frac{\mathrm{d}v}{v}\le g(e^k), ∫ e k e k + 1 g ( v ) v d v ≤ g ( e k ) , and summing over these k k k
∫ u 1 ∞ g ( v ) d v v < ∞ . (148) \int_{u_1}^{\infty}g(v)\frac{\mathrm{d}v}{v}<\infty. \tag*{(148)} ∫ u 1 ∞ g ( v ) v d v < ∞. (148) For u ≥ 2 u 1 u\ge2u_1 u ≥ 2 u 1 g g g
g ( u ) log 2 ≤ ∫ u / 2 u g ( v ) d v v , g(u)\log2\le\int_{u/2}^{u}g(v)\frac{\mathrm{d}v}{v}, g ( u ) log 2 ≤ ∫ u /2 u g ( v ) v d v , so (A.19) implies that g ( u ) → 0 g(u) \to0 g ( u ) → 0 u → ∞ u \to\infty u → ∞
For real ℓ ≥ 4 u 0 \ell\ge4u_0 ℓ ≥ 4 u 0
R ( ℓ ) = ∑ i ≥ 1 g ( ℓ 4 ( 4 3 ) i ) . (149) \mathcal{R}(\ell) = \sum_{i \ge1} g\left(\frac{\ell}{4}\left(\frac{4}{3}\right)^i\right). \tag*{(149)} R ( ℓ ) = i ≥ 1 ∑ g ( 4 ℓ ( 3 4 ) i ) . (149) For every i ≥ 1 i \ge1 i ≥ 1 ( ℓ / 4 ) ( 4 / 3 ) i − 1 (\ell/4)(4/3)^{i-1} ( ℓ /4 ) ( 4/3 ) i − 1 ( ℓ / 4 ) ( 4 / 3 ) i (\ell/4)(4/3)^i ( ℓ /4 ) ( 4/3 ) i u 0 u_0 u 0 g g g
g ( ℓ 4 ( 4 3 ) i ) log 4 3 ≤ ∫ ( ℓ / 4 ) ( 4 / 3 ) i − 1 ( ℓ / 4 ) ( 4 / 3 ) i g ( v ) d v v . g\left(\frac{\ell}{4}\left(\frac{4}{3}\right)^i\right)\log\frac{4}{3} \le\int_{(\ell/4)(4/3)^{i-1}}^{(\ell/4)(4/3)^i} g(v)\frac{\mathrm{d}v}{v}. g ( 4 ℓ ( 3 4 ) i ) log 3 4 ≤ ∫ ( ℓ /4 ) ( 4/3 ) i − 1 ( ℓ /4 ) ( 4/3 ) i g ( v ) v d v . Summing over i i i
R ( ℓ ) log 4 3 ≤ ∫ ℓ / 4 ∞ g ( v ) d v v . \mathcal{R}(\ell)\log\frac{4}{3} \le\int_{\ell/4}^{\infty} g(v)\frac{\mathrm{d}v}{v}. R ( ℓ ) log 3 4 ≤ ∫ ℓ /4 ∞ g ( v ) v d v . and by (A.19) the right side tends to zero as ℓ → ∞ \ell\to\infty ℓ → ∞
Next, we construct the tree. Fix an integer ℓ ≥ max ( n 0 , 4 u 0 , 2 ) \ell\ge\max(n_0,4u_0,2) ℓ ≥ max ( n 0 , 4 u 0 , 2 ) n ≥ ℓ 2 n \ge\ell^2 n ≥ ℓ 2 n = k ℓ + j n=k\ell+j n = k ℓ + j k = ⌊ n / ℓ ⌋ k=\lfloor n/\ell\rfloor k = ⌊ n / ℓ ⌋ 0 ≤ j < ℓ 0 \le j<\ell 0 ≤ j < ℓ k ≥ ℓ > j k \ge\ell>j k ≥ ℓ > j n n n j j j ℓ + 1 \ell+1 ℓ + 1 k − j k-j k − j ℓ \ell ℓ
We construct a rooted binary tree whose leaves correspond to the k k k ν \nu ν k ν k_\nu k ν ν \nu ν N ν N_\nu N ν k ν ≥ 2 k_\nu\ge2 k ν ≥ 2 ⌈ k ν / 2 ⌉ \lceil k_\nu/2\rceil ⌈ k ν /2 ⌉ ⌊ k ν / 2 ⌋ \lfloor k_\nu/2\rfloor ⌊ k ν /2 ⌋ n n n ℓ \ell ℓ ℓ + 1 \ell+1 ℓ + 1 ht ( ν ) \operatorname{ht}(\nu) ht ( ν ) ν \nu ν ν \nu ν
We claim that the children ν 1 \nu_1 ν 1 ν 2 \nu_2 ν 2 ν \nu ν
N ν 4 ≤ N ν i ≤ 3 N ν 4 , i = 1 , 2. (150) \frac{N_\nu}{4} \le N_{\nu_i} \le\frac{3N_\nu}{4}, \qquad i=1,2. \tag*{(150)} 4 N ν ≤ N ν i ≤ 4 3 N ν , i = 1 , 2. (150) Indeed, each child has at least ⌊ k ν / 2 ⌋ ≥ 1 \lfloor k_\nu/2\rfloor\ge1 ⌊ k ν /2 ⌋ ≥ 1 ⌈ k ν / 2 ⌉ ≤ 2 ⌊ k ν / 2 ⌋ \lceil k_\nu/2\rceil\le2\lfloor k_\nu/2\rfloor ⌈ k ν /2 ⌉ ≤ 2 ⌊ k ν /2 ⌋ ℓ \ell ℓ ℓ + 1 ≤ 3 ℓ / 2 \ell+1 \le3\ell/2 ℓ + 1 ≤ 3 ℓ /2
N ν 1 N ν 2 ≤ ⌈ k ν / 2 ⌉ ( ℓ + 1 ) ⌊ k ν / 2 ⌋ ℓ ≤ 2 ⋅ 3 2 = 3 , \frac{N_{\nu_1}}{N_{\nu_2}} \le\frac{\lceil k_\nu/2\rceil(\ell+1)}{\lfloor k_\nu/2\rfloor\ell} \le2\cdot\frac{3}{2}=3, N ν 2 N ν 1 ≤ ⌊ k ν /2 ⌋ ℓ ⌈ k ν /2 ⌉ ( ℓ + 1 ) ≤ 2 ⋅ 2 3 = 3 , and the same bound holds for N ν 2 / N ν 1 N_{\nu_2}/N_{\nu_1} N ν 2 / N ν 1 N ν 1 + N ν 2 = N ν N_{\nu_1}+N_{\nu_2}=N_\nu N ν 1 + N ν 2 = N ν
Now let ν = ν 0 , ν 1 , … , ν ht ( ν ) \nu=\nu_0,\nu_1,\ldots,\nu_{\operatorname{ht}(\nu)} ν = ν 0 , ν 1 , … , ν ht ( ν ) ν \nu ν N ν i − 1 ≥ ( 4 / 3 ) N ν i N_{\nu_{i-1}} \ge(4/3)N_{\nu_i} N ν i − 1 ≥ ( 4/3 ) N ν i i i i
N ν ≥ ( 4 3 ) ht ( ν ) ℓ . (151) N_\nu\ge\left(\frac{4}{3}\right)^{\operatorname{ht}(\nu)}\ell. \tag*{(151)} N ν ≥ ( 3 4 ) ht ( ν ) ℓ . (151) Figure A.1. The tree of the proof of Lemma A.5 for k = 5 k=5 k = 5 k ν k_\nu k ν k ν k_\nu k ν ⌈ k ν / 2 ⌉ \lceil k_\nu/2\rceil ⌈ k ν /2 ⌉ ⌊ k ν / 2 ⌋ \lfloor k_\nu/2\rfloor ⌊ k ν /2 ⌋ ℓ \ell ℓ ℓ + 1 \ell+1 ℓ + 1 4 / 3 4/3 4/3 R ( ℓ ) R(\ell) R ( ℓ )
We now show by induction on the height that every node ν \nu ν
ζ N ν ≥ ζ ℓ − g ( ℓ ) − ∑ i = 1 ht ( ν ) g ( ℓ 4 ( 4 3 ) i ) . (152) \zeta_{N_\nu} \ge\zeta_\ell- g(\ell) - \sum_{i=1}^{\operatorname{ht}(\nu)} g\left(\frac{\ell}{4}\left(\frac{4}{3}\right)^i\right). \tag*{(152)} ζ N ν ≥ ζ ℓ − g ( ℓ ) − i = 1 ∑ ht ( ν ) g ( 4 ℓ ( 3 4 ) i ) . (152) A node of height 0 is a leaf of size ℓ \ell ℓ ℓ + 1 \ell+1 ℓ + 1 N 1 = ℓ ≥ n 0 N_1=\ell\ge n_0 N 1 = ℓ ≥ n 0 ζ ℓ + 1 ≥ ζ ℓ − g ( ℓ ) \zeta_{\ell+1}\ge\zeta_\ell-g(\ell) ζ ℓ + 1 ≥ ζ ℓ − g ( ℓ )
Now let ν \nu ν ν 1 \nu_1 ν 1 ν 2 \nu_2 ν 2 ν 1 \nu_1 ν 1 ν 2 \nu_2 ν 2 ht ( ν ) − 1 \operatorname{ht}(\nu)-1 ht ( ν ) − 1 ℓ ≥ n 0 \ell\ge n_0 ℓ ≥ n 0
ζ N ν ≥ min ( ζ N ν 1 , ζ N ν 2 ) − g ( min ( N ν 1 , N ν 2 ) ) ≥ ζ ℓ − g ( ℓ ) − ∑ i = 1 ht ( ν ) − 1 g ( ℓ 4 ( 4 3 ) i ) − g ( min ( N ν 1 , N ν 2 ) ) . \begin{aligned}
\zeta_{N_\nu} &\ge\min(\zeta_{N_{\nu_1}},\zeta_{N_{\nu_2}})-g\left(\min(N_{\nu_1},N_{\nu_2})\right) \\
&\ge\zeta_\ell-g(\ell)-\sum_{i=1}^{\operatorname{ht}(\nu)-1}g\left(\frac{\ell}{4}\left(\frac{4}{3}\right)^i\right)-g\left(\min(N_{\nu_1},N_{\nu_2})\right).
\end{aligned} ζ N ν ≥ min ( ζ N ν 1 , ζ N ν 2 ) − g ( min ( N ν 1 , N ν 2 ) ) ≥ ζ ℓ − g ( ℓ ) − i = 1 ∑ ht ( ν ) − 1 g ( 4 ℓ ( 3 4 ) i ) − g ( min ( N ν 1 , N ν 2 ) ) . The bounds (A.21) and (A.22) give
min ( N ν 1 , N ν 2 ) ≥ N ν 4 ≥ ℓ 4 ( 4 3 ) ht ( ν ) ≥ u 0 , \min(N_{\nu_1},N_{\nu_2})\ge\frac{N_\nu}{4}\ge\frac{\ell}{4}\left(\frac{4}{3}\right)^{\operatorname{ht}(\nu)}\ge u_0, min ( N ν 1 , N ν 2 ) ≥ 4 N ν ≥ 4 ℓ ( 3 4 ) ht ( ν ) ≥ u 0 , and since g g g [ u 0 , ∞ ) [u_0,\infty) [ u 0 , ∞ ) − g ( ( ℓ / 4 ) ( 4 / 3 ) ht ( ν ) ) -g\left((\ell/4)(4/3)^{\operatorname{ht}(\nu)}\right) − g ( ( ℓ /4 ) ( 4/3 ) ht ( ν ) )
Applying (A.23) at the root, we obtain ζ n ≥ ζ ℓ − g ( ℓ ) − R ( ℓ ) \zeta_n\ge\zeta_\ell-g(\ell)-R(\ell) ζ n ≥ ζ ℓ − g ( ℓ ) − R ( ℓ ) n ≥ ℓ 2 n\ge\ell^2 n ≥ ℓ 2 n → ∞ n\to\infty n → ∞
lim inf n → ∞ ζ n ≥ ζ ℓ − g ( ℓ ) − R ( ℓ ) \liminf_{n\to\infty}\zeta_n\ge\zeta_\ell-g(\ell)-R(\ell) n → ∞ lim inf ζ n ≥ ζ ℓ − g ( ℓ ) − R ( ℓ ) for every integer ℓ ≥ max ( n 0 , 4 u 0 , 2 ) \ell\ge\max(n_0,4u_0,2) ℓ ≥ max ( n 0 , 4 u 0 , 2 ) g ( ℓ ) → 0 g(\ell) \to0 g ( ℓ ) → 0 R ( ℓ ) → 0 \mathcal{R}(\ell) \to0 R ( ℓ ) → 0 ℓ → ∞ \ell\to\infty ℓ → ∞ lim inf n ξ n ≥ lim sup ℓ ξ ℓ \liminf_n \xi_n \ge\limsup_\ell\xi_\ell lim inf n ξ n ≥ lim sup ℓ ξ ℓ
The minimum form of the size comparison and the tree with blocks of sizes ℓ \ell ℓ ℓ + 1 \ell+1 ℓ + 1 [51 Lemma 5.4] , which assumes polynomially small errors. For the classical additive form of near-superadditivity, sufficient conditions of this summable type go back to de Bruijn and Erdős [25 Theorem 23] ; see also [1 arXiv:1006.3786v1, Section 4.3, Lemma 8 and Remark 3] . A minimum inequality is weaker than the additive one, and these classical results do not apply directly.
Remark A.6. The summability assumption in Lemma A.5 and the adjacent-size inequality in (A.14) cannot be dropped from this convergence argument. First, let δ ~ : [ 1 , ∞ ) → ( 0 , ∞ ) \widetilde{\delta} : [1,\infty) \to(0,\infty) δ : [ 1 , ∞ ) → ( 0 , ∞ ) ∫ 1 ∞ δ ~ ( v ) d v / v = ∞ \int_1^\infty\widetilde{\delta}(v)\,\mathrm{d}v/v = \infty ∫ 1 ∞ δ ( v ) d v / v = ∞ t ≥ 1 t \ge1 t ≥ 1 F ~ ( t ) = ∫ 1 t δ ~ ( v ) d v / v \widetilde{F}(t) = \int_1^t \widetilde{\delta}(v)\,\mathrm{d}v/v F ( t ) = ∫ 1 t δ ( v ) d v / v n ≥ 1 n \ge1 n ≥ 1 y n = 1 + sin ( F ~ ( n ) ) / 2 y_n = 1 + \sin(\widetilde{F}(n))/2 y n = 1 + sin ( F ( n )) /2 3 ≤ N 1 ≤ N 2 3 \le N_1 \le N_2 3 ≤ N 1 ≤ N 2
∣ y N 1 + N 2 − y N 2 ∣ ≤ 1 2 ∫ N 2 N 1 + N 2 δ ~ ( v ) d v v ≤ log 2 2 δ ~ ( N 2 ) ≤ log 2 2 δ ~ ( N 1 ) , \lvert y_{N_1+N_2} - y_{N_2} \rvert\le\frac{1}{2}\int_{N_2}^{N_1+N_2} \widetilde{\delta}(v)\,\frac{\mathrm{d}v}{v} \le\frac{\log2}{2}\widetilde{\delta}(N_2) \le\frac{\log2}{2}\widetilde{\delta}(N_1), ∣ y N 1 + N 2 − y N 2 ∣ ≤ 2 1 ∫ N 2 N 1 + N 2 δ ( v ) v d v ≤ 2 log 2 δ ( N 2 ) ≤ 2 log 2 δ ( N 1 ) , and ∣ y N 1 + 1 − y N 1 ∣ ≤ δ ~ ( N 1 ) / ( 2 N 1 ) ≤ δ ~ ( N 1 ) \lvert y_{N_1+1} - y_{N_1} \rvert\le\widetilde{\delta}(N_1)/(2N_1) \le\widetilde{\delta}(N_1) ∣ y N 1 + 1 − y N 1 ∣ ≤ δ ( N 1 ) / ( 2 N 1 ) ≤ δ ( N 1 ) ( y n ) (y_n) ( y n ) g = δ ~ g = \widetilde{\delta} g = δ n 0 = 3 n_0 = 3 n 0 = 3 F ~ ( n ) → ∞ \widetilde{F}(n) \to\infty F ( n ) → ∞ F ~ ( n + 1 ) − F ~ ( n ) ≤ δ ~ ( 1 ) / n → 0 \widetilde{F}(n+1) - \widetilde{F}(n) \le\widetilde{\delta}(1)/n \to0 F ( n + 1 ) − F ( n ) ≤ δ ( 1 ) / n → 0 F ~ ( n ) \widetilde{F}(n) F ( n ) lim sup n y n = 3 / 2 \limsup_n y_n = 3/2 lim sup n y n = 3/2 lim inf n y n = 1 / 2 \liminf_n y_n = 1/2 lim inf n y n = 1/2 δ ~ \widetilde{\delta} δ ∫ 1 ∞ δ ~ ( v ) d v / v \int_1^\infty\widetilde{\delta}(v)\,\mathrm{d}v/v ∫ 1 ∞ δ ( v ) d v / v ∑ k δ ~ ( e k ) = ∞ \sum_k \widetilde{\delta}(e^k) = \infty ∑ k δ ( e k ) = ∞ δ ~ ( v ) = 1 / log ( e v ) \widetilde{\delta}(v) = 1/\log(ev) δ ( v ) = 1/ log ( e v ) n / log n n/\log n n / log n V V V [35 arXiv:1810.11723v1, Theorem 6] .
Second, we define b ~ n = 1 { n is even } \widetilde{b}_n = \mathbf{1}\{n\text{ is even}\} b n = 1 { n is even } n ≥ 1 n \ge1 n ≥ 1 b ~ N 1 + N 2 ≥ min ( b ~ N 1 , b ~ N 2 ) \widetilde{b}_{N_1+N_2} \ge\min(\widetilde{b}_{N_1},\widetilde{b}_{N_2}) b N 1 + N 2 ≥ min ( b N 1 , b N 2 ) N 1 , N 2 ≥ 1 N_1,N_2 \ge1 N 1 , N 2 ≥ 1 N 1 + N 2 N_1+N_2 N 1 + N 2 N 1 N_1 N 1 N 2 N_2 N 2 ( b ~ n ) (\widetilde{b}_n) ( b n ) g ≥ 0 g \ge0 g ≥ 0 g g g
Transfer to the uniform and binomial random graphs
Recall that, for integers n ≥ 1 n \ge1 n ≥ 1 0 ≤ m ≤ ( n 2 ) 0 \le m \le\binom{n}{2} 0 ≤ m ≤ ( 2 n ) G ( n , m ) G(n,m) G ( n , m ) [ n ] [n] [ n ] m m m p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] G ( n , p ) G(n,p) G ( n , p ) [ n ] [n] [ n ] p p p
Lemma A.7. (a) For all large n n n 0 ≤ m ≤ Λ n 0 \le m \le\Lambda n 0 ≤ m ≤ Λ n
P ( G ( n , m ) is not colorable ) ≤ e 2 Λ 2 + 1 P ( H ≤ m ) , \mathbb{P}(G(n,m)\text{ is not colorable}) \le e^{2\Lambda^2+1}\mathbb{P}(H \le m), P ( G ( n , m ) is not colorable ) ≤ e 2 Λ 2 + 1 P ( H ≤ m ) , P ( G ( n , m ) is colorable ) ≤ e 2 Λ 2 + 1 P ( H > m ) . (153) \mathbb{P}(G(n,m)\text{ is colorable}) \le e^{2\Lambda^2+1}\mathbb{P}(H > m). \tag*{(153)} P ( G ( n , m ) is colorable ) ≤ e 2 Λ 2 + 1 P ( H > m ) . (153) (b) For all integers n ≥ 2 n \ge2 n ≥ 2 P ( G ( n , m ) is colorable ) \mathbb{P}(G(n,m)\text{ is colorable}) P ( G ( n , m ) is colorable ) m ∈ { 0 , … , ( n 2 ) } m \in\{0,\ldots,\binom{n}{2}\} m ∈ { 0 , … , ( 2 n ) } 0 ≤ m ≤ ( n 2 ) 0 \le m \le\binom{n}{2} 0 ≤ m ≤ ( 2 n ) p ∈ [ 0 , 1 ] p \in[0,1] p ∈ [ 0 , 1 ] X X X Bin ( ( n 2 ) , p ) \operatorname{Bin}(\binom{n}{2},p) Bin ( ( 2 n ) , p )
P ( G ( n , p ) is colorable ) ≥ P ( G ( n , m ) is colorable ) P ( X ≤ m ) , \mathbb{P}(G(n,p)\text{ is colorable}) \ge\mathbb{P}(G(n,m)\text{ is colorable})\mathbb{P}(X \le m), P ( G ( n , p ) is colorable ) ≥ P ( G ( n , m ) is colorable ) P ( X ≤ m ) , P ( G ( n , p ) is colorable ) ≤ P ( G ( n , m ) is colorable ) + P ( X < m ) . (154) \mathbb{P}(G(n,p)\text{ is colorable}) \le\mathbb{P}(G(n,m)\text{ is colorable})+\mathbb{P}(X < m). \tag*{(154)} P ( G ( n , p ) is colorable ) ≤ P ( G ( n , m ) is colorable ) + P ( X < m ) . (154) Proof. We begin with (a). Conditioning on a simplicity event of probability bounded away from zero is a standard transfer from nonsimple to simple random graphs; see [10 Section 2, paragraph on nonsimple graphs] . Let b = ( n 2 ) b=\binom{n}{2} b = ( 2 n ) D m \mathcal{D}_m D m e 1 , … , e m e_1,\ldots,e_m e 1 , … , e m n n n m ≤ Λ n m \le\Lambda n m ≤ Λ n m / b ≤ 2 Λ / ( n − 1 ) ≤ 1 / 2 m/b \le2\Lambda/(n-1) \le1/2 m / b ≤ 2Λ/ ( n − 1 ) ≤ 1/2 log ( 1 − x ) ≥ − x − x 2 \log(1-x) \ge-x-x^2 log ( 1 − x ) ≥ − x − x 2 x ∈ [ 0 , 1 / 2 ] x \in[0,1/2] x ∈ [ 0 , 1/2 ]
P ( D m ) = ∏ i = 1 m − 1 ( 1 − i b ) ≥ exp ( − ∑ i = 1 m − 1 i b − ∑ i = 1 m − 1 i 2 b 2 ) ≥ exp ( − m 2 2 b − m 3 b 2 ) ≥ e − 2 Λ 2 − 1 , \begin{aligned}
\mathbb{P}(\mathcal{D}_m)=\prod_{i=1}^{m-1}\left(1-\frac{i}{b}\right) \\
&\ge\exp\left(-\sum_{i=1}^{m-1}\frac{i}{b}-\sum_{i=1}^{m-1}\frac{i^2}{b^2}\right) \\
&\ge\exp\left(-\frac{m^2}{2b}-\frac{m^3}{b^2}\right) \\
&\ge e^{-2\Lambda^2-1},
\end{aligned} P ( D m ) = i = 1 ∏ m − 1 ( 1 − b i ) ≥ exp ( − i = 1 ∑ m − 1 b i − i = 1 ∑ m − 1 b 2 i 2 ) ≥ exp ( − 2 b m 2 − b 2 m 3 ) ≥ e − 2 Λ 2 − 1 , where in the last step we used m 2 / ( 2 b ) = m 2 / ( n ( n − 1 ) ) ≤ 2 Λ 2 m^2/(2b)=m^2/(n(n-1))\le2\Lambda^2 m 2 / ( 2 b ) = m 2 / ( n ( n − 1 )) ≤ 2 Λ 2 m 3 / b 2 ≤ 4 Λ 3 n / ( n − 1 ) 2 ≤ 1 m^3/b^2\le4\Lambda^3n/(n-1)^2\le1 m 3 / b 2 ≤ 4 Λ 3 n / ( n − 1 ) 2 ≤ 1
Conditionally on D m \mathcal{D}_m D m ( e 1 , … , e m ) (e_1,\ldots,e_m) ( e 1 , … , e m ) m m m [ n ] [n] [ n ] { e 1 , … , e m } \{e_1,\ldots,e_m\} { e 1 , … , e m } G ( n , m ) G(n,m) G ( n , m ) G m G_m G m B \mathcal{B} B G m G_m G m
P ( G ( n , m ) is not colorable ) = P ( B ∣ D m ) ≤ P ( B ) P ( D m ) ≤ e 2 Λ 2 + 1 P ( H ≤ m ) , \mathbb{P}(G(n,m)\text{ is not colorable})=\mathbb{P}(\mathcal{B}\mid\mathcal{D}_m)\le\frac{\mathbb{P}(\mathcal{B})}{\mathbb{P}(\mathcal{D}_m)}\le e^{2\Lambda^2+1}\mathbb{P}(H\le m), P ( G ( n , m ) is not colorable ) = P ( B ∣ D m ) ≤ P ( D m ) P ( B ) ≤ e 2 Λ 2 + 1 P ( H ≤ m ) , and the second bound in (A.24) follows in the same way from the complement of B \mathcal{B} B
We now prove (b). Let m < ( n 2 ) m<\binom{n}{2} m < ( 2 n ) G ( n , m ) G(n,m) G ( n , m ) [ n ] [n] [ n ] G ( n , m ) G(n,m) G ( n , m ) G ( n , m + 1 ) G(n,m+1) G ( n , m + 1 ) X X X G ( n , p ) G(n,p) G ( n , p ) X X X Bin ( ( n 2 ) , p ) \operatorname{Bin}\left(\binom{n}{2},p\right) Bin ( ( 2 n ) , p ) X X X G ( n , p ) G(n,p) G ( n , p ) G ( n , X ) G(n,X) G ( n , X ) [34 Lemma 1.1] , so
P ( G ( n , p ) is colorable ) = ∑ j = 0 ( n 2 ) P ( X = j ) P ( G ( n , j ) is colorable ) . \mathbb{P}(G(n,p)\text{ is colorable})=\sum_{j=0}^{\binom{n}{2}}\mathbb{P}(X=j)\mathbb{P}(G(n,j)\text{ is colorable}). P ( G ( n , p ) is colorable ) = j = 0 ∑ ( 2 n ) P ( X = j ) P ( G ( n , j ) is colorable ) . Bounding the terms with j ≥ m j\ge m j ≥ m P ( X = j ) P ( G ( n , m ) is colorable ) \mathbb{P}(X=j)\mathbb{P}(G(n,m)\text{ is colorable}) P ( X = j ) P ( G ( n , m ) is colorable ) j < m j<m j < m P ( X = j ) \mathbb{P}(X=j) P ( X = j ) j > m j>m j > m P ( X = j ) P ( G ( n , m ) is colorable ) \mathbb{P}(X=j)\mathbb{P}(G(n,m)\text{ is colorable}) P ( X = j ) P ( G ( n , m ) is colorable )
Proof of Theorem 4.1
We assume the hypotheses recalled at the beginning of this appendix, and we proceed in four steps.
Step 1 (one level). Fix η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] ξ n \xi_n ξ n g g g g g g K ′ > 0 K'>0 K ′ > 0 δ n > 0 \delta_n>0 δ n > 0 [ e 2 , ∞ ) [e^2,\infty) [ e 2 , ∞ ) δ ( u ) \delta(u) δ ( u ) u u u ( log u ) / u (\log u)/\sqrt{u} ( log u ) / u u ≥ e 2 u\ge e^2 u ≥ e 2
∑ k ≥ 1 g ( e k ) = K ′ ∑ k ≥ 1 δ ⌊ e k ⌋ + ( Λ + 6 ) ∑ k ≥ 1 k e − k / 2 < ∞ . \sum_{k\ge1}g(e^k)=K'\sum_{k\ge1}\delta_{\lfloor e^k\rfloor}+(\Lambda+6)\sum_{k\ge1}ke^{-k/2}<\infty. k ≥ 1 ∑ g ( e k ) = K ′ k ≥ 1 ∑ δ ⌊ e k ⌋ + ( Λ + 6 ) k ≥ 1 ∑ k e − k /2 < ∞. The sequence ( ξ n ) n ≥ 2 q (\xi_n)_{n\ge2q} ( ξ n ) n ≥ 2 q [ 0 , Λ ] [0,\Lambda] [ 0 , Λ ] N 1 , N 2 ≥ n 0 N_1,N_2\ge n_0 N 1 , N 2 ≥ n 0 u 0 = e 2 u_0=e^2 u 0 = e 2 ( ξ n ) (\xi_n) ( ξ n ) α ( η ) \alpha(\eta) α ( η )
Lemma 3.17 implies that c 0 ≤ α ( η ) ≤ c 1 c_0\le\alpha(\eta)\le c_1 c 0 ≤ α ( η ) ≤ c 1 c 0 < 1 / 2 c_0<1/2 c 0 < 1/2 c 1 > γ q c_1>\gamma_q c 1 > γ q α ( η ) ∈ [ 1 / 2 , γ q ] \alpha(\eta)\in[1/2,\gamma_q] α ( η ) ∈ [ 1/2 , γ q ] δ ⌊ e k ⌋ → 0 \delta_{\lfloor e^k\rfloor}\to0 δ ⌊ e k ⌋ → 0 ( δ n ) (\delta_n) ( δ n ) δ n → 0 \delta_n\to0 δ n → 0 U n ( η ) / n = L n ( η ) / n + w n ( η ) / n U_n(\eta)/n=L_n(\eta)/n+w_n(\eta)/n U n ( η ) / n = L n ( η ) / n + w n ( η ) / n α ( η ) \alpha(\eta) α ( η )
Step 2 (all levels share the limit). Let 0 < η − < η + ≤ η 0 0<\eta_-<\eta_+\le\eta_0 0 < η − < η + ≤ η 0 m m m P ( V ≤ m ) ≥ η + \mathbb{P}(V\le m)\ge\eta_+ P ( V ≤ m ) ≥ η + P ( V ≤ m ) ≥ η − \mathbb{P}(V\le m)\ge\eta_- P ( V ≤ m ) ≥ η − L n ( η − ) ≤ L n ( η + ) L_n(\eta_-)\le L_n(\eta_+) L n ( η − ) ≤ L n ( η + )
P ( V ≥ L n ( η + ) ) = 1 − P ( V ≤ L n ( η + ) − 1 ) > 1 − η + ≥ η − , \mathbb{P}(V\ge L_n(\eta_+))=1-\mathbb{P}(V\le L_n(\eta_+)-1)>1-\eta_+\ge\eta_-, P ( V ≥ L n ( η + )) = 1 − P ( V ≤ L n ( η + ) − 1 ) > 1 − η + ≥ η − , and L n ( η + ) ≤ M L_n(\eta_+)\le M L n ( η + ) ≤ M L n ( η + ) L_n(\eta_+) L n ( η + ) U n ( η − ) U_n(\eta_-) U n ( η − )
L n ( η − ) n ≤ L n ( η + ) n ≤ U n ( η − ) n \frac{L_n(\eta_-)}{n}\le\frac{L_n(\eta_+)}{n}\le\frac{U_n(\eta_-)}{n} n L n ( η − ) ≤ n L n ( η + ) ≤ n U n ( η − ) for all n ≥ 2 q n\ge2q n ≥ 2 q
By Step 1, the outer terms converge to α ( η − ) \alpha(\eta_-) α ( η − ) α ( η + ) = α ( η − ) \alpha(\eta_+)=\alpha(\eta_-) α ( η + ) = α ( η − ) α q \alpha_q α q L n ( η ) / n → α q L_n(\eta)/n\to\alpha_q L n ( η ) / n → α q U n ( η ) / n → α q U_n(\eta)/n\to\alpha_q U n ( η ) / n → α q η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] α q ∈ [ 1 / 2 , γ q ] \alpha_q\in[1/2,\gamma_q] α q ∈ [ 1/2 , γ q ]
Step 3 (the edge process). Let 0 ≤ c < α q 0 \le c < \alpha_q 0 ≤ c < α q η ∈ ( 0 , η 0 ] \eta\in(0,\eta_0] η ∈ ( 0 , η 0 ] L n ( η ) / n → α q > c L_n(\eta)/n \to\alpha_q > c L n ( η ) / n → α q > c ⌊ c n ⌋ ≤ L n ( η ) \lfloor cn\rfloor\le L_n(\eta) ⌊ c n ⌋ ≤ L n ( η ) n n n n n n P ( H > ⌊ c n ⌋ ) > 1 − η \mathbb{P}(H > \lfloor cn\rfloor) > 1-\eta P ( H > ⌊ c n ⌋) > 1 − η lim inf n P ( H > ⌊ c n ⌋ ) ≥ 1 − η \liminf_n \mathbb{P}(H > \lfloor cn\rfloor) \ge1-\eta lim inf n P ( H > ⌊ c n ⌋) ≥ 1 − η η → 0 \eta\to0 η → 0 P ( H > ⌊ c n ⌋ ) → 1 \mathbb{P}(H > \lfloor cn\rfloor) \to1 P ( H > ⌊ c n ⌋) → 1
Now let c > α q c > \alpha_q c > α q c < Λ c < \Lambda c < Λ U n ( η ) / n → α q < c U_n(\eta)/n \to\alpha_q < c U n ( η ) / n → α q < c U n ( η ) < ⌊ c n ⌋ ≤ M U_n(\eta) < \lfloor cn\rfloor\le M U n ( η ) < ⌊ c n ⌋ ≤ M n n n n n n P ( H > ⌊ c n ⌋ ) < η \mathbb{P}(H > \lfloor cn\rfloor) < \eta P ( H > ⌊ c n ⌋) < η n → ∞ n \to\infty n → ∞ η → 0 \eta\to0 η → 0 P ( H > ⌊ c n ⌋ ) → 0 \mathbb{P}(H > \lfloor cn\rfloor) \to0 P ( H > ⌊ c n ⌋) → 0 c ≥ Λ c \ge\Lambda c ≥ Λ c ′ ∈ ( α q , Λ ) c' \in(\alpha_q,\Lambda) c ′ ∈ ( α q , Λ ) P ( H > m ) \mathbb{P}(H > m) P ( H > m ) m m m P ( H > ⌊ c n ⌋ ) ≤ P ( H > ⌊ c ′ n ⌋ ) → 0 \mathbb{P}(H > \lfloor cn\rfloor) \le\mathbb{P}(H > \lfloor c'n\rfloor) \to0 P ( H > ⌊ c n ⌋) ≤ P ( H > ⌊ c ′ n ⌋) → 0
Step 4 (the uniform and binomial random graphs). Let 0 ≤ c < α q 0 \le c < \alpha_q 0 ≤ c < α q α q ≤ γ q < Λ \alpha_q \le\gamma_q < \Lambda α q ≤ γ q < Λ ⌊ c n ⌋ ≤ Λ n \lfloor cn\rfloor\le\Lambda n ⌊ c n ⌋ ≤ Λ n
P ( G ( n , ⌊ c n ⌋ ) is not colorable ) ≤ e 2 Λ 2 + 1 P ( H ≤ ⌊ c n ⌋ ) → 0. \mathbb{P}(G(n,\lfloor cn\rfloor)\text{ is not colorable}) \le e^{2\Lambda^2+1}\mathbb{P}(H \le\lfloor cn\rfloor) \to0. P ( G ( n , ⌊ c n ⌋) is not colorable ) ≤ e 2 Λ 2 + 1 P ( H ≤ ⌊ c n ⌋) → 0. For α q < c < Λ \alpha_q < c < \Lambda α q < c < Λ P ( G ( n , ⌊ c n ⌋ ) is colorable ) → 0 \mathbb{P}(G(n,\lfloor cn\rfloor)\text{ is colorable}) \to0 P ( G ( n , ⌊ c n ⌋) is colorable ) → 0 c ≥ Λ c \ge\Lambda c ≥ Λ c ′ ∈ ( α q , Λ ) c' \in(\alpha_q,\Lambda) c ′ ∈ ( α q , Λ ) n n n ⌊ c n ⌋ ≤ ( n 2 ) \lfloor cn\rfloor\le\binom{n}{2} ⌊ c n ⌋ ≤ ( 2 n )
Now let d ≥ 0 d \ge0 d ≥ 0 n n n d / n ≤ 1 d/n \le1 d / n ≤ 1 X X X Bin ( ( n 2 ) , d / n ) \operatorname{Bin}(\binom{n}{2},d/n) Bin ( ( 2 n ) , d / n ) E X = d ( n − 1 ) / 2 \mathbb{E}X = d(n-1)/2 E X = d ( n − 1 ) /2 Var X ≤ d ( n − 1 ) / 2 \operatorname{Var}X \le d(n-1)/2 Var X ≤ d ( n − 1 ) /2
Suppose first that d < 2 α q d < 2\alpha_q d < 2 α q c ∈ ( d / 2 , α q ) c \in(d/2,\alpha_q) c ∈ ( d /2 , α q ) m = ⌊ c n ⌋ m = \lfloor cn\rfloor m = ⌊ c n ⌋ m + 1 − E X ≥ ( c − d / 2 ) n > 0 m+1-\mathbb{E}X \ge(c-d/2)n > 0 m + 1 − E X ≥ ( c − d /2 ) n > 0
P ( X > m ) ≤ P ( X − E X ≥ ( c − d / 2 ) n ) ≤ d n 2 ( c − d / 2 ) 2 n 2 → 0. \mathbb{P}(X > m) \le\mathbb{P}(X-\mathbb{E}X \ge(c-d/2)n) \le\frac{dn}{2(c-d/2)^2n^2} \to0. P ( X > m ) ≤ P ( X − E X ≥ ( c − d /2 ) n ) ≤ 2 ( c − d /2 ) 2 n 2 d n → 0. The first inequality in (A.25) and the first part of this step then give
P ( G ( n , d / n ) is colorable ) ≥ P ( G ( n , m ) is colorable ) P ( X ≤ m ) → 1. \mathbb{P}(G(n,d/n)\text{ is colorable}) \ge\mathbb{P}(G(n,m)\text{ is colorable})\mathbb{P}(X \le m) \to1. P ( G ( n , d / n ) is colorable ) ≥ P ( G ( n , m ) is colorable ) P ( X ≤ m ) → 1. Suppose now that d > 2 α q d > 2\alpha_q d > 2 α q c ∈ ( α q , d / 2 ) c \in(\alpha_q,d/2) c ∈ ( α q , d /2 ) m = ⌊ c n ⌋ m = \lfloor cn\rfloor m = ⌊ c n ⌋ E X − m ≥ ( d / 2 − c ) n − d / 2 \mathbb{E}X-m \ge(d/2-c)n-d/2 E X − m ≥ ( d /2 − c ) n − d /2 n n n P ( X < m ) → 0 \mathbb{P}(X < m) \to0 P ( X < m ) → 0
P ( G ( n , d / n ) is colorable ) ≤ P ( G ( n , m ) is colorable ) + P ( X < m ) → 0. \mathbb{P}(G(n,d/n)\text{ is colorable}) \le\mathbb{P}(G(n,m)\text{ is colorable})+\mathbb{P}(X < m) \to0. P ( G ( n , d / n ) is colorable ) ≤ P ( G ( n , m ) is colorable ) + P ( X < m ) → 0. This completes the proof of Theorem 4.1.
We note that α q \alpha_q α q q q q P ( H > ⌊ c n ⌋ ) \mathbb{P}(H > \lfloor cn\rfloor) P ( H > ⌊ c n ⌋) c < α q c < \alpha_q c < α q c > α q c > \alpha_q c > α q Λ \Lambda Λ η \eta η ( δ n ) (\delta_n) ( δ n ) Λ \Lambda Λ α q \alpha_q α q
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