Introduction
We study how a local normalization constraint affects the spectrum of a random operator on a graph. At each vertex, a random probability vector distributes unit mass among the outgoing directions. An edge weight uses one mass from each endpoint. Edges meeting at a vertex are therefore dependent, even when the vertex vectors are sampled independently. The same construction defines a reflection quantum walk in an environment that is sampled once and then kept fixed.
Let G G G d d d d ≥ 3 d \ge3 d ≥ 3 v v v
X v = ( X v , u ) u ∼ v ∼ Dirichlet ( α , … , α ) , α > 0. X_v = (X_{v,u})_{u\sim v} \sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha), \qquad\alpha> 0. X v = ( X v , u ) u ∼ v ∼ Dirichlet ( α , … , α ) , α > 0. The coordinate X v , u X_{v,u} X v , u v v v u u u d d d α ↓ 0 \alpha\downarrow0 α ↓ 0 α → ∞ \alpha\to\infty α → ∞
T v , u = X v , u X u , v , u ∼ v . T_{v,u} = \sqrt{X_{v,u}X_{u,v}}, \qquad u\sim v. T v , u = X v , u X u , v , u ∼ v . It is sparse and self-adjoint, unlike the dense Dirichlet Markov matrices studied in [9 ] . The full model also allows complex phases. On a tree these phases can be removed by a diagonal unitary transformation, so they do not change the tree spectral law.
Choose amplitudes q v , u q_{v,u} q v , u ∣ q v , u ∣ 2 = X v , u \lvert q_{v,u}\rvert^2 = X_{v,u} ∣ q v , u ∣ 2 = X v , u q v = ( q v , u ) u ∼ v q_v = (q_{v,u})_{u\sim v} q v = ( q v , u ) u ∼ v quenched setting.
The associated self-adjoint operator T T T discriminant . Its spectral values give the inherited part of the unitary spectrum through x ↦ x ± i 1 − x 2 x \mapsto x \pm i\sqrt{1-x^2} x ↦ x ± i 1 − x 2 [27 ] . When the amplitudes are chosen positive, X v , u X_{v,u} X v , u [26 ] , and the quantum walk is its reflection/Szegedy-type quantization [24 , 28 ] .
The central question is how the local dependence changes the spectral behavior as the degree grows. Our main result identifies two distinct scales for the deterministic expected root spectral laws. The first classifies their variance-normalized weak limits. In the regime where this limit collapses to a point mass at zero, the second classifies the spectral mass carrying the still nonzero second moment. Thus different Dirichlet parameter sequences can have the same ordinary spectral limit but different rescaled, second-moment-weighted limits.
Main results. We take the large-graph limit first, with d d d α \alpha α d → ∞ d \to\infty d → ∞
Fixed degree: the tree law and its support. Consider a graph sequence in which the fraction of vertices with a tree neighborhood of any fixed radius tends to one. The empirical spectral measures of the discriminants converge almost surely to a deterministic measure μ d , α \mu_{d,\alpha} μ d , α tree law (Theorem 3.1 ). We compute every moment by summing the weights of closed walks. The formula retains the mixed Dirichlet moments at each vertex, so it accounts for the dependence between incident edges (Theorem 3.3 ).
For every α > 0 \alpha> 0 α > 0 [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] Theorems 3.6 –3.7 ). Despite this common support, the endpoint laws have smaller support: as α ↓ 0 \alpha\downarrow0 α ↓ 0 { − 1 , 0 , 1 } \{-1, 0, 1\} { − 1 , 0 , 1 } α → ∞ \alpha\to\infty α → ∞
High degree: the spectral law and the mass carrying its second moment. The second moment of μ d , α \mu_{d,\alpha} μ d , α 1 / d 1/d 1/ d d \sqrt{d} d α = α d \alpha= \alpha_d α = α d d α d 2 d\alpha_d^2 d α d 2 [ 0 , ∞ ] [0, \infty] [ 0 , ∞ ] δ 0 \delta_0 δ 0
The critical laws and their two endpoint laws form a continuous family in the first Wasserstein distance. At degree d d d d α 2 d\alpha^2 d α 2 α > 0 \alpha> 0 α > 0 Proposition 4.24 ).
For a finite positive limit d α d 2 → τ d\alpha_d^2 \to\tau d α d 2 → τ Theorem 4.10 ).
The case d α d 2 → 0 d\alpha_d^2 \to0 d α d 2 → 0
To describe this mass, we weight the tree law by the square of the spectral variable, normalize the result to a probability measure, and change the observation scale. These energy-biased laws converge if and only if d α d d\alpha_d d α d [ 0 , ∞ ] [0, \infty] [ 0 , ∞ ] d α d 2 → 0 d\alpha_d^2 \to0 d α d 2 → 0 Theorem 4.19 and Corollary 4.20 ). For example, all choices α d = c / d \alpha_d = c/d α d = c / d c > 0 c > 0 c > 0 δ 0 \delta_0 δ 0 c c c
Quantum-walk eigenphases and tree recursions. At high degree, the full unitary spectral law is dominated by structural atoms at ± 1 \pm1 ± 1 O ( d − 1 / 2 ) O(d^{-1/2}) O ( d − 1/2 ) α \alpha α O ( d − 1 ) O(d^{-1}) O ( d − 1 )
Removing an edge of the tree also gives a recursion for branch Green functions. The branch Green function remains dependent on the Dirichlet mass assigned to the parent direction. Their product satisfies a closed recursion. We prove uniqueness of its law as a random analytic function with positive imaginary part and the resolvent bound. We also prove the corresponding uniqueness result for the critical Poisson model (Theorems A.4 and 4.13 ).
Related work. The spectral relation between reflection walks and self-adjoint discriminants comes from abstract quantum-walk theory [19 , 27 ] . For sparse random operators, local weak convergence and resolvent methods yield spectral limits [6 , 8 ] . Closed-walk expansions provide another established approach to sparse and weighted random matrices [18 , 30 ] . We use these methods while keeping the mixed moments imposed by the Dirichlet constraint.
Poisson-weighted trees and recursive distributional equations are also established tools for random-operator limits [4 , 5 , 7 ] . In particular, Jung [14 ] represents spectral limits of certain random matrix ensembles using Poisson-weighted trees, with matrix entries independent up to Hermitian symmetry. Here, the incident coefficients share Dirichlet coordinates, and their mixed moments enter the closed-walk expansion. The resulting critical law has a specific Bessel intensity, while a second scaling recovers the spectral energy that is invisible in the collapsed weak limit.
The Kesten–McKay-to-semicircle transition gives the comparison without Dirichlet disorder [10 , 11 , 17 , 22 ] . Other work on tree operators concerns spectral stability, resonant delocalization, and trees of finite cone type [2 , 3 , 16 ] . Our support theorem identifies where the spectrum lies, without distinguishing pure point, absolutely continuous, and singular continuous parts. The Dirichlet Markov ensemble of Chafaï [9 ] and classical random walks in Dirichlet environment [26 ] provide further comparisons. Here we study the sparse self-adjoint discriminant and its unitary reflection walk.
The way randomness enters a quantum walk is important. Hamza and Joye study a different static-disorder model on homogeneous trees [12 ] . Quillen resamples a full Haar coin at every time step and studies the averaged channel [25 ] ; our environment is held fixed. In a 2026 preprint (arXiv:2607.02356), Maltsev and Osman study mesoscopic linear statistics for quantum graphs, including a Haar-scattering ensemble. Our local coins, by contrast, are reflections generated by rank-one projections. Higuchi and Segawa [13 ] use Dirichlet boundary conditions and finite-depth approximations on infinite trees. Their use of “Dirichlet” refers to a boundary condition, not to the probability distribution at a vertex.
Section 2 defines the model and its spectral mapping. Section 3 proves the fixed-degree results. Section 4 establishes the two high-degree limits, the Poisson representation, and the uniform approximation. Section 5 discusses further questions, and Appendix A develops the finite-degree recursion.
Notation. We use N = { 1 , 2 , … } \mathbb{N} = \{1,2,\ldots\} N = { 1 , 2 , … } T = { z ∈ C : ∣ z ∣ = 1 } \mathbb{T} = \{z \in\mathbb{C} : \lvert z\rvert= 1\} T = { z ∈ C : ∣ z ∣ = 1 } C + = { z ∈ C : Im z > 0 } \mathbb{C}_{+} = \{z \in\mathbb{C} : \operatorname{Im} z > 0\} C + = { z ∈ C : Im z > 0 } B ( S ) \mathcal{B}(S) B ( S ) S S S f # μ f_{\#}\mu f # μ μ \mu μ f f f μ d , α \mu_{d,\alpha} μ d , α ν d , α \nu_{d,\alpha} ν d , α γ \gamma γ H γ H_{\gamma} H γ e ( γ ) = ∣ E ( H γ ) ∣ e(\gamma)=\lvert E(H_{\gamma})\rvert e ( γ ) = ∣ E ( H γ )∣ α r \alpha_{r} α r x r x^{r} x r ϵ d , α \epsilon_{d,\alpha} ϵ d , α τ d \tau_{d} τ d θ d \theta_{d} θ d s d s_{d} s d R \mathbb{R} R W 1 ( μ , ν ) = inf E ∣ X − Y ∣ W_{1}(\mu,\nu)=\inf\mathbb{E}\lvert X-Y\rvert W 1 ( μ , ν ) = inf E ∣ X − Y ∣ μ \mu μ ν \nu ν ∣ x − y ∣ \lvert x-y\rvert ∣ x − y ∣ W 1 W_{1} W 1 [29 Chapter 6] .
Random reflection model and discriminant reduction
The random reflection model
Fix an integer d ≥ 3 d \ge3 d ≥ 3 G = ( V , E ) G = (V,E) G = ( V , E ) d d d
A → = { ( v , u ) : { v , u } ∈ E } A^{\to} = \{(v,u) : \{v,u\} \in E\} A → = {( v , u ) : { v , u } ∈ E } be its set of directed edges. The walk space is
H G = ℓ 2 ( A → ) . \mathcal{H}_G = \ell^2(A^{\to}). H G = ℓ 2 ( A → ) . Fix an ordering of the d d d
ω v = ( X v , ξ v ) , X v = ( X v , u ) u ∼ v , ξ v = ( ξ v , u ) u ∼ v , \omega_v = (X_v,\xi_v), \qquad X_v = (X_{v,u})_{u \sim v}, \qquad\xi_v = (\xi_{v,u})_{u \sim v}, ω v = ( X v , ξ v ) , X v = ( X v , u ) u ∼ v , ξ v = ( ξ v , u ) u ∼ v , are independent and identically distributed over the vertices. The marginal law of the squared amplitudes is
X v ∼ Dirichlet ( α , … , α ) , α > 0 , X_v \sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha), \qquad\alpha> 0, X v ∼ Dirichlet ( α , … , α ) , α > 0 , and ξ v ∈ T d \xi_v \in\mathbb{T}^d ξ v ∈ T d X v X_v X v X v X_v X v
q v , u = X v , u ξ v , u , q v = ( q v , u ) u ∼ v . q_{v,u} = \sqrt{X_{v,u}}\,\xi_{v,u}, \qquad q_v = (q_{v,u})_{u \sim v}. q v , u = X v , u ξ v , u , q v = ( q v , u ) u ∼ v . Then ∥ q v ∥ = 1 \lVert q_v \rVert= 1 ∥ q v ∥ = 1
C v = 2 q v q v ∗ − I d , C_v = 2q_v q_v^* - I_d, C v = 2 q v q v ∗ − I d , and the global coin is C = ⨁ v ∈ V C v C = \bigoplus_{v \in V} C_v C = ⨁ v ∈ V C v
( S ψ ) ( v , u ) = ψ ( u , v ) . (S\psi)(v,u) = \psi(u,v). ( S ψ ) ( v , u ) = ψ ( u , v ) . Thus S = S ∗ = S − 1 S = S^* = S^{-1} S = S ∗ = S − 1
The canonical amplitude-disorder model is obtained by taking ξ v , u = 1 \xi_{v,u} = 1 ξ v , u = 1 α = 1 / 2 \alpha= 1/2 α = 1/2 α = 1 \alpha= 1 α = 1
The phrase weak amplitude disorder refers to the limit α → ∞ \alpha\to\infty α → ∞ ξ v , u = 1 \xi_{v,u} = 1 ξ v , u = 1
Discriminant and spectral mapping
The next identities reduce the spectral analysis of the unitary walk to a self-adjoint operator on vertices. The characteristic-polynomial formula also keeps track of the contributions at the endpoints ± 1 \pm1 ± 1
Define K : H G → ℓ 2 ( V ) K:\mathcal{H}_{G}\to\ell^{2}(V) K : H G → ℓ 2 ( V )
( K ψ ) ( v ) = ∑ u ∼ v q v , u ‾ ψ ( v , u ) . (K\psi)(v)=\sum_{u\sim v}\overline{q_{v,u}}\psi(v,u). ( K ψ ) ( v ) = u ∼ v ∑ q v , u ψ ( v , u ) . Then
( K ∗ f ) ( v , u ) = q v , u f ( v ) , K K ∗ = I ℓ 2 ( V ) . (K^{*}f)(v,u)=q_{v,u}f(v),\qquad KK^{*}=I_{\ell^{2}(V)}. ( K ∗ f ) ( v , u ) = q v , u f ( v ) , K K ∗ = I ℓ 2 ( V ) . and
C = 2 K ∗ K − I H G . C=2K^{*}K-I_{\mathcal{H}_{G}}. C = 2 K ∗ K − I H G . The discriminant is
Proposition 2.2 (Self-adjoint reduction). The operator T T T ℓ 2 ( V ) \ell^{2}(V) ℓ 2 ( V )
T v , u = { q v , u ‾ q u , v , u ∼ v , 0 , u ≁ v . T_{v,u}=
\begin{cases}
\overline{q_{v,u}}q_{u,v}, & u\sim v,\\
0, & u\not\sim v.
\end{cases} T v , u = { q v , u q u , v , 0 , u ∼ v , u ∼ v . In particular,
∣ T v , u ∣ = X v , u X u , v . |T_{v,u}|=\sqrt{X_{v,u}X_{u,v}}. ∣ T v , u ∣ = X v , u X u , v . Proof. Since S = S ∗ S=S^{*} S = S ∗ T ∗ = K S ∗ K ∗ = T T^{*}=KS^{*}K^{*}=T T ∗ = K S ∗ K ∗ = T ∥ T ∥ ≤ ∥ K ∥ ∥ S ∥ ∥ K ∗ ∥ = 1 \|T\|\leq\|K\|\|S\|\|K^{*}\|=1 ∥ T ∥ ≤ ∥ K ∥∥ S ∥∥ K ∗ ∥ = 1
⟨ δ v , T δ u ⟩ = q v , u ‾ q u , v , \langle\delta_{v},T\delta_{u}\rangle=\overline{q_{v,u}}q_{u,v}, ⟨ δ v , T δ u ⟩ = q v , u q u , v , and the entry is zero otherwise. In particular, T u , v = T v , u ‾ T_{u,v}=\overline{T_{v,u}} T u , v = T v , u
Lemma 2.3 (Gauge reduction on a tree). Let G G G o o o D D D ℓ 2 ( V ) \ell^{2}(V) ℓ 2 ( V ) D δ o = δ o D\delta_{o}=\delta_{o} D δ o = δ o
( D ∗ T D ) v , u = ∣ T v , u ∣ = X v , u X u , v ( u ∼ v ) . (D^{*}TD)_{v,u}=|T_{v,u}|=\sqrt{X_{v,u}X_{u,v}}\qquad(u\sim v). ( D ∗ T D ) v , u = ∣ T v , u ∣ = X v , u X u , v ( u ∼ v ) . Consequently, the spectrum of T T T X X X
Proof. On each edge with a v , u : = ∣ T v , u ∣ > 0 a_{v,u}:=|T_{v,u}|>0 a v , u := ∣ T v , u ∣ > 0 T v , u = a v , u e i ϑ v , u T_{v,u}=a_{v,u}e^{i\vartheta_{v,u}} T v , u = a v , u e i ϑ v , u ϑ u , v = − ϑ v , u \vartheta_{u,v}=-\vartheta_{v,u} ϑ u , v = − ϑ v , u
Let n = ∣ V ∣ n=|V| n = ∣ V ∣ m = ∣ E ∣ = d n / 2 m=|E|=dn/2 m = ∣ E ∣ = d n /2
Proposition 2.4 (Characteristic polynomial and trace identity). For every z ∈ C z\in\mathbb{C} z ∈ C
det ( z I 2 m − U ) = ( z 2 − 1 ) m − n det ( ( z 2 + 1 ) I n − 2 z T ) . \det(zI_{2m}-U)=(z^2-1)^{m-n}\det\left((z^2+1)I_n-2zT\right). det ( z I 2 m − U ) = ( z 2 − 1 ) m − n det ( ( z 2 + 1 ) I n − 2 z T ) . If T k \mathbf{T}_k T k
Tr U k = ( m − n ) ( 1 + ( − 1 ) k ) + 2 Tr T k ( T ) , k ≥ 1. \operatorname{Tr}U^k=(m-n)\left(1+(-1)^k\right)+2\operatorname{Tr}\mathbf{T}_k(T),\qquad k\geq1. Tr U k = ( m − n ) ( 1 + ( − 1 ) k ) + 2 Tr T k ( T ) , k ≥ 1. Proof. For z 2 ≠ 1 z^2\ne1 z 2 = 1
z I − U = z I + S − 2 S K ∗ K . zI-U=zI+S-2SK^*K. z I − U = z I + S − 2 S K ∗ K . Since
( z I + S ) − 1 = z I − S z 2 − 1 , (zI+S)^{-1}=\frac{zI-S}{z^2-1}, ( z I + S ) − 1 = z 2 − 1 z I − S , and the flip–flop shift is a direct sum of m m m det ( z I + S ) = ( z 2 − 1 ) m \det(zI+S)=(z^2-1)^m det ( z I + S ) = ( z 2 − 1 ) m
det ( z I − U ) = det ( z I + S ) det ( I n − 2 K ( z I + S ) − 1 S K ∗ ) = ( z 2 − 1 ) m det ( I n − 2 ( z T − I n ) z 2 − 1 ) = ( z 2 − 1 ) m − n det ( ( z 2 + 1 ) I n − 2 z T ) . \begin{aligned}
\det(zI-U)&=\det(zI+S)\det\left(I_n-2K(zI+S)^{-1}SK^*\right)\\
&=(z^2-1)^m\det\left(I_n-\frac{2(zT-I_n)}{z^2-1}\right)\\
&=(z^2-1)^{m-n}\det\left((z^2+1)I_n-2zT\right).
\end{aligned} det ( z I − U ) = det ( z I + S ) det ( I n − 2 K ( z I + S ) − 1 S K ∗ ) = ( z 2 − 1 ) m det ( I n − z 2 − 1 2 ( z T − I n ) ) = ( z 2 − 1 ) m − n det ( ( z 2 + 1 ) I n − 2 z T ) . Both sides are polynomials, so the identity extends to z = ± 1 z=\pm1 z = ± 1
If λ = cos θ ∈ [ − 1 , 1 ] \lambda=\cos\theta\in[-1,1] λ = cos θ ∈ [ − 1 , 1 ] T T T e ± i θ e^{\pm i\theta} e ± i θ Tr U k \operatorname{Tr}U^k Tr U k
e i k θ + e − i k θ = 2 T k ( λ ) . e^{ik\theta}+e^{-ik\theta}=2\mathbf{T}_k(\lambda). e ik θ + e − ik θ = 2 T k ( λ ) . The factor ( z 2 − 1 ) m − n (z^2-1)^{m-n} ( z 2 − 1 ) m − n
Corollary 2.5 (Exact relation between empirical laws). Let
μ T = 1 n ∑ j = 1 n δ λ j ( T ) \mu_T=\frac{1}{n}\sum_{j=1}^{n}\delta_{\lambda_j(T)} μ T = n 1 j = 1 ∑ n δ λ j ( T ) and let ν U \nu_U ν U U U U 2 m = d n 2m=dn 2 m = d n
ϕ ± ( x ) = x ± i 1 − x 2 , \phi_{\pm}(x)=x\pm i\sqrt{1-x^2}, ϕ ± ( x ) = x ± i 1 − x 2 , one has
ν U = d − 2 2 d ( δ 1 + δ − 1 ) + 1 d ( ϕ + ) # μ T + 1 d ( ϕ − ) # μ T . \nu_U=\frac{d-2}{2d}(\delta_1+\delta_{-1})+\frac{1}{d}(\phi_+)_{\#}\mu_T+\frac{1}{d}(\phi_-)_{\#}\mu_T. ν U = 2 d d − 2 ( δ 1 + δ − 1 ) + d 1 ( ϕ + ) # μ T + d 1 ( ϕ − ) # μ T . When T T T ± 1 \pm1 ± 1 ϕ + \phi_+ ϕ + ϕ − \phi_- ϕ − ( z ∓ 1 ) 2 (z\mp1)^2 ( z ∓ 1 ) 2
Definition 2.6 (Normalized inherited spectral law). The normalized inherited spectral law is defined by the two inverse Joukowsky branches:
ν T i n h = 1 2 ( ϕ + ) # μ T + 1 2 ( ϕ − ) # μ T . \nu_T^{\mathrm{inh}} = \frac{1}{2}(\phi_+)_{\#}\mu_T + \frac{1}{2}(\phi_-)_{\#}\mu_T. ν T inh = 2 1 ( ϕ + ) # μ T + 2 1 ( ϕ − ) # μ T . Consequently,
ν U = ( 1 − 2 d ) δ 1 + δ − 1 2 + 2 d ν T i n h . \nu_U = \left(1-\frac{2}{d}\right)\frac{\delta_1+\delta_{-1}}{2}+\frac{2}{d}\nu_T^{\mathrm{inh}}. ν U = ( 1 − d 2 ) 2 δ 1 + δ − 1 + d 2 ν T inh . The first term is the structural contribution from the flip–flop construction; the second is the inherited contribution. Multiplicities follow the characteristic-polynomial factorization, so both branches are counted even when they coincide at an endpoint.
The inherited part has total weight 2 / d 2/d 2/ d
Infinite-volume spectral law
Quenched convergence on locally tree-like graphs
Let I ⊂ N \mathcal{I} \subset\mathbb{N} I ⊂ N G n = ( V n , E n ) G_n=(V_n,E_n) G n = ( V n , E n ) n ∈ I n \in\mathcal{I} n ∈ I d d d ∣ V n ∣ = n |V_n|=n ∣ V n ∣ = n n ≥ d + 1 n \ge d+1 n ≥ d + 1 d n dn d n n ∈ I n \in\mathcal{I} n ∈ I n → ∞ n \to\infty n → ∞ I \mathcal{I} I B R ( G , v ) B_R(G,v) B R ( G , v ) R R R T d \mathbb{T}_d T d d d d o o o ≃ \simeq ≃ R R R
1 n # { v ∈ V n : B R ( G n , v ) ≄ B R ( T d , o ) } ⟶ 0. \frac{1}{n}\#\{v \in V_n : B_R(G_n,v) \not\simeq B_R(\mathbb{T}_d,o)\} \longrightarrow0. n 1 # { v ∈ V n : B R ( G n , v ) ≃ B R ( T d , o )} ⟶ 0. To treat the entire graph sequence on one probability space, place the marked stars on the product space
∏ n ∈ I ∏ v ∈ V n Ω n , v , \prod_{n\in\mathcal{I}}\prod_{v\in V_n}\Omega_{n,v}, n ∈ I ∏ v ∈ V n ∏ Ω n , v , with independent copies of the common one-vertex law, and let T n T_n T n
On the infinite d d d T d \mathbb{T}_d T d A → ( T d ) A^{\to}(\mathbb{T}_d) A → ( T d )
H ∞ = ℓ 2 ( A → ( T d ) ) , C ∞ = ⨁ v ∈ T d ( 2 q v q v ∗ − I d ) , \mathcal{H}_{\infty}=\ell^2(A^{\to}(\mathbb{T}_d)),\qquad C_{\infty}=\bigoplus_{v\in\mathbb{T}_d}(2q_vq_v^*-I_d), H ∞ = ℓ 2 ( A → ( T d )) , C ∞ = v ∈ T d ⨁ ( 2 q v q v ∗ − I d ) , and define the flip–flop shift S ∞ S_{\infty} S ∞ K ∞ K_{\infty} K ∞
U ∞ = S ∞ C ∞ and T ∞ = K ∞ S ∞ K ∞ ∗ . U_{\infty}=S_{\infty}C_{\infty}\qquad\text{and}\qquad T_{\infty}=K_{\infty}S_{\infty}K_{\infty}^*. U ∞ = S ∞ C ∞ and T ∞ = K ∞ S ∞ K ∞ ∗ . The direct-sum coin and the shift are unitary, K ∞ K ∞ ∗ = I K_{\infty}K_{\infty}^{*}=I K ∞ K ∞ ∗ = I U ∞ U_{\infty} U ∞ T ∞ T_{\infty} T ∞
μ d , α ( B ) = E [ ⟨ δ o , 1 B ( T ∞ ) δ o ⟩ ] . \mu_{d,\alpha}(B)=\mathbb{E}\left[\langle\delta_o,\mathbf{1}_B(T_{\infty})\delta_o\rangle\right]. μ d , α ( B ) = E [ ⟨ δ o , 1 B ( T ∞ ) δ o ⟩ ] . Measurability of the rooted spectral measure follows from the standard theory of measurable bounded local operators; see [6 , 8 ] .
Theorem 3.1 (Quenched limiting spectral law). For almost every realization of the vertex marks,
μ T n = 1 n ∑ j = 1 n δ λ j ( T n ) ⟹ μ d , α . \mu_{T_n}=\frac{1}{n}\sum_{j=1}^{n}\delta_{\lambda_j(T_n)}\Longrightarrow\mu_{d,\alpha}. μ T n = n 1 j = 1 ∑ n δ λ j ( T n ) ⟹ μ d , α . The limit is independent of the admissible conditional phase law.
Proof. Fix k ≥ 1 k\ge1 k ≥ 1
M n , k = 1 n Tr T n k M_{n,k}=\frac{1}{n}\operatorname{Tr}T_n^k M n , k = n 1 Tr T n k and
G n , k = { v ∈ V n : B ⌈ k / 2 ⌉ ( G n , v ) ≃ B ⌈ k / 2 ⌉ ( T d , o ) } . \mathcal{G}_{n,k}=\left\{v\in V_n:B_{\lceil k/2\rceil}(G_n,v)\simeq B_{\lceil k/2\rceil}(T_d,o)\right\}. G n , k = { v ∈ V n : B ⌈ k /2 ⌉ ( G n , v ) ≃ B ⌈ k /2 ⌉ ( T d , o ) } . A closed walk of length k k k v v v v ∈ G n , k v\in\mathcal{G}_{n,k} v ∈ G n , k
E ⟨ δ v , T n k δ v ⟩ = m k ( d , α ) : = E ⟨ δ o , T ∞ k δ o ⟩ = ∫ − 1 1 x k μ d , α ( d x ) . \mathbb{E}\langle\delta_v,T_n^k\delta_v\rangle=m_k(d,\alpha):=\mathbb{E}\langle\delta_o,T_{\infty}^k\delta_o\rangle=\int_{-1}^{1}x^k\,\mu_{d,\alpha}(\mathrm{d}x). E ⟨ δ v , T n k δ v ⟩ = m k ( d , α ) := E ⟨ δ o , T ∞ k δ o ⟩ = ∫ − 1 1 x k μ d , α ( d x ) . Since both the diagonal entry and m k ( d , α ) m_k(d,\alpha) m k ( d , α )
∣ E M n , k − m k ( d , α ) ∣ ≤ 2 ( 1 − ∣ G n , k ∣ n ) ⟶ 0. \left|\mathbb{E}M_{n,k}-m_k(d,\alpha)\right|\leq2\left(1-\frac{|\mathcal{G}_{n,k}|}{n}\right)\longrightarrow0. ∣ E M n , k − m k ( d , α ) ∣ ≤ 2 ( 1 − n ∣ G n , k ∣ ) ⟶ 0. Let T ~ n \widetilde{T}_n T n v v v v v v A = T n − T ~ n A=T_n-\widetilde{T}_n A = T n − T n b b b v v v
A = δ v b ∗ + b δ v ∗ . A=\delta_v b^{*}+b\delta_v^{*}. A = δ v b ∗ + b δ v ∗ . This decomposition gives rank A ≤ 2 \operatorname{rank}A\leq2 rank A ≤ 2 ∥ ⋅ ∥ 1 \|\cdot\|_1 ∥ ⋅ ∥ 1 ∥ A ∥ ≤ ∥ T n ∥ + ∥ T ~ n ∥ ≤ 2 \|A\|\leq\|T_n\|+\|\widetilde{T}_n\|\leq2 ∥ A ∥ ≤ ∥ T n ∥ + ∥ T n ∥ ≤ 2 ∥ A ∥ 1 ≤ 4 \|A\|_1\leq4 ∥ A ∥ 1 ≤ 4
T n k − T ~ n k = ∑ j = 0 k − 1 T n j A T ~ n k − 1 − j T_n^k-\widetilde{T}_n^k=\sum_{j=0}^{k-1}T_n^jA\widetilde{T}_n^{k-1-j} T n k − T n k = j = 0 ∑ k − 1 T n j A T n k − 1 − j therefore gives
∣ Tr T n k − Tr T ~ n k ∣ ≤ ∥ T n k − T ~ n k ∥ 1 ≤ 4 k . \left|\operatorname{Tr} T_n^k-\operatorname{Tr}\widetilde{T}_n^k\right|\leq\left\|T_n^k-\widetilde{T}_n^k\right\|_1\leq4k. Tr T n k − Tr T n k ≤ T n k − T n k 1 ≤ 4 k . McDiarmid’s bounded-difference inequality [21 Lemma (1.2), p. 149] gives, for every t > 0 t>0 t > 0
P ( ∣ M n , k − E M n , k ∣ > t ) ≤ 2 exp ( − n t 2 8 k 2 ) . \mathbb{P}\left(\left|M_{n,k}-\mathbb{E}M_{n,k}\right|>t\right)\leq2\exp\left(-\frac{nt^2}{8k^2}\right). P ( ∣ M n , k − E M n , k ∣ > t ) ≤ 2 exp ( − 8 k 2 n t 2 ) . For each k k k M n , k → m k ( d , α ) M_{n,k}\to m_k(d,\alpha) M n , k → m k ( d , α ) k k k μ T n \mu_{T_n} μ T n [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
The same phase cancellation shows that the limiting law does not depend on the admissible conditional phase distribution.
Define the probability measure on the unit circle
ν d , α = d − 2 2 d ( δ 1 + δ − 1 ) + 1 d ( ϕ + ) # μ d , α + 1 d ( ϕ − ) # μ d , α , \nu_{d,\alpha}=\frac{d-2}{2d}\left(\delta_1+\delta_{-1}\right)+\frac{1}{d}(\phi_+)_\#\mu_{d,\alpha}+\frac{1}{d}(\phi_-)_\#\mu_{d,\alpha}, ν d , α = 2 d d − 2 ( δ 1 + δ − 1 ) + d 1 ( ϕ + ) # μ d , α + d 1 ( ϕ − ) # μ d , α , where ϕ ± ( x ) = x ± i 1 − x 2 \phi_\pm(x)=x\pm i\sqrt{1-x^2} ϕ ± ( x ) = x ± i 1 − x 2
Corollary 3.2 (Limiting law of the quantum walks). Let ν U n \nu_{U_n} ν U n U n U_n U n d ∣ V n ∣ d\lvert V_n\rvert d ∣ V n ∣ Theorem 3.1 ,
ν U n ⟹ ν d , α almost surely . \nu_{U_n}\Longrightarrow\nu_{d,\alpha}\qquad\text{almost surely}. ν U n ⟹ ν d , α almost surely . The measure ν d , α \nu_{d,\alpha} ν d , α
Proof. Apply Corollary 2.5 and preservation of weak convergence under the continuous maps ϕ + \phi_+ ϕ + ϕ − \phi_- ϕ −
Dirichlet closed-walk calculus
A diagonal entry of a power of the discriminant is a sum over closed walks. On a tree, each edge is crossed equally often in both directions, so its phase cancels. The remaining calculation uses independence across vertices and the Dirichlet mixed moments within each vertex.
For j ≥ 0 j\geq0 j ≥ 0
m j ( d , α ) = ∫ − 1 1 x j μ d , α ( d x ) . m_j(d,\alpha)=\int_{-1}^{1}x^j\,\mu_{d,\alpha}(\mathrm{d}x). m j ( d , α ) = ∫ − 1 1 x j μ d , α ( d x ) . For a rooted closed walk
γ = ( v 0 = o , v 1 , … , v 2 k = o ) \gamma=(v_0=o,v_1,\ldots,v_{2k}=o) γ = ( v 0 = o , v 1 , … , v 2 k = o ) on T d \mathbb{T}_d T d V ( γ ) V(\gamma) V ( γ ) E ( γ ) E(\gamma) E ( γ ) e = { v , u } e=\{v,u\} e = { v , u } v v v X v , e : = X v , u X_{v,e}:=X_{v,u} X v , e := X v , u r e ( γ ) r_e(\gamma) r e ( γ ) e e e γ \gamma γ e e e 2 r e ( γ ) 2r_e(\gamma) 2 r e ( γ )
R v ( γ ) = ∑ e ∈ E ( γ ) e ∋ v r e ( γ ) . R_v(\gamma)=\sum_{\substack{e\in E(\gamma)\\ e\ni v}}r_e(\gamma). R v ( γ ) = e ∈ E ( γ ) e ∋ v ∑ r e ( γ ) . Consequently,
∑ e ∈ E ( γ ) r e ( γ ) = k , ∑ v ∈ V ( γ ) R v ( γ ) = 2 k . \sum_{e\in E(\gamma)} r_e(\gamma)=k,\qquad\sum_{v\in V(\gamma)} R_v(\gamma)=2k. e ∈ E ( γ ) ∑ r e ( γ ) = k , v ∈ V ( γ ) ∑ R v ( γ ) = 2 k . We use the rising factorial
( a ) r = a ( a + 1 ) ⋯ ( a + r − 1 ) , ( a ) 0 = 1. (a)_r=a(a+1)\cdots(a+r-1),\qquad(a)_0=1. ( a ) r = a ( a + 1 ) ⋯ ( a + r − 1 ) , ( a ) 0 = 1. Theorem 3.3 (All-order moment formula). Each rooted closed-walk sequence in T d \mathbb{T}_d T d μ d , α \mu_{d,\alpha} μ d , α
m 2 k + 1 ( d , α ) = 0 , m_{2k+1}(d,\alpha)=0, m 2 k + 1 ( d , α ) = 0 , while
m 2 k ( d , α ) = ∑ γ : o → o ∣ γ ∣ = 2 k ∏ v ∈ V ( γ ) ∏ e ∈ E ( γ ) e ∋ v ( α ) r e ( γ ) ( d α ) R v ( γ ) = ∑ γ : o → o ∣ γ ∣ = 2 k ∏ e ∈ E ( γ ) ( α ) r e ( γ ) 2 ∏ v ∈ V ( γ ) ( d α ) R v ( γ ) . \begin{aligned}
m_{2k}(d,\alpha)
&=\sum_{\substack{\gamma:o\to o\\|\gamma|=2k}}\prod_{v\in V(\gamma)}
\frac{\prod_{\substack{e\in E(\gamma)\\e\ni v}}(\alpha)_{r_e(\gamma)}}{(d\alpha)^{R_v(\gamma)}}\\
&=\sum_{\substack{\gamma:o\to o\\|\gamma|=2k}}
\frac{\prod_{e\in E(\gamma)}(\alpha)_{r_e(\gamma)}^2}
{\prod_{v\in V(\gamma)}(d\alpha)^{R_v(\gamma)}}.
\end{aligned} m 2 k ( d , α ) = γ : o → o ∣ γ ∣ = 2 k ∑ v ∈ V ( γ ) ∏ ( d α ) R v ( γ ) ∏ e ∈ E ( γ ) e ∋ v ( α ) r e ( γ ) = γ : o → o ∣ γ ∣ = 2 k ∑ ∏ v ∈ V ( γ ) ( d α ) R v ( γ ) ∏ e ∈ E ( γ ) ( α ) r e ( γ ) 2 . In particular,
m 2 ( d , α ) = 1 d m_2(d,\alpha)=\frac{1}{d} m 2 ( d , α ) = d 1 and
m 4 ( d , α ) = 2 α ( d − 1 ) d 2 ( d α + 1 ) + ( α + 1 ) 2 d ( d α + 1 ) 2 . m_4(d,\alpha)=\frac{2\alpha(d-1)}{d^2(d\alpha+1)}+\frac{(\alpha+1)^2}{d(d\alpha+1)^2}. m 4 ( d , α ) = d 2 ( d α + 1 ) 2 α ( d − 1 ) + d ( d α + 1 ) 2 ( α + 1 ) 2 . Proof. Let J J J
( J f ) ( v ) = ( − 1 ) dist ( o , v ) f ( v ) . (Jf)(v)=(-1)^{\operatorname{dist}(o,v)}f(v). ( J f ) ( v ) = ( − 1 ) dist ( o , v ) f ( v ) . Every edge joins opposite parity classes, so J T ∞ J = − T ∞ JT_\infty J=-T_\infty J T ∞ J = − T ∞ J δ o = δ o J\delta_o=\delta_o J δ o = δ o μ d , α \mu_{d,\alpha} μ d , α
∏ e = { x , y } ( X x , y X y , x ) r e ( γ ) . \prod_{e=\{x,y\}}(X_{x,y}X_{y,x})^{r_e(\gamma)}. e = { x , y } ∏ ( X x , y X y , x ) r e ( γ ) . The vertex marks are independent. At a fixed vertex, the Dirichlet moment identity gives
E [ ∏ e ∈ E ( γ ) e ∋ v X v , e r e ] = ∏ e ∈ E ( γ ) e ∋ v ( α ) r e ( d α ) R v ( γ ) . \mathbb{E}\left[\prod_{\substack{e\in E(\gamma)\\e\ni v}}X_{v,e}^{r_e}\right]
=\frac{\prod_{\substack{e\in E(\gamma)\\e\ni v}}(\alpha)_{r_e}}{(d\alpha)^{R_v(\gamma)}}. E e ∈ E ( γ ) e ∋ v ∏ X v , e r e = ( d α ) R v ( γ ) ∏ e ∈ E ( γ ) e ∋ v ( α ) r e . Independence across vertices lets us multiply these factors. Summing over closed walks gives the first expression; the second follows because each traced edge has exactly two endpoints.
At length two, there are d d d o → u → o o \to u \to o o → u → o d − 2 d^{-2} d − 2 m 2 ( d , α ) = 1 / d m_2(d,\alpha)=1/d m 2 ( d , α ) = 1/ d o → u → o → w → o o \to u \to o \to w \to o o → u → o → w → o u ≠ w u \ne w u = w o → u → w → u → o o \to u \to w \to u \to o o → u → w → u → o w ≠ o w \ne o w = o d ( d − 1 ) d(d-1) d ( d − 1 )
α d 3 ( d α + 1 ) . \frac{\alpha}{d^3(d\alpha+1)}. d 3 ( d α + 1 ) α . The alternating type o → u → o → u → o o \to u \to o \to u \to o o → u → o → u → o d d d
[ α + 1 d ( d α + 1 ) ] 2 . \left[\frac{\alpha+1}{d(d\alpha+1)}\right]^2. [ d ( d α + 1 ) α + 1 ] 2 . Adding the three contributions gives the stated formula for m 4 m_4 m 4
A general rare-path criterion and the Dirichlet specialization
Full spectrum can arise from rare paths on which two directions at each interior vertex receive nearly half the mass each. The remaining mass is small, so the path is weakly coupled to the rest of the tree. This mechanism applies beyond the Dirichlet distribution. Let
Δ d − 1 = { x ∈ [ 0 , 1 ] d : ∑ j = 1 d x j = 1 } . \Delta_{d-1}=\left\{x\in[0,1]^d:\sum_{j=1}^{d}x_j=1\right\}. Δ d − 1 = { x ∈ [ 0 , 1 ] d : j = 1 ∑ d x j = 1 } . and let L \mathcal{L} L Δ d − 1 \Delta_{d-1} Δ d − 1 L \mathcal{L} L half-split support condition if, for every η ∈ ( 0 , 1 / 4 ) \eta\in(0,1/4) η ∈ ( 0 , 1/4 )
L ( { x : ∣ x 1 − 1 2 ∣ < η , ∣ x 2 − 1 2 ∣ < η } ) > 0. (1) \mathcal{L}\left(\left\{x:\left|x_1-\frac{1}{2}\right|<\eta,\ \left|x_2-\frac{1}{2}\right|<\eta\right\}\right)>0. \tag*{(1)} L ( { x : x 1 − 2 1 < η , x 2 − 2 1 < η } ) > 0. (1) Attach independent L \mathcal{L} L T d \mathbb{T}_d T d T L T_{\mathcal{L}} T L
( T L ) u , v = X u , v X v , u . (T_{\mathcal{L}})_{u,v}=\sqrt{X_{u,v}X_{v,u}}. ( T L ) u , v = X u , v X v , u . It is a self-adjoint contraction by the same discriminant construction as in Section 2. Write
μ L ( B ) = E ⟨ δ o , 1 B ( T L ) δ o ⟩ . \mu_{\mathcal{L}}(B)=\mathbb{E}\langle\delta_o,\mathbf{1}_B(T_{\mathcal{L}})\delta_o\rangle. μ L ( B ) = E ⟨ δ o , 1 B ( T L ) δ o ⟩ . The symmetric Dirichlet law satisfies (H). For every η > 0 \eta>0 η > 0
For a simple path
P = ( v 0 , v 1 , … , v L , v L + 1 ) P=(v_0,v_1,\ldots,v_L,v_{L+1}) P = ( v 0 , v 1 , … , v L , v L + 1 ) and 0 < η < 1 / 4 0<\eta<1/4 0 < η < 1/4 G ( P , η ) \mathcal{G}(P,\eta) G ( P , η ) 1 ≤ j ≤ L 1\le j\le L 1 ≤ j ≤ L
∣ X v j , v j − 1 − 1 2 ∣ < η , \left|X_{v_j,v_{j-1}}-\frac{1}{2}\right|<\eta, X v j , v j − 1 − 2 1 < η , ∣ X v j , v j + 1 − 1 2 ∣ < η . \left|X_{v_j,v_{j+1}}-\frac{1}{2}\right|<\eta. X v j , v j + 1 − 2 1 < η . The simplex constraint then gives
∑ w ∼ v j w ∉ { v j − 1 , v j + 1 } X v j , w < 2 η . \sum_{\substack{w\sim v_j\\ w\notin\{v_{j-1},v_{j+1}\}}} X_{v_j,w}<2\eta. w ∼ v j w ∈ / { v j − 1 , v j + 1 } ∑ X v j , w < 2 η . Lemma 3.4 (Occurrence of rare paths). Assume (H). For every fixed L ≥ 2 L\ge2 L ≥ 2 η ∈ ( 0 , 1 / 4 ) \eta\in(0,1/4) η ∈ ( 0 , 1/4 ) G ( P , η ) \mathcal{G}(P,\eta) G ( P , η ) G ( P , η ) \mathcal{G}(P,\eta) G ( P , η )
Proof. At each interior vertex, the two path directions correspond to distinct coordinates. By permutation invariance and (H), the required inequalities hold with positive probability. Let p η > 0 p_\eta>0 p η > 0
P ( G ( P , η ) ) = p η L > 0. \mathbb{P}(\mathcal{G}(P,\eta))=p_\eta^L>0. P ( G ( P , η )) = p η L > 0. For a deterministic family of pairwise vertex-disjoint paths, the corresponding events depend on disjoint sets of marks and are therefore independent. The second Borel–Cantelli lemma proves the assertion. □ \square □
For L ≥ 1 L\ge1 L ≥ 1 J L J_L J L ℓ 2 ( { 1 , … , L } ) \ell^2(\{1,\ldots,L\}) ℓ 2 ({ 1 , … , L })
( J L f ) j = 1 2 f j − 1 + 1 2 f j + 1 , f 0 = f L + 1 = 0. (J_Lf)_j=\frac{1}{2}f_{j-1}+\frac{1}{2}f_{j+1},\qquad f_0=f_{L+1}=0. ( J L f ) j = 2 1 f j − 1 + 2 1 f j + 1 , f 0 = f L + 1 = 0. Its normalized eigenvectors and eigenvalues are
φ j ( L , r ) = 2 L + 1 sin ( r π j L + 1 ) , λ L , r = cos ( r π L + 1 ) , 1 ≤ r ≤ L . \varphi_j^{(L,r)}=\sqrt{\frac{2}{L+1}}\sin\left(\frac{r\pi j}{L+1}\right),\qquad\lambda_{L,r}=\cos\left(\frac{r\pi}{L+1}\right),\qquad1\le r\le L. φ j ( L , r ) = L + 1 2 sin ( L + 1 r π j ) , λ L , r = cos ( L + 1 r π ) , 1 ≤ r ≤ L . Lemma 3.5 (Rare-path quasimodes). Suppose G ( P , η ) \mathcal{G}(P,\eta) G ( P , η ) Extend φ ( L , r ) \varphi^{(L,r)} φ ( L , r ) v 1 , … , v L v_1,\ldots,v_L v 1 , … , v L ψ P , L , r \psi_{P,L,r} ψ P , L , r
∥ ( T L − λ ) ψ P , L , r ∥ ≤ ∣ λ L , r − λ ∣ + 2 η + 2 η + 4 L + 1 \left\|(T_{\mathcal{L}}-\lambda)\psi_{P,L,r}\right\|\le|\lambda_{L,r}-\lambda|+2\eta+\sqrt{2\eta+\frac{4}{L+1}} ∥ ( T L − λ ) ψ P , L , r ∥ ≤ ∣ λ L , r − λ ∣ + 2 η + 2 η + L + 1 4 for every λ ∈ R \lambda\in\mathbb{R} λ ∈ R
Proof. Let Q P Q_P Q P span { δ v 1 , … , δ v L } \operatorname{span}\{\delta_{v_1},\ldots,\delta_{v_L}\} span { δ v 1 , … , δ v L }
∣ X v j , v j + 1 X v j + 1 , v j − 1 2 ∣ < η , 1 ≤ j < L , \left|\sqrt{X_{v_j,v_{j+1}}X_{v_{j+1},v_j}}-\frac{1}{2}\right|<\eta,\qquad1\le j<L, X v j , v j + 1 X v j + 1 , v j − 2 1 < η , 1 ≤ j < L , because the geometric mean of two positive numbers lies between them. The compression of T L T_{\mathcal{L}} T L J L J_L J L 2 η 2\eta 2 η
∥ Q P ( T L − λ ) ψ P , L , r ∥ ≤ 2 η + ∣ λ L , r − λ ∣ . \left\|Q_P(T_{\mathcal{L}}-\lambda)\psi_{P,L,r}\right\|\le2\eta+|\lambda_{L,r}-\lambda|. ∥ Q P ( T L − λ ) ψ P , L , r ∥ ≤ 2 η + ∣ λ L , r − λ ∣. An outside vertex is adjacent to at most one of v 1 , … , v L v_1,\ldots,v_L v 1 , … , v L
X v j , w X w , v j ≤ X v j , w . X_{v_j,w}X_{w,v_j}\le X_{v_j,w}. X v j , w X w , v j ≤ X v j , w . Writing ψ = ψ P , L , r \psi=\psi_{P,L,r} ψ = ψ P , L , r
∥ ( I − Q P ) T L ψ ∥ 2 = ∑ j = 1 L ∣ φ j ( L , r ) ∣ 2 ∑ w ∉ { v 1 , … , v L } w ∼ v j X v j , w X w , v j . \left\|(I-Q_P)T_{\mathcal{L}}\psi\right\|^2=\sum_{j=1}^{L}\left|\varphi_j^{(L,r)}\right|^2\sum_{\substack{w\notin\{v_1,\ldots,v_L\}\\w\sim v_j}}X_{v_j,w}X_{w,v_j}. ∥ ( I − Q P ) T L ψ ∥ 2 = j = 1 ∑ L φ j ( L , r ) 2 w ∈ / { v 1 , … , v L } w ∼ v j ∑ X v j , w X w , v j . Off-path branches contribute at most 2 η 2\eta 2 η
∣ φ 1 ( L , r ) ∣ 2 + ∣ φ L ( L , r ) ∣ 2 ≤ 4 L + 1 , \left|\varphi_1^{(L,r)}\right|^2+\left|\varphi_L^{(L,r)}\right|^2\leq\frac{4}{L+1}, φ 1 ( L , r ) 2 + φ L ( L , r ) 2 ≤ L + 1 4 , since every edge weight is at most one. Combining the bound on the path support with the leakage bound proves the claim.
Theorem 3.6 (Rare-path full-spectrum criterion). If L \mathcal{L} L
σ ( T L ) = σ e s s ( T L ) = [ − 1 , 1 ] almost surely. \sigma(T_{\mathcal{L}})=\sigma_{\mathrm{ess}}(T_{\mathcal{L}})=[-1,1]\qquad\text{almost surely.} σ ( T L ) = σ ess ( T L ) = [ − 1 , 1 ] almost surely. Proof. The contraction property gives σ ( T L ) ⊂ [ − 1 , 1 ] \sigma(T_{\mathcal{L}})\subset[-1,1] σ ( T L ) ⊂ [ − 1 , 1 ] D D D [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] λ ∈ D \lambda\in D λ ∈ D L n ↑ ∞ L_n\uparrow\infty L n ↑ ∞ 1 ≤ r n ≤ L n 1\leq r_n\leq L_n 1 ≤ r n ≤ L n η n ↓ 0 \eta_n\downarrow0 η n ↓ 0
λ L n , r n ⟶ λ . \lambda_{L_n,r_n}\longrightarrow\lambda. λ L n , r n ⟶ λ . Choose a deterministic family ( P n , k ) n , k ≥ 1 (P_{n,k})_{n,k\geq1} ( P n , k ) n , k ≥ 1 P n , k P_{n,k} P n , k L n + 2 L_n+2 L n + 2 T d \mathbb{T}_d T d n n n η n \eta_n η n k k k ψ n \psi_n ψ n ψ n ⇀ 0 \psi_n\rightharpoonup0 ψ n ⇀ 0
∥ ( T L − λ ) ψ n ∥ ≤ ∣ λ L n , r n − λ ∣ + 2 η n + 2 η n + 4 L n + 1 ⟶ 0. \left\|(T_{\mathcal{L}}-\lambda)\psi_n\right\|\leq\left|\lambda_{L_n,r_n}-\lambda\right|+2\eta_n+\sqrt{2\eta_n+\frac{4}{L_n+1}}\longrightarrow0. ∥ ( T L − λ ) ψ n ∥ ≤ ∣ λ L n , r n − λ ∣ + 2 η n + 2 η n + L n + 1 4 ⟶ 0. Thus ( ψ n ) (\psi_n) ( ψ n ) D D D n n n [ − 1 , 1 ] ⊂ σ e s s ( T L ) [-1,1]\subset\sigma_{\mathrm{ess}}(T_{\mathcal{L}}) [ − 1 , 1 ] ⊂ σ ess ( T L )
Theorem 3.7 (Full support of the averaged tree law). Under the hypotheses of Theorem 3.6,
supp μ L = [ − 1 , 1 ] . \operatorname{supp}\mu_{\mathcal{L}}=[-1,1]. supp μ L = [ − 1 , 1 ] . For the symmetric Dirichlet environment, this gives supp μ d , α = [ − 1 , 1 ] \operatorname{supp}\mu_{d,\alpha}=[-1,1] supp μ d , α = [ − 1 , 1 ] The corresponding infinite-tree quantum walk satisfies
σ ( U ∞ ) = T almost surely, \sigma(U_\infty)=\mathbb{T}\qquad\text{almost surely,} σ ( U ∞ ) = T almost surely, and the limiting unitary law ν d , α \nu_{d,\alpha} ν d , α has support T \mathbb{T} T
Proof. Let I I I [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] P ω ( I ) = 1 I ( T L ( ω ) ) P_\omega(I)=\mathbb{1}_I(T_{\mathcal{L}}(\omega)) P ω ( I ) = 1 I ( T L ( ω )) μ L ( I ) = 0 \mu_{\mathcal{L}}(I)=0 μ L ( I ) = 0 v v v
E ⟨ δ v , P ω ( I ) δ v ⟩ = 0. \mathbb{E}\langle\delta_v,P_\omega(I)\delta_v\rangle=0. E ⟨ δ v , P ω ( I ) δ v ⟩ = 0. The integrand is nonnegative. Since the vertex set is countable, almost surely
⟨ δ v , P ω ( I ) δ v ⟩ = 0 for all v . \langle\delta_v,P_\omega(I)\delta_v\rangle=0 \qquad\text{for all }v. ⟨ δ v , P ω ( I ) δ v ⟩ = 0 for all v . As P ω ( I ) P_\omega(I) P ω ( I )
⟨ δ v , P ω ( I ) δ v ⟩ = ∥ P ω ( I ) δ v ∥ 2 , \langle\delta_v,P_\omega(I)\delta_v\rangle=\lVert P_\omega(I)\delta_v\rVert^2, ⟨ δ v , P ω ( I ) δ v ⟩ = ∥ P ω ( I ) δ v ∥ 2 , so P ω ( I ) δ v = 0 P_\omega(I)\delta_v=0 P ω ( I ) δ v = 0 P ω ( I ) = 0 P_\omega(I)=0 P ω ( I ) = 0 Theorem 3.6 . Every open interval intersecting [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] μ L \mu_{\mathcal{L}} μ L
For the Dirichlet reflection walk, K ∞ K ∞ ∗ = I K_\infty K_\infty^*=I K ∞ K ∞ ∗ = I U ∞ = S ∞ ( 2 K ∞ ∗ K ∞ − I ) U_\infty=S_\infty(2K_\infty^*K_\infty-I) U ∞ = S ∞ ( 2 K ∞ ∗ K ∞ − I ) [27 ] are satisfied. Away from the exceptional points, that theorem identifies the inherited unitary spectrum with the inverse Joukowsky image of ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) T ∖ { ± 1 } \mathbb{T}\setminus\{\pm1\} T ∖ { ± 1 } supp ν d , α = T \operatorname{supp}\nu_{d,\alpha}=\mathbb{T} supp ν d , α = T
Remark 3.8. The rare-path theorem applies to every permutation-invariant simplex law that satisfies the half-split condition. The simplex constraint controls leakage when the two path coordinates are close to one half. For the Dirichlet law, the explicit closed-walk factors additionally produce the scale d α d 2 d\alpha_d^2 d α d 2 § 4 .
Disorder endpoints
Here the degree remains fixed. As α ↓ 0 \alpha\downarrow0 α ↓ 0 α → ∞ \alpha\to\infty α → ∞
Let A T d A_{\mathbb{T}_d} A T d T d \mathbb{T}_d T d μ d K M \mu_d^{\mathrm{KM}} μ d KM A T d / d A_{\mathbb{T}_d}/d A T d / d
ρ d K M ( x ) = 4 ( d − 1 ) − d 2 x 2 2 π ( 1 − x 2 ) 1 { ∣ x ∣ ≤ 2 d − 1 / d } . \rho_d^{\mathrm{KM}}(x)=\frac{\sqrt{4(d-1)-d^2x^2}}{2\pi(1-x^2)}\mathbb{1}_{\{|x|\le2\sqrt{d-1}/d\}}. ρ d KM ( x ) = 2 π ( 1 − x 2 ) 4 ( d − 1 ) − d 2 x 2 1 { ∣ x ∣ ≤ 2 d − 1 / d } . Theorem 3.9 (Strong- and weak-amplitude-disorder limits). For fixed d ≥ 3 d\ge3 d ≥ 3
μ d , α ⟹ ( 1 − 1 d ) δ 0 + 1 2 d ( δ − 1 + δ 1 ) ( α ↓ 0 ) , \mu_{d,\alpha}\Longrightarrow\left(1-\frac{1}{d}\right)\delta_0+\frac{1}{2d}(\delta_{-1}+\delta_1)\qquad(\alpha\downarrow0), μ d , α ⟹ ( 1 − d 1 ) δ 0 + 2 d 1 ( δ − 1 + δ 1 ) ( α ↓ 0 ) , and
μ d , α ⟹ μ d K M ( α → ∞ ) . \mu_{d,\alpha}\Longrightarrow\mu_d^{\mathrm{KM}}\qquad(\alpha\to\infty). μ d , α ⟹ μ d KM ( α → ∞ ) . Proof. Fix k ≥ 1 k\ge1 k ≥ 1 α ↓ 0 \alpha\downarrow0 α ↓ 0 r ≥ 1 r\ge1 r ≥ 1
( α ) r ∼ α ( r − 1 ) ! , ( d α ) r ∼ d α ( r − 1 ) ! . (\alpha)_r\sim\alpha(r-1)!,\qquad(d\alpha)_r\sim d\alpha(r-1)!. ( α ) r ∼ α ( r − 1 )! , ( d α ) r ∼ d α ( r − 1 )! . Let the traced tree of a closed walk have e e e v v v deg H ( v ) \deg_H(v) deg H ( v ) v v v Theorem 3.3 is then of order α deg H ( v ) − 1 \alpha^{\deg_H(v)-1} α d e g H ( v ) − 1 e e e e + 1 e+1 e + 1 ∑ v deg H ( v ) = 2 e \sum_v \deg_H(v)=2e ∑ v deg H ( v ) = 2 e α e − 1 \alpha^{e-1} α e − 1 o ( 1 ) o(1) o ( 1 ) 2 k 2k 2 k o ( 1 ) o(1) o ( 1 ) k k k d d d
( α ) k ( d α ) k ⟶ 1 d . \frac{(\alpha)_k}{(d\alpha)_k} \longrightarrow\frac{1}{d}. ( d α ) k ( α ) k ⟶ d 1 . Consequently,
m 2 k ( d , α ) ⟶ d ( 1 d ) 2 = 1 d , m 2 k + 1 ( d , α ) = 0. m_{2k}(d,\alpha) \longrightarrow d\left(\frac{1}{d}\right)^2=\frac{1}{d}, \qquad m_{2k+1}(d,\alpha)=0. m 2 k ( d , α ) ⟶ d ( d 1 ) 2 = d 1 , m 2 k + 1 ( d , α ) = 0. These are the moments of the stated matching law. To interpret the limit, note that a symmetric Dirichlet vector converges as α ↓ 0 \alpha\downarrow0 α ↓ 0
As α → ∞ \alpha\to\infty α → ∞
∏ e ∋ v ( α ) r e ( d α ) R v ⟶ d − R v . \frac{\prod_{e\ni v}(\alpha)_{r_e}}{(d\alpha)_{R_v}}\longrightarrow d^{-R_v}. ( d α ) R v ∏ e ∋ v ( α ) r e ⟶ d − R v . Since ∑ v R v = 2 k \sum_v R_v=2k ∑ v R v = 2 k 2 k 2k 2 k d − 2 k d^{-2k} d − 2 k A T d / d \mathrm{AT}_d/d AT d / d [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
High-degree discriminant and eigenphase transitions
We now study the high-degree asymptotics of the infinite-tree law μ d , α \mu_{d,\alpha} μ d , α m 2 ( d , α ) = 1 / d m_2(d,\alpha)=1/d m 2 ( d , α ) = 1/ d d \sqrt{d} d Theorem 3.1 , these results describe the large-graph limit at fixed d d d d d d
For the full unitary law, Corollary 3.2 and the preceding decomposition imply, uniformly in α \alpha α
d T V ( ν d , α , δ 1 + δ − 1 2 ) ≤ 2 d . d_{\mathrm{TV}}\left(\nu_{d,\alpha},\frac{\delta_1+\delta_{-1}}{2}\right)\leq\frac{2}{d}. d TV ( ν d , α , 2 δ 1 + δ − 1 ) ≤ d 2 . where
d T V ( μ , ν ) = sup B ∈ B ( C ) ∣ μ ( B ) − ν ( B ) ∣ . d_{\mathrm{TV}}(\mu,\nu)=\sup_{B\in\mathcal{B}(\mathbb{C})}|\mu(B)-\nu(B)|. d TV ( μ , ν ) = B ∈ B ( C ) sup ∣ μ ( B ) − ν ( B ) ∣. Thus the full limiting unitary law converges to the equally weighted structural atoms as d → ∞ d\to\infty d → ∞
To compare different degrees, we group closed walks by the finite tree pattern they trace. This separates the number of embeddings into the regular tree from the expected weight of each embedded walk.
Fix an integer k ≥ 1 k\geq1 k ≥ 1 W k \mathcal{W}_k W k 2 k 2k 2 k 0 0 0
For γ ∈ W k \gamma\in\mathcal{W}_k γ ∈ W k H γ H_\gamma H γ
e ( γ ) = ∣ E ( H γ ) ∣ . e(\gamma)=|E(H_\gamma)|. e ( γ ) = ∣ E ( H γ ) ∣. Let c v ( γ ) c_v(\gamma) c v ( γ ) v v v r e ( γ ) ≥ 1 r_e(\gamma)\geq1 r e ( γ ) ≥ 1 e e e
R v ( γ ) = ∑ e ∈ E ( H γ ) e ∋ v r e ( γ ) . R_v(\gamma)=\sum_{\substack{e\in E(H_\gamma)\\e\ni v}}r_e(\gamma). R v ( γ ) = e ∈ E ( H γ ) e ∋ v ∑ r e ( γ ) . Since every edge is traversed equally often in both directions,
∑ e ∈ E ( H γ ) r e ( γ ) = k , ∑ v ∈ V ( H γ ) R v ( γ ) = 2 k . \sum_{e\in E(H_\gamma)}r_e(\gamma)=k,\qquad\sum_{v\in V(H_\gamma)}R_v(\gamma)=2k. e ∈ E ( H γ ) ∑ r e ( γ ) = k , v ∈ V ( H γ ) ∑ R v ( γ ) = 2 k . For the falling factorial, we write
x r ‾ = x ( x − 1 ) ⋯ ( x − r + 1 ) , x 0 ‾ = 1 , x^{\underline{r}}=x(x-1)\cdots(x-r+1),\qquad x^{\underline{0}}=1, x r = x ( x − 1 ) ⋯ ( x − r + 1 ) , x 0 = 1 , reserving ( a ) r (a)_r ( a ) r
Cat k = 1 k + 1 ( 2 k k ) . \operatorname{Cat}_k=\frac{1}{k+1}\binom{2k}{k}. Cat k = k + 1 1 ( k 2 k ) . Proposition 4.1 (Grouped moment expansion). For d ≥ 2 k d \ge2k d ≥ 2 k
m 2 k ( d , α ) = ∑ γ ∈ W k N d ( γ ) ∏ e ∈ E ( H γ ) ( α ) r e ( γ ) ∏ v ∈ V ( H γ ) ( d α ) R v ( γ ) . m_{2k}(d,\alpha)=\sum_{\gamma\in\mathcal{W}_k}N_d(\gamma)\frac{\prod_{e\in E(H_\gamma)}(\alpha)_{r_e(\gamma)}}{\prod_{v\in V(H_\gamma)}(d\alpha)_{R_v(\gamma)}}. m 2 k ( d , α ) = γ ∈ W k ∑ N d ( γ ) ∏ v ∈ V ( H γ ) ( d α ) R v ( γ ) ∏ e ∈ E ( H γ ) ( α ) r e ( γ ) . where
N d ( γ ) = d c o ( γ ) ∏ v ∈ V ( H γ ) v ≠ o ( d − 1 ) c v ( γ ) . N_d(\gamma)=d^{c_o(\gamma)}\prod_{\substack{v\in V(H_\gamma)\\v\ne o}}(d-1)^{c_v(\gamma)}. N d ( γ ) = d c o ( γ ) v ∈ V ( H γ ) v = o ∏ ( d − 1 ) c v ( γ ) . Proof. The sufficient condition d ≥ 2 k d\ge2k d ≥ 2 k γ \gamma γ H γ H_\gamma H γ T d T_d T d c o ( γ ) c_o(\gamma) c o ( γ ) d d d d c o ( γ ) d^{c_o(\gamma)} d c o ( γ ) d − 1 d-1 d − 1 N d ( γ ) N_d(\gamma) N d ( γ )
∏ v ∈ V ( H γ ) ∏ e ∈ E ( H γ ) e ∋ v ( α ) r e ( γ ) = ∏ e ∈ E ( H γ ) ( α ) r e ( γ ) 2 . \prod_{v\in V(H_\gamma)}\prod_{\substack{e\in E(H_\gamma)\\e\ni v}}(\alpha)_{r_e(\gamma)}
=
\prod_{e\in E(H_\gamma)}(\alpha)_{r_e(\gamma)}^2. v ∈ V ( H γ ) ∏ e ∈ E ( H γ ) e ∋ v ∏ ( α ) r e ( γ ) = e ∈ E ( H γ ) ∏ ( α ) r e ( γ ) 2 . Define
ε d , α = ( 1 + α ) 2 d α 2 = ( 1 + α − 1 ) 2 d . \varepsilon_{d,\alpha}=\frac{(1+\alpha)^2}{d\alpha^2}=\frac{(1+\alpha^{-1})^2}{d}. ε d , α = d α 2 ( 1 + α ) 2 = d ( 1 + α − 1 ) 2 . The leading classes traverse each edge once in each direction; their number is Catalan. The next bound controls both their deviation from unit weight after rescaling and the contribution of walks that repeat edges.
Lemma 4.2 (Catalan sector and a uniform remainder bound). For every fixed k ≥ 1 k\ge1 k ≥ 1 C k < ∞ C_k<\infty C k < ∞ k k k d ≥ 2 k d\ge2k d ≥ 2 k ε d , α ≤ 1 \varepsilon_{d,\alpha}\le1 ε d , α ≤ 1
∣ d k m 2 k ( d , α ) − Cat k ∣ ≤ C k ε d , α . \left|d^k m_{2k}(d,\alpha)-\operatorname{Cat}_k\right|\le C_k\varepsilon_{d,\alpha}. d k m 2 k ( d , α ) − Cat k ≤ C k ε d , α . Proof. For γ ∈ W k \gamma\in\mathcal{W}_k γ ∈ W k A d , α ( γ ) A_{d,\alpha}(\gamma) A d , α ( γ ) ℓ = e ( γ ) \ell=e(\gamma) ℓ = e ( γ )
First suppose that ℓ < k \ell<k ℓ < k H γ H_\gamma H γ
N d ( γ ) ≤ d ℓ . N_d(\gamma)\le d^\ell. N d ( γ ) ≤ d ℓ . Moreover,
( d α ) R v ≥ ( d α ) R v . (d\alpha)_{R_v}\ge(d\alpha)^{R_v}. ( d α ) R v ≥ ( d α ) R v . For 1 ≤ r ≤ k 1\le r\le k 1 ≤ r ≤ k
( α ) r = α r ∏ j = 1 r − 1 ( 1 + j α ) ≤ α r [ k ( 1 + α − 1 ) ] r − 1 . (\alpha)_r=\alpha^r\prod_{j=1}^{r-1}\left(1+\frac{j}{\alpha}\right)
\le\alpha^r\left[k(1+\alpha^{-1})\right]^{r-1}. ( α ) r = α r j = 1 ∏ r − 1 ( 1 + α j ) ≤ α r [ k ( 1 + α − 1 ) ] r − 1 . Using ∑ f r f = k \sum_f r_f = k ∑ f r f = k ∑ v R v = 2 k \sum_v R_v = 2k ∑ v R v = 2 k
0 ≤ d k A d , α ( γ ) ≤ k 2 ( k − ℓ ) ε d , α k − ℓ . 0 \le d^k A_{d,\alpha}(\gamma) \le k^{2(k-\ell)}\varepsilon_{d,\alpha}^{k-\ell}. 0 ≤ d k A d , α ( γ ) ≤ k 2 ( k − ℓ ) ε d , α k − ℓ . Since W k \mathcal{W}_k W k ε d , α ≤ 1 \varepsilon_{d,\alpha} \le1 ε d , α ≤ 1 ℓ < k \ell< k ℓ < k C k ( 1 ) ε d , α C_k^{(1)}\varepsilon_{d,\alpha} C k ( 1 ) ε d , α k k k
Now suppose that ℓ = k \ell= k ℓ = k r f = 1 r_f = 1 r f = 1 f ∈ E ( H γ ) f \in E(H_\gamma) f ∈ E ( H γ ) 2 k 2k 2 k Cat k \operatorname{Cat}_k Cat k
For a class in this sector, the root contribution satisfies
d c o α c o ( d α ) c o = ∏ j = 1 c o − 1 ( d − j ) α d α + j . \frac{d^{c_o}\alpha^{c_o}}{(d\alpha)_{c_o}} = \prod_{j=1}^{c_o-1} \frac{(d-j)\alpha}{d\alpha+j}. ( d α ) c o d c o α c o = j = 1 ∏ c o − 1 d α + j ( d − j ) α . whereas a nonroot vertex with c v c_v c v
( d − 1 ) c v α c v + 1 ( d α ) c v + 1 = 1 d ∏ j = 1 c v ( d − j ) α d α + j . \frac{(d-1)_{c_v}\alpha^{c_v+1}}{(d\alpha)_{c_v+1}} = \frac{1}{d}\prod_{j=1}^{c_v} \frac{(d-j)\alpha}{d\alpha+j}. ( d α ) c v + 1 ( d − 1 ) c v α c v + 1 = d 1 j = 1 ∏ c v d α + j ( d − j ) α . There are k k k d k d^k d k
d k A d , α ( γ ) = ∏ j = 1 c o ( γ ) − 1 ( d − j ) α d α + j × ∏ v ≠ o ∏ j = 1 c v ( γ ) ( d − j ) α d α + j . d^k A_{d,\alpha}(\gamma) =
\prod_{j=1}^{c_o(\gamma)-1} \frac{(d-j)\alpha}{d\alpha+j}
\times\prod_{v\ne o}\prod_{j=1}^{c_v(\gamma)} \frac{(d-j)\alpha}{d\alpha+j}. d k A d , α ( γ ) = j = 1 ∏ c o ( γ ) − 1 d α + j ( d − j ) α × v = o ∏ j = 1 ∏ c v ( γ ) d α + j ( d − j ) α . There are
c o ( γ ) − 1 + ∑ v ≠ o c v ( γ ) = k − 1 c_o(\gamma)-1+\sum_{v\ne o}c_v(\gamma)=k-1 c o ( γ ) − 1 + v = o ∑ c v ( γ ) = k − 1 nontrivial factors. Each belongs to [ 0 , 1 ] [0,1] [ 0 , 1 ]
0 ≤ 1 − ( d − j ) α d α + j = j ( α + 1 ) d α + j ≤ k 1 + α − 1 d ≤ k ε d , α . 0 \le1-\frac{(d-j)\alpha}{d\alpha+j}
=\frac{j(\alpha+1)}{d\alpha+j}
\le k\frac{1+\alpha^{-1}}{d}
\le k\varepsilon_{d,\alpha}. 0 ≤ 1 − d α + j ( d − j ) α = d α + j j ( α + 1 ) ≤ k d 1 + α − 1 ≤ k ε d , α . The inequality 1 − ∏ i ( 1 − u i ) ≤ ∑ i u i 1-\prod_i(1-u_i)\le\sum_i u_i 1 − ∏ i ( 1 − u i ) ≤ ∑ i u i u i ∈ [ 0 , 1 ] u_i\in[0,1] u i ∈ [ 0 , 1 ]
∣ d k A d , α ( γ ) − 1 ∣ ≤ k 2 ε d , α . \left|d^k A_{d,\alpha}(\gamma)-1\right|\le k^2\varepsilon_{d,\alpha}. d k A d , α ( γ ) − 1 ≤ k 2 ε d , α . Sum over the Cat k \operatorname{Cat}_k Cat k
Let
μ s c ( d x ) = 1 2 π 4 − x 2 1 { ∣ x ∣ ≤ 2 } d x \mu_{\mathrm{sc}}(dx)=\frac{1}{2\pi}\sqrt{4-x^2}\,\mathbf{1}_{\{|x|\le2\}}\,dx μ sc ( d x ) = 2 π 1 4 − x 2 1 { ∣ x ∣ ≤ 2 } d x be the standard semicircle law.
Theorem 4.3 (High-degree semicircle regime). Let d → ∞ d \to\infty d → ∞ and let α d > 0 \alpha_d > 0 α d > 0 If
d α d 2 → ∞ , d\alpha_d^2 \to\infty, d α d 2 → ∞ , then
( d ⋅ ) # μ d , α d ⟹ μ s c . (\sqrt{d}\cdot)_{\#}\mu_{d,\alpha_d} \Longrightarrow\mu_{\mathrm{sc}}. ( d ⋅ ) # μ d , α d ⟹ μ sc . More precisely, for every fixed k ≥ 1 k \ge1 k ≥ 1 and all sufficiently large d d d
d k m 2 k ( d , α d ) = Cat k + O k ( ( 1 + α d ) 2 d α d 2 ) , d^k m_{2k}(d,\alpha_d)=\operatorname{Cat}_k+O_k\left(\frac{(1+\alpha_d)^2}{d\alpha_d^2}\right), d k m 2 k ( d , α d ) = Cat k + O k ( d α d 2 ( 1 + α d ) 2 ) , whereas all odd moments vanish. In particular, for fixed α > 0 \alpha>0 α > 0 the error is O k , α ( d − 1 ) O_{k,\alpha}(d^{-1}) O k , α ( d − 1 )
Proof. Because d → ∞ d \to\infty d → ∞
d α d 2 → ∞ ⟺ ε d , α d → 0. d\alpha_d^2 \to\infty\quad\Longleftrightarrow\quad\varepsilon_{d,\alpha_d}\to0. d α d 2 → ∞ ⟺ ε d , α d → 0. Indeed,
ε d , α d = 1 d + 2 d α d + 1 d α d 2 , \varepsilon_{d,\alpha_d}=\frac{1}{d}+\frac{2}{d\alpha_d}+\frac{1}{d\alpha_d^2}, ε d , α d = d 1 + d α d 2 + d α d 2 1 , and d α d = d d α d 2 → ∞ d\alpha_d=\sqrt{d}\sqrt{d\alpha_d^2}\to\infty d α d = d d α d 2 → ∞
Let
μ ^ d = ( d ⋅ ) # μ d , α d . \widehat{\mu}_d=(\sqrt{d}\cdot)_{\#}\mu_{d,\alpha_d}. μ d = ( d ⋅ ) # μ d , α d . Since
∫ x 2 μ ^ d ( d x ) = d m 2 ( d , α d ) = 1 , \int x^2\,\widehat{\mu}_d(dx)=dm_2(d,\alpha_d)=1, ∫ x 2 μ d ( d x ) = d m 2 ( d , α d ) = 1 , the family ( μ ^ d ) (\widehat{\mu}_d) ( μ d ) ν \nu ν k k k k + 1 k+1 k + 1
sup d ∫ ∣ x ∣ 2 k + 2 μ ^ d ( d x ) < ∞ \sup_d\int|x|^{2k+2}\,\widehat{\mu}_d(dx)<\infty d sup ∫ ∣ x ∣ 2 k + 2 μ d ( d x ) < ∞ after discarding finitely many indices. Hence ( ∣ x ∣ 2 k ) (|x|^{2k}) ( ∣ x ∣ 2 k )
∫ x 2 k ν ( d x ) = Cat k . \int x^{2k}\,\nu(dx)=\operatorname{Cat}_k. ∫ x 2 k ν ( d x ) = Cat k . Symmetry gives the vanishing odd moments. Since the semicircle law is moment-determinate, ν = μ s c \nu=\mu_{\mathrm{sc}} ν = μ sc
The second moment is fixed by the normalization and cannot distinguish the high-degree regimes. The fourth moment already detects the scale that leads to the semicircle law.
Proposition 4.4 (Fourth-moment characterization of the scale). Let d → ∞ d \to\infty d → ∞ and α d > 0 \alpha_d>0 α d > 0 Then
d 2 m 4 ( d , α d ) ⟶ 2 ⟺ d α d 2 ⟶ ∞ . d^2m_4(d,\alpha_d)\longrightarrow2 \quad\Longleftrightarrow\quad d\alpha_d^2\longrightarrow\infty. d 2 m 4 ( d , α d ) ⟶ 2 ⟺ d α d 2 ⟶ ∞. Thus the scale in Theorem 4.3 is necessary and sufficient for convergence of the rescaled fourth moment to its semicircle value.
Proof. The exact fourth-moment formula gives
d 2 m 4 ( d , α ) − 2 = ( α + 1 ) { d ( 1 − α ) − 2 } ( d α + 1 ) 2 . d^{2}m_{4}(d,\alpha)-2=\frac{(\alpha+1)\{d(1-\alpha)-2\}}{(d\alpha+1)^{2}}. d 2 m 4 ( d , α ) − 2 = ( d α + 1 ) 2 ( α + 1 ) { d ( 1 − α ) − 2 } . The implication from right to left also follows from Theorem 4.3 . For the converse, suppose that d α d 2 d\alpha_{d}^{2} d α d 2 M M M α d ≤ M / d → 0 \alpha_{d}\le\sqrt{M/d}\to0 α d ≤ M / d → 0 α d ≤ 1 / 2 \alpha_{d}\le1/2 α d ≤ 1/2 d ≥ 8 d\ge8 d ≥ 8
d ( 1 − α d ) − 2 ≥ d 4 . d(1-\alpha_{d})-2\ge\frac{d}{4}. d ( 1 − α d ) − 2 ≥ 4 d . Moreover,
( d α d + 1 ) 2 ≤ 2 d 2 α d 2 + 2 ≤ 2 d ( M + 1 ) . (d\alpha_{d}+1)^{2}\le2d^{2}\alpha_{d}^{2}+2\le2d(M+1). ( d α d + 1 ) 2 ≤ 2 d 2 α d 2 + 2 ≤ 2 d ( M + 1 ) . Therefore, along this subsequence,
d 2 m 4 ( d , α d ) − 2 ≥ 1 8 ( M + 1 ) , d^{2}m_{4}(d,\alpha_{d})-2\ge\frac{1}{8(M+1)}, d 2 m 4 ( d , α d ) − 2 ≥ 8 ( M + 1 ) 1 , which rules out convergence to zero. In particular, if d α d 2 → τ ∈ ( 0 , ∞ ) d\alpha_{d}^{2}\to\tau\in(0,\infty) d α d 2 → τ ∈ ( 0 , ∞ ) d 2 m 4 ( d , α d ) → 2 + τ − 1 d^{2}m_{4}(d,\alpha_{d})\to2+\tau^{-1} d 2 m 4 ( d , α d ) → 2 + τ − 1
At a finite positive limit of d α d 2 d\alpha_{d}^{2} d α d 2
Theorem 4.5 (Critical high-degree family). Let d → ∞ d\to\infty d → ∞
d α d 2 ⟶ τ ∈ ( 0 , ∞ ) . d\alpha_{d}^{2}\longrightarrow\tau\in(0,\infty). d α d 2 ⟶ τ ∈ ( 0 , ∞ ) . Then there exists a symmetric probability measure μ τ \mu_{\tau} μ τ
( d ) # μ d , α d ⟹ μ τ , (\sqrt{d})_{\#}\mu_{d,\alpha_{d}}\Longrightarrow\mu_{\tau}, ( d ) # μ d , α d ⟹ μ τ , and all moments converge. The limiting odd moments vanish, and the even moments are
M 2 k ( τ ) = ∑ γ ∈ W k τ e ( γ ) − k ∏ e ∈ E ( H γ ) ( ( r e ( γ ) − 1 ) ! ) 2 . M_{2k}(\tau)=\sum_{\gamma\in\mathcal{W}_{k}}\tau^{e(\gamma)-k}\prod_{e\in E(H_{\gamma})}\bigl((r_{e}(\gamma)-1)!\bigr)^{2}. M 2 k ( τ ) = γ ∈ W k ∑ τ e ( γ ) − k e ∈ E ( H γ ) ∏ ( ( r e ( γ ) − 1 )! ) 2 . The measure μ τ \mu_{\tau} μ τ
Proof. For each fixed moment order, take d d d d ≥ 2 k d\ge2k d ≥ 2 k
α d → 0 , d α d = d d α d 2 → ∞ . \alpha_{d}\to0,\qquad d\alpha_{d}=\sqrt{d}\sqrt{d\alpha_{d}^{2}}\to\infty. α d → 0 , d α d = d d α d 2 → ∞. For a fixed class γ ∈ W k \gamma\in\mathcal{W}_{k} γ ∈ W k 2 k 2k 2 k
N d ( γ ) d e ( γ ) ( d α d 2 ) e ( γ ) − k × ∏ e ∈ E ( H γ ) [ ( α d ) r e ( γ ) / α d ] 2 ∏ v ∈ V ( H γ ) [ ( d α d ) R v ( γ ) / ( d α d ) R v ( γ ) ] . \frac{N_{d}(\gamma)}{d^{e(\gamma)}}(d\alpha_{d}^{2})^{e(\gamma)-k}
\times
\frac{\prod_{e\in E(H_{\gamma})}\left[(\alpha_{d})_{r_{e}(\gamma)}/\alpha_{d}\right]^{2}}
{\prod_{v\in V(H_{\gamma})}\left[(d\alpha_{d})_{R_{v}(\gamma)}/(d\alpha_{d})^{R_{v}(\gamma)}\right]}. d e ( γ ) N d ( γ ) ( d α d 2 ) e ( γ ) − k × ∏ v ∈ V ( H γ ) [ ( d α d ) R v ( γ ) / ( d α d ) R v ( γ ) ] ∏ e ∈ E ( H γ ) [ ( α d ) r e ( γ ) / α d ] 2 . For every fixed positive integer r r r
N d ( γ ) d e ( γ ) ⟶ 1 , ( d α d ) r ( d α d ) r ⟶ 1 , ( α d ) r α d ⟶ ( r − 1 ) ! . \frac{N_d(\gamma)}{d^{e(\gamma)}} \longrightarrow1,\qquad\frac{(d\alpha_d)_r}{(d\alpha_d)^r} \longrightarrow1,\qquad\frac{(\alpha_d)_r}{\alpha_d} \longrightarrow(r-1)!. d e ( γ ) N d ( γ ) ⟶ 1 , ( d α d ) r ( d α d ) r ⟶ 1 , α d ( α d ) r ⟶ ( r − 1 )! . Since W k \mathcal{W}_k W k 2 k 2k 2 k M 2 k ( τ ) M_{2k}(\tau) M 2 k ( τ )
We next establish moment determinacy. Ordering children by first discovery associates a rooted plane tree with each canonical walk class. Fix such a tree with e e e r 1 + ⋯ + r e = k r_1+\cdots+r_e=k r 1 + ⋯ + r e = k r j r_j r j j j j
( 2 k ) ! ∏ j = 1 e ( r j ! ) 2 . \frac{(2k)!}{\prod_{j=1}^{e}(r_j!)^2}. ∏ j = 1 e ( r j ! ) 2 ( 2 k )! . Multiplying by the factorial part of the critical weight gives
( 2 k ) ! ∏ j = 1 e ( r j ! ) 2 ∏ j = 1 e ( ( r j − 1 ) ! ) 2 ≤ ( 2 k ) ! . \frac{(2k)!}{\prod_{j=1}^{e}(r_j!)^2}\prod_{j=1}^{e}((r_j-1)!)^2\leq(2k)!. ∏ j = 1 e ( r j ! ) 2 ( 2 k )! j = 1 ∏ e (( r j − 1 )! ) 2 ≤ ( 2 k )! . Rooted plane trees with e e e Cat e \operatorname{Cat}_e Cat e ( k − 1 e − 1 ) \binom{k-1}{e-1} ( e − 1 k − 1 ) Cat e ≤ 4 e \operatorname{Cat}_e\leq4^e Cat e ≤ 4 e
M 2 k ( τ ) ≤ ( 2 k ) ! ∑ e = 1 k Cat e ( k − 1 e − 1 ) τ e − k M_{2k}(\tau)\leq(2k)!\sum_{e=1}^{k}\operatorname{Cat}_e\binom{k-1}{e-1}\tau^{e-k} M 2 k ( τ ) ≤ ( 2 k )! e = 1 ∑ k Cat e ( e − 1 k − 1 ) τ e − k ≤ 4 ( 2 k ) ! ( 4 + τ − 1 ) k − 1 . \leq4(2k)!(4+\tau^{-1})^{k-1}. ≤ 4 ( 2 k )! ( 4 + τ − 1 ) k − 1 . Together with ( 2 k ) ! ≤ ( 2 k ) 2 k (2k)!\leq(2k)^{2k} ( 2 k )! ≤ ( 2 k ) 2 k c τ > 0 c_\tau>0 c τ > 0
M 2 k ( τ ) − 1 / ( 2 k ) ≥ c τ k . M_{2k}(\tau)^{-1/(2k)}\geq\frac{c_\tau}{k}. M 2 k ( τ ) − 1/ ( 2 k ) ≥ k c τ . Hence
∑ k ≥ 1 M 2 k ( τ ) − 1 / ( 2 k ) = ∞ , \sum_{k\geq1}M_{2k}(\tau)^{-1/(2k)}=\infty, k ≥ 1 ∑ M 2 k ( τ ) − 1/ ( 2 k ) = ∞ , so Carleman’s condition holds.
Set μ ^ d = ( d ⋅ ) # μ d , α d \widehat{\mu}_d=(\sqrt{d}\mathbin{\cdot})_{\#}\mu_{d,\alpha_d} μ d = ( d ⋅ ) # μ d , α d ν \nu ν p ≥ 0 p\geq0 p ≥ 0 ( 2 p + 2 ) (2p+2) ( 2 p + 2 )
sup d ∫ ∣ x ∣ 2 p + 2 μ ^ d ( d x ) < ∞ \sup_d\int|x|^{2p+2}\widehat{\mu}_d(\mathrm{d}x)<\infty d sup ∫ ∣ x ∣ 2 p + 2 μ d ( d x ) < ∞ along the subsequence, after finitely many terms are discarded. Thus ∣ x ∣ 2 p |x|^{2p} ∣ x ∣ 2 p ν \nu ν M j ( τ ) M_j(\tau) M j ( τ ) μ τ \mu_\tau μ τ
Finally, the unique class supported on one edge contributes
τ 1 − k ( ( k − 1 ) ! ) 2 \tau^{1-k}((k-1)!)^2 τ 1 − k (( k − 1 )! ) 2 to M 2 k ( τ ) M_{2k}(\tau) M 2 k ( τ )
M 2 k ( τ ) 1 / ( 2 k ) ≥ τ ( 1 − k ) / ( 2 k ) ( ( k − 1 ) ! ) 1 / k ⟶ ∞ . M_{2k}(\tau)^{1/(2k)} \ge\tau^{(1-k)/(2k)}((k-1)!)^{1/k} \longrightarrow\infty. M 2 k ( τ ) 1/ ( 2 k ) ≥ τ ( 1 − k ) / ( 2 k ) (( k − 1 )! ) 1/ k ⟶ ∞. If μ τ \mu_\tau μ τ [ − R , R ] [-R,R] [ − R , R ] R R R
Poisson-weighted tree representation at criticality
We now give an operator representation of the critical laws. The Poisson intensity of the squared edge weights reproduces the factorial factors in their closed-walk moments. Because each vertex has infinitely many children, we first establish self-adjointness and measurability, and then identify the expected root spectral measure. The resulting tree also gives a recursive equation for its Green function. Let K 0 K_0 K 0
Proposition 4.6 (Bessel Lévy intensity). For τ > 0 \tau> 0 τ > 0
L τ ( d y ) = 2 τ y K 0 ( 2 τ y ) d y , y > 0. L_\tau(\mathrm{d}y) = \frac{2\tau}{y}K_0(2\sqrt{\tau y})\,\mathrm{d}y,\qquad y>0. L τ ( d y ) = y 2 τ K 0 ( 2 τ y ) d y , y > 0. Then, for every real r > 0 r>0 r > 0
∫ 0 ∞ y r L τ ( d y ) = τ 1 − r Γ ( r ) 2 . \int_0^\infty y^r L_\tau(\mathrm{d}y)=\tau^{1-r}\Gamma(r)^2. ∫ 0 ∞ y r L τ ( d y ) = τ 1 − r Γ ( r ) 2 . In particular,
∫ y L τ ( d y ) = 1 , ∫ y L τ ( d y ) = π τ . \int y L_\tau(\mathrm{d}y)=1,\qquad\int\sqrt{y}L_\tau(\mathrm{d}y)=\pi\sqrt{\tau}. ∫ y L τ ( d y ) = 1 , ∫ y L τ ( d y ) = π τ . The measure has infinite total mass, is finite on ( ε , ∞ ) (\varepsilon,\infty) ( ε , ∞ ) ε > 0 \varepsilon>0 ε > 0
∫ ( 1 ∧ y ) L τ ( d y ) < ∞ . \int(1\wedge y)L_\tau(\mathrm{d}y)<\infty. ∫ ( 1 ∧ y ) L τ ( d y ) < ∞. Proof. With x = 2 τ y x=2\sqrt{\tau y} x = 2 τ y
∫ 0 ∞ y r L τ ( d y ) = ( 4 τ ) 1 − r ∫ 0 ∞ x 2 r − 1 K 0 ( x ) d x = τ 1 − r Γ ( r ) 2 . \begin{aligned}
\int_0^\infty y^r L_\tau(\mathrm{d}y)
=(4\tau)^{1-r}\int_0^\infty x^{2r-1}K_0(x)\,\mathrm{d}x \\
&=\tau^{1-r}\Gamma(r)^2.
\end{aligned} ∫ 0 ∞ y r L τ ( d y ) = ( 4 τ ) 1 − r ∫ 0 ∞ x 2 r − 1 K 0 ( x ) d x = τ 1 − r Γ ( r ) 2 . where we used the standard modified-Bessel integral [23 Eq. 10.43.19] ,
∫ 0 ∞ x μ − 1 K 0 ( x ) d x = 2 μ − 2 Γ ( μ / 2 ) 2 , μ > 0. \int_0^\infty x^{\mu-1}K_0(x)\,\mathrm{d}x=2^{\mu-2}\Gamma(\mu/2)^2,\qquad\mu>0. ∫ 0 ∞ x μ − 1 K 0 ( x ) d x = 2 μ − 2 Γ ( μ /2 ) 2 , μ > 0. As y ↓ 0 y\downarrow0 y ↓ 0 L τ L_\tau L τ τ y − 1 log ( 1 / y ) + O τ ( y − 1 ) \tau y^{-1}\log(1/y)+O_\tau(y^{-1}) τ y − 1 log ( 1/ y ) + O τ ( y − 1 ) ∫ 0 1 y L τ ( d y ) < ∞ \int_0^1 yL_\tau(\mathrm{d}y)<\infty ∫ 0 1 y L τ ( d y ) < ∞ ∫ 0 1 y L τ ( d y ) < ∞ \int_0^1\sqrt{y}L_\tau(\mathrm{d}y)<\infty ∫ 0 1 y L τ ( d y ) < ∞ K 0 K_0 K 0 ( ε , ∞ ) (\varepsilon,\infty) ( ε , ∞ )
Let
V = ⋃ m ≥ 0 N m , N 0 = { ∅ } , \mathcal{V}=\bigcup_{m\geq0}\mathbb{N}^{m},\qquad\mathbb{N}^{0}=\{\varnothing\}, V = m ≥ 0 ⋃ N m , N 0 = { ∅ } , be the Ulam–Harris tree. Independently at each v ∈ V v\in\mathcal{V} v ∈ V
Π v = ∑ j ≥ 1 δ Y v , j \Pi_v=\sum_{j\geq1}\delta_{Y_{v,j}} Π v = j ≥ 1 ∑ δ Y v , j be a Poisson point process on ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) L τ \mathsf{L}_{\tau} L τ L τ ( ( ε , ∞ ) ) < ∞ \mathsf{L}_{\tau}((\varepsilon,\infty))<\infty L τ (( ε , ∞ )) < ∞ ε > 0 \varepsilon>0 ε > 0 L τ \mathsf{L}_{\tau} L τ
Y v , 1 > Y v , 2 > ⋯ ↓ 0. Y_{v,1}>Y_{v,2}>\cdots\downarrow0. Y v , 1 > Y v , 2 > ⋯ ↓ 0. Join v v v v j vj v j w v , v j = Y v , j w_{v,vj}=\sqrt{Y_{v,j}} w v , v j = Y v , j
Q v = ∑ j ≥ 1 Y v , j . Q_v=\sum_{j\geq1}Y_{v,j}. Q v = j ≥ 1 ∑ Y v , j . By Campbell’s formula [20 ] and Proposition 4.6 , E Q v = 1 \mathbb{E}Q_v=1 E Q v = 1 Q v < ∞ Q_v<\infty Q v < ∞ ℓ 2 \ell^2 ℓ 2 Q ∅ Q_{\varnothing} Q ∅ w par ( v ) , v 2 + Q v w_{\operatorname{par}(v),v}^{2}+Q_v w par ( v ) , v 2 + Q v c 00 ( V ) c_{00}(\mathcal{V}) c 00 ( V ) c 00 ( V ) c_{00}(\mathcal{V}) c 00 ( V ) ℓ 2 ( V ) \ell^2(\mathcal{V}) ℓ 2 ( V ) A τ , min A_{\tau,\min} A τ , m i n
Lemma 4.7 (Finite-exhaustion criterion for essential self-adjointness). Let T = ( V , E ) \mathcal{T}=(V,E) T = ( V , E ) A min A_{\min} A m i n c 00 ( V ) c_{00}(V) c 00 ( V ) w u , v = w v , u w_{u,v}=w_{v,u} w u , v = w v , u κ < ∞ \kappa<\infty κ < ∞ S n ↑ V S_n\uparrow V S n ↑ V
∑ u ∉ S n : u ∼ v ∣ w u , v ∣ 2 ≤ κ ( n ≥ 1 , v ∈ S n ) . \sum_{\substack{u\notin S_n:u\sim v}}|w_{u,v}|^2\leq\kappa\qquad(n\geq1,\ v\in S_n). u ∈ / S n : u ∼ v ∑ ∣ w u , v ∣ 2 ≤ κ ( n ≥ 1 , v ∈ S n ) . Then A min A_{\min} A m i n
Proof. This is the criterion in [7 Lemma A.3] . We recall its proof. It suffices to show that both deficiency spaces at ± i \pm i ± i φ ∈ Dom ( A min ∗ ) \varphi\in\operatorname{Dom}(A_{\min}^{*}) φ ∈ Dom ( A m i n ∗ ) A min ∗ φ = i φ A_{\min}^{*}\varphi=i\varphi A m i n ∗ φ = i φ v ∈ V v\in V v ∈ V
i φ ( v ) = ∑ u ∼ v w v , u φ ( u ) . i\varphi(v)=\sum_{u\sim v}w_{v,u}\varphi(u). i φ ( v ) = u ∼ v ∑ w v , u φ ( u ) . For a finite set S ⊂ V S\subset V S ⊂ V φ ( v ) ‾ \overline{\varphi(v)} φ ( v ) v ∈ S v\in S v ∈ S
∑ v ∈ S ∣ φ ( v ) ∣ 2 = Im ∑ v ∈ S , u ∉ S u ∼ v w v , u φ ( v ) ‾ φ ( u ) , \sum_{v\in S}|\varphi(v)|^2=\operatorname{Im}\sum_{\substack{v\in S,\ u\notin S\\u\sim v}}w_{v,u}\overline{\varphi(v)}\varphi(u), v ∈ S ∑ ∣ φ ( v ) ∣ 2 = Im v ∈ S , u ∈ / S u ∼ v ∑ w v , u φ ( v ) φ ( u ) , because the contributions from edges internal to S S S
∑ v ∈ S ∣ φ ( v ) ∣ 2 ≤ ∑ v ∈ S ∣ φ ( v ) ∣ ∑ u ∉ S : u ∼ v ∣ w v , u ∣ ∣ φ ( u ) ∣ . \sum_{v\in S}|\varphi(v)|^2\leq\sum_{v\in S}|\varphi(v)|\sum_{\substack{u\notin S:u\sim v}}|w_{v,u}|\,|\varphi(u)|. v ∈ S ∑ ∣ φ ( v ) ∣ 2 ≤ v ∈ S ∑ ∣ φ ( v ) ∣ u ∈ / S : u ∼ v ∑ ∣ w v , u ∣ ∣ φ ( u ) ∣. Take S = S n S=S_n S = S n
∑ v ∈ S n ∣ φ ( v ) ∣ 2 ≤ κ ( ∑ v ∈ S n ∣ φ ( v ) ∣ 2 ) 1 / 2 ( ∑ u ∉ S n ∣ φ ( u ) ∣ 2 ) 1 / 2 . \sum_{v\in S_n}|\varphi(v)|^2\le\sqrt{\kappa}\left(\sum_{v\in S_n}|\varphi(v)|^2\right)^{1/2}\left(\sum_{u\notin S_n}|\varphi(u)|^2\right)^{1/2}. v ∈ S n ∑ ∣ φ ( v ) ∣ 2 ≤ κ ( v ∈ S n ∑ ∣ φ ( v ) ∣ 2 ) 1/2 u ∈ / S n ∑ ∣ φ ( u ) ∣ 2 1/2 . Connectedness of S n S_n S n S n S_n S n
∑ v ∈ S n ∣ φ ( v ) ∣ 2 ≤ κ ∑ u ∉ S n ∣ φ ( u ) ∣ 2 . \sum_{v\in S_n}|\varphi(v)|^2\le\kappa\sum_{u\notin S_n}|\varphi(u)|^2. v ∈ S n ∑ ∣ φ ( v ) ∣ 2 ≤ κ u ∈ / S n ∑ ∣ φ ( u ) ∣ 2 . As S n ↑ V S_n\uparrow V S n ↑ V ∥ φ ∥ 2 2 \|\varphi\|_2^2 ∥ φ ∥ 2 2 φ = 0 \varphi=0 φ = 0 A min ∗ φ = − i φ A_{\min}^*\varphi=-i\varphi A m i n ∗ φ = − i φ
Lemma 4.8 (Self-adjoint Poisson tree operator). The minimal operator A τ , min A_{\tau,\min} A τ , m i n
Proof. For every vertex v v v
S v = ∑ j ≥ 1 Y v , j . S_v=\sum_{j\ge1}\sqrt{Y_{v,j}}. S v = j ≥ 1 ∑ Y v , j . Campbell’s formula and Proposition 4.6 give E S v = π τ < ∞ \mathbb{E}S_v=\pi\sqrt{\tau}<\infty E S v = π τ < ∞ S v < ∞ S_v<\infty S v < ∞
E S v 2 = ( ∫ y L τ ( d y ) ) 2 + ∫ y L τ ( d y ) = π 2 τ + 1. \mathbb{E}S_v^2=\left(\int\sqrt{y}\,L_\tau(\mathrm{d}y)\right)^2+\int yL_\tau(\mathrm{d}y)=\pi^2\tau+1. E S v 2 = ( ∫ y L τ ( d y ) ) 2 + ∫ y L τ ( d y ) = π 2 τ + 1. For κ > 0 \kappa>0 κ > 0
N κ ( v ) = min { t ∈ N ∪ { 0 } : ∑ j > t Y v , j ≤ κ } . N_\kappa(v)=\min\left\{t\in\mathbb{N}\cup\{0\}:\sum_{j>t}Y_{v,j}\le\kappa\right\}. N κ ( v ) = min { t ∈ N ∪ { 0 } : j > t ∑ Y v , j ≤ κ } . Because the sequence ( Y v , j ) j (\sqrt{Y_{v,j}})_j ( Y v , j ) j Y v , j ≤ S v / j \sqrt{Y_{v,j}}\le S_v/j Y v , j ≤ S v / j
∑ j > t Y v , j ≤ S v 2 ∑ j > t 1 j 2 ≤ S v 2 t , t ≥ 1. \sum_{j>t}Y_{v,j}\le S_v^2\sum_{j>t}\frac{1}{j^2}\le\frac{S_v^2}{t},\qquad t\ge1. j > t ∑ Y v , j ≤ S v 2 j > t ∑ j 2 1 ≤ t S v 2 , t ≥ 1. It follows that
N κ ( v ) ≤ 1 { Q v > κ } ( 1 + ⌈ S v 2 κ ⌉ ) , N_\kappa(v)\le\mathbf{1}_{\{Q_v>\kappa\}}\left(1+\left\lceil\frac{S_v^2}{\kappa}\right\rceil\right), N κ ( v ) ≤ 1 { Q v > κ } ( 1 + ⌈ κ S v 2 ⌉ ) , and therefore
E N κ ( v ) ≤ 2 P ( Q v > κ ) + E S v 2 κ ⟶ 0 ( κ → ∞ ) . \mathbb{E}N_\kappa(v)\le2\mathbb{P}(Q_v>\kappa)+\frac{\mathbb{E}S_v^2}{\kappa}\longrightarrow0\qquad(\kappa\to\infty). E N κ ( v ) ≤ 2 P ( Q v > κ ) + κ E S v 2 ⟶ 0 ( κ → ∞ ) . Choose κ \kappa κ E N κ ( v ) < 1 \mathbb{E}N_\kappa(v)<1 E N κ ( v ) < 1 N κ ( v ) N_\kappa(v) N κ ( v ) N κ ( v ) N_\kappa(v) N κ ( v ) V \mathcal{V} V ( S n ) (S_n) ( S n ) V \mathcal{V} V
If v ∈ S n v \in S_n v ∈ S n S n S_n S n N κ ( v ) N_{\kappa}(v) N κ ( v ) S n S_n S n v ∈ S n v \in S_n v ∈ S n
∑ u ∉ S n : u ∼ v ∣ ( A τ , min ) u , v ∣ 2 = ∑ j : v j ∉ S n Y v , j ≤ ∑ j > N κ ( v ) Y v , j ≤ κ . \sum_{u \notin S_n: u\sim v} \left|(A_{\tau,\min})_{u,v}\right|^2
=
\sum_{j:v_j\notin S_n}Y_{v,j}
\le
\sum_{j>N_{\kappa}(v)}Y_{v,j}
\le\kappa. u ∈ / S n : u ∼ v ∑ ∣ ( A τ , m i n ) u , v ∣ 2 = j : v j ∈ / S n ∑ Y v , j ≤ j > N κ ( v ) ∑ Y v , j ≤ κ . This is the subcritical Galton–Watson exhaustion of [7 Proposition A.2] . The boundary estimate verifies the hypotheses of Lemma 4.7 , which proves the claim. □
To relate closed-walk sums to powers of the operator, we need to recognize which vectors belong to its domain. On the event of Lemma 4.8 , define the maximal formal realization by
Dom ( A τ , max ) = { f ∈ ℓ 2 ( V ) : ( ∑ u ∼ v w v , u f ( u ) ) v ∈ V ∈ ℓ 2 ( V ) } . \operatorname{Dom}(A_{\tau,\max})=
\left\{
f\in\ell^2(\mathcal{V}):
\left(\sum_{u\sim v}w_{v,u}f(u)\right)_{v\in\mathcal{V}}\in\ell^2(\mathcal{V})
\right\}. Dom ( A τ , m a x ) = { f ∈ ℓ 2 ( V ) : ( u ∼ v ∑ w v , u f ( u ) ) v ∈ V ∈ ℓ 2 ( V ) } . The coordinate sums are absolutely convergent by Cauchy–Schwarz and the square-summability of the rows.
Lemma 4.9 (Minimal and maximal realizations). Almost surely,
A τ , min ∗ = A τ , max . A_{\tau,\min}^{*}=A_{\tau,\max}. A τ , m i n ∗ = A τ , m a x . Consequently, if A τ A_\tau A τ A τ , min A_{\tau,\min} A τ , m i n A τ = A τ , max A_\tau=A_{\tau,\max} A τ = A τ , m a x
Proof. Let g ∈ Dom ( A τ , min ∗ ) g\in\operatorname{Dom}(A_{\tau,\min}^{*}) g ∈ Dom ( A τ , m i n ∗ ) h = A τ , min ∗ g h=A_{\tau,\min}^{*}g h = A τ , m i n ∗ g δ v \delta_v δ v
h ( v ) = ∑ u ∼ v w v , u g ( u ) . h(v)=\sum_{u\sim v}w_{v,u}g(u). h ( v ) = u ∼ v ∑ w v , u g ( u ) . Cauchy–Schwarz gives absolute convergence of the series. Since h ∈ ℓ 2 h\in\ell^2 h ∈ ℓ 2 g ∈ Dom ( A τ , max ) g\in\operatorname{Dom}(A_{\tau,\max}) g ∈ Dom ( A τ , m a x ) g ∈ Dom ( A τ , max ) g\in\operatorname{Dom}(A_{\tau,\max}) g ∈ Dom ( A τ , m a x ) f f f supp f \operatorname{supp} f supp f
⟨ A τ , min f , g ⟩ = ∑ u ∈ supp f f ( u ) ‾ ∑ v ∼ u w u , v g ( v ) = ⟨ f , A τ , max g ⟩ . \langle A_{\tau,\min}f,g\rangle
=
\sum_{u\in\operatorname{supp} f}\overline{f(u)}
\sum_{v\sim u}w_{u,v}g(v)
=
\langle f,A_{\tau,\max}g\rangle. ⟨ A τ , m i n f , g ⟩ = u ∈ supp f ∑ f ( u ) v ∼ u ∑ w u , v g ( v ) = ⟨ f , A τ , m a x g ⟩ . Thus g ∈ Dom ( A τ , min ∗ ) g\in\operatorname{Dom}(A_{\tau,\min}^{*}) g ∈ Dom ( A τ , m i n ∗ )
A τ = A τ , min = A τ , min ∗ = A τ , max . A_\tau=A_{\tau,\min}=A_{\tau,\min}^{*}=A_{\tau,\max}. A τ = A τ , m i n = A τ , m i n ∗ = A τ , m a x . To establish measurability of the root spectral measure and Green function, we approximate the operator by finite principal compressions. For N ≥ 1 N\ge1 N ≥ 1
V N = { ( v 1 , … , v m ) : 0 ≤ m ≤ N , 1 ≤ v i ≤ N } , \mathcal{V}_N=\left\{(v_1,\ldots,v_m):0\le m\le N,\ 1\le v_i\le N\right\}, V N = { ( v 1 , … , v m ) : 0 ≤ m ≤ N , 1 ≤ v i ≤ N } , and let P N P_N P N ℓ 2 ( V N ) \ell^2(\mathcal{V}_N) ℓ 2 ( V N ) A τ A_\tau A τ A τ , N A_{\tau,N} A τ , N ℓ 2 ( V ) \ell^2(\mathcal{V}) ℓ 2 ( V ) A ~ τ , N = P N A τ P N \widetilde{A}_{\tau,N}=P_NA_\tau P_N A τ , N = P N A τ P N f ∈ c 00 ( V ) f\in c_{00}(\mathcal{V}) f ∈ c 00 ( V ) supp f ⊂ V N \operatorname{supp}f\subset\mathcal{V}_N supp f ⊂ V N A ~ τ , N f = P N A τ f → A τ f \widetilde{A}_{\tau,N}f=P_NA_\tau f\to A_\tau f A τ , N f = P N A τ f → A τ f ℓ 2 \ell^2 ℓ 2 z ∈ C ∖ R z\in\mathbb{C}\setminus\mathbb{R} z ∈ C ∖ R
( A ~ τ , N − z ) − 1 ( A τ − z ) f − f = ( A ~ τ , N − z ) − 1 ( A τ − A ~ τ , N ) f ⟶ 0. (\widetilde{A}_{\tau,N}-z)^{-1}(A_\tau-z)f-f
=(\widetilde{A}_{\tau,N}-z)^{-1}(A_\tau-\widetilde{A}_{\tau,N})f\longrightarrow0. ( A τ , N − z ) − 1 ( A τ − z ) f − f = ( A τ , N − z ) − 1 ( A τ − A τ , N ) f ⟶ 0. Since c 00 ( V ) c_{00}(\mathcal{V}) c 00 ( V ) A τ A_\tau A τ ( A τ − z ) c 00 ( V ) (A_\tau-z)c_{00}(\mathcal{V}) ( A τ − z ) c 00 ( V ) ∣ Im z ∣ − 1 |\operatorname{Im}z|^{-1} ∣ Im z ∣ − 1 A ~ τ , N \widetilde{A}_{\tau,N} A τ , N A τ A_\tau A τ
Define the root spectral measure by
ρ τ , o ( B ) = ⟨ δ ∅ , 1 B ( A τ ) δ ∅ ⟩ . \rho_{\tau,o}(B)=\langle\delta_{\varnothing},\mathbb{1}_B(A_\tau)\delta_{\varnothing}\rangle. ρ τ , o ( B ) = ⟨ δ ∅ , 1 B ( A τ ) δ ∅ ⟩ . The root belongs to every V N \mathcal{V}_N V N ρ τ , o \rho_{\tau,o} ρ τ , o ρ τ , o \rho_{\tau,o} ρ τ , o Hol ( C + ) \operatorname{Hol}(\mathbb{C}_+) Hol ( C + ) z ↦ ⟨ δ ∅ , ( A τ − z ) − 1 δ ∅ ⟩ z\mapsto\langle\delta_{\varnothing},(A_\tau-z)^{-1}\delta_{\varnothing}\rangle z ↦ ⟨ δ ∅ , ( A τ − z ) − 1 δ ∅ ⟩
Theorem 4.10 (Poisson-tree representation of the critical law). For every τ > 0 \tau>0 τ > 0
μ τ ( B ) = E ρ τ , o ( B ) , B ∈ B ( R ) . \mu_\tau(B)=\mathbb{E}\rho_{\tau,o}(B),\qquad B\in\mathcal{B}(\mathbb{R}). μ τ ( B ) = E ρ τ , o ( B ) , B ∈ B ( R ) . Thus the critical high-degree law is the expected root spectral measure of an explicitly weighted Poisson tree.
Proof. For each k ≥ 0 k\geq0 k ≥ 0
F k , N = A τ , N k δ ∅ F_{k,N}=A_{\tau,N}^{k}\delta_{\varnothing} F k , N = A τ , N k δ ∅ by zero to V \mathcal{V} V F k , N F_{k,N} F k , N N N N k k k
F k ( v ) = lim N → ∞ F k , N ( v ) . F_k(v)=\lim_{N\to\infty}F_{k,N}(v). F k ( v ) = N → ∞ lim F k , N ( v ) . Its coordinate at v v v k k k ∅ \varnothing ∅ v v v k ≥ 1 k\geq1 k ≥ 1 N N N k k k ∥ F k , N ∥ 2 \lVert F_{k,N}\rVert^2 ∥ F k , N ∥ 2 2 k 2k 2 k V N \mathcal{V}_N V N [20 Chapter 4] ) give
E ∥ F k ∥ 2 = ∑ γ ∈ W k ∏ e ∈ E ( H γ ) ∫ 0 ∞ y r e ( γ ) L τ ( d y ) = ∑ γ ∈ W k τ e ( γ ) − k ∏ e ∈ E ( H γ ) ( ( r e ( γ ) − 1 ) ! ) 2 = M 2 k ( τ ) . \begin{aligned}
\mathbb{E}\lVert F_k\rVert^2
=\sum_{\gamma\in\mathcal{W}_k}\prod_{e\in E(H_\gamma)}\int_0^\infty y^{r_e(\gamma)}L_\tau(dy) \\
&=\sum_{\gamma\in\mathcal{W}_k}\tau^{e(\gamma)-k}\prod_{e\in E(H_\gamma)}\bigl((r_e(\gamma)-1)!\bigr)^2=M_{2k}(\tau).
\end{aligned} E ∥ F k ∥ 2 = γ ∈ W k ∑ e ∈ E ( H γ ) ∏ ∫ 0 ∞ y r e ( γ ) L τ ( d y ) = γ ∈ W k ∑ τ e ( γ ) − k e ∈ E ( H γ ) ∏ ( ( r e ( γ ) − 1 )! ) 2 = M 2 k ( τ ) . This expectation is finite. Together with F 0 = δ ∅ F_0=\delta_{\varnothing} F 0 = δ ∅ F k ∈ ℓ 2 ( V ) F_k\in\ell^2(\mathcal{V}) F k ∈ ℓ 2 ( V ) k ≥ 0 k\geq0 k ≥ 0
For every k ≥ 0 k\geq0 k ≥ 0 A τ , max F k = F k + 1 A_{\tau,\max}F_k=F_{k+1} A τ , m a x F k = F k + 1 F k + 1 ∈ ℓ 2 F_{k+1}\in\ell^2 F k + 1 ∈ ℓ 2 F k ∈ Dom ( A τ ) F_k\in\operatorname{Dom}(A_\tau) F k ∈ Dom ( A τ )
A τ F k = F k + 1 . A_\tau F_k=F_{k+1}. A τ F k = F k + 1 . Induction yields
δ ∅ ∈ Dom ( A τ k ) , A τ k δ ∅ = F k . \delta_{\varnothing}\in\operatorname{Dom}(A_\tau^k),\qquad A_\tau^k\delta_{\varnothing}=F_k. δ ∅ ∈ Dom ( A τ k ) , A τ k δ ∅ = F k . Consequently,
E ∫ R x 2 k ρ τ , o ( d x ) = E ∥ A τ k δ ∅ ∥ 2 2 = M 2 k ( τ ) . \mathbb{E}\int_{\mathbb{R}}x^{2k}\,\rho_{\tau,o}(\mathrm{d}x)=\mathbb{E}\lVert A_\tau^k\delta_{\varnothing}\rVert_2^2=M_{2k}(\tau). E ∫ R x 2 k ρ τ , o ( d x ) = E ∥ A τ k δ ∅ ∥ 2 2 = M 2 k ( τ ) . Bipartite symmetry makes all odd moments vanish. Thus the probability measures E ρ τ , o \mathbb{E}\rho_{\tau,o} E ρ τ , o μ τ \mu_\tau μ τ
Remark 4.11 (Poisson-tree recursion). The Poisson-tree representation in Theorem 4.10 links the critical law to a recursive operator structure. Removing the root separates independent copies of the same rooted tree, yielding the cavity equation below.
The recursion must determine the law of an entire Green function, not just its value at one spectral parameter. We therefore work with random analytic functions and compare their joint values at finitely many points. Let
H = { h ∈ Hol ( C + , C + ) : ∣ h ( z ) ∣ ≤ ( Im z ) − 1 for all z ∈ C + } , (2) \mathscr{H}=\{h\in\operatorname{Hol}(\mathbb{C}_+,\mathbb{C}_+): |h(z)|\leq(\operatorname{Im}z)^{-1}\text{ for all }z\in\mathbb{C}_+\}, \tag*{(2)} H = { h ∈ Hol ( C + , C + ) : ∣ h ( z ) ∣ ≤ ( Im z ) − 1 for all z ∈ C + } , (2) with the topology of locally uniform convergence. The ambient space Hol ( C + ) \operatorname{Hol}(\mathbb{C}_+) Hol ( C + ) H \mathscr{H} H C + \mathbb{C}_+ C + C + \mathbb{C}_+ C +
Lemma 4.12 (Identification of analytic laws). Let Q Q Q Q ′ Q' Q ′ H \mathscr{H} H D ⊂ C + D\subset\mathbb{C}_+ D ⊂ C + C + \mathbb{C}_+ C + Q Q Q Q ′ Q' Q ′ D D D Q = Q ′ Q=Q' Q = Q ′
Proof. The evaluation map
E D : H ⟶ C D , E D ( h ) = ( h ( z ) ) z ∈ D , E_D:\mathscr{H}\longrightarrow\mathbb{C}^{D},\qquad E_D(h)=(h(z))_{z\in D}, E D : H ⟶ C D , E D ( h ) = ( h ( z ) ) z ∈ D , is Borel, and the identity theorem makes it injective. Since H \mathscr{H} H [15 Corollary 15.2, p. 89] shows that E D ( H ) E_D(\mathscr{H}) E D ( H ) E D E_D E D D D D ( E D ) # Q = ( E D ) # Q ′ (E_D)_\#Q=(E_D)_\#Q' ( E D ) # Q = ( E D ) # Q ′ Q = Q ′ Q=Q' Q = Q ′
For a probability law Q Q Q H \mathscr{H} H { Y j } j ≥ 1 \{Y_j\}_{j\geq1} { Y j } j ≥ 1 L τ L_\tau L τ h 1 , h 2 , … h_1,h_2,\ldots h 1 , h 2 , … Q Q Q
Φ Y ( h 1 , h 2 , … ) ( z ) = − ( z + ∑ j ≥ 1 Y j h j ( z ) ) − 1 . \Phi_Y(h_1,h_2,\ldots)(z)=-\left(z+\sum_{j\geq1}Y_jh_j(z)\right)^{-1}. Φ Y ( h 1 , h 2 , … ) ( z ) = − ( z + j ≥ 1 ∑ Y j h j ( z ) ) − 1 . For every compact K ⊂ C + K \subset\mathbb{C}_{+} K ⊂ C + η K = inf z ∈ K Im z > 0 \eta_K = \inf_{z\in K} \operatorname{Im} z > 0 η K = inf z ∈ K Im z > 0
∑ j ≥ 1 sup z ∈ K ∣ Y j h j ( z ) ∣ ≤ 1 η K ∑ j ≥ 1 Y j < ∞ . \sum_{j\geq1}\sup_{z\in K}|Y_jh_j(z)| \leq\frac{1}{\eta_K}\sum_{j\geq1}Y_j < \infty. j ≥ 1 ∑ z ∈ K sup ∣ Y j h j ( z ) ∣ ≤ η K 1 j ≥ 1 ∑ Y j < ∞. The series therefore converges absolutely and locally uniformly. Positivity of the denominator’s imaginary part gives Φ Y ( h 1 , h 2 , … ) ∈ H \Phi_{\mathbf{Y}}(h_1,h_2,\ldots)\in\mathscr{H} Φ Y ( h 1 , h 2 , … ) ∈ H Hol ( C + ) \operatorname{Hol}(\mathbb{C}_{+}) Hol ( C + ) F τ ( Q ) \mathcal{F}_{\tau}(Q) F τ ( Q ) H \mathscr{H} H
Theorem 4.13 (Critical Poisson cavity recursion). Let
G τ ( z ) = ⟨ δ ∅ , ( A τ − z ) − 1 δ ∅ ⟩ , z ∈ C + , G_{\tau}(z)=\langle\delta_{\varnothing},(A_{\tau}-z)^{-1}\delta_{\varnothing}\rangle,\qquad z\in\mathbb{C}_{+}, G τ ( z ) = ⟨ δ ∅ , ( A τ − z ) − 1 δ ∅ ⟩ , z ∈ C + , viewed as a random element of H \mathscr{H} H Q τ Q_{\tau} Q τ H \mathscr{H} H
Q τ = F τ ( Q τ ) . Q_{\tau}=\mathcal{F}_{\tau}(Q_{\tau}). Q τ = F τ ( Q τ ) . Equivalently, if G τ , j G_{\tau,j} G τ , j G τ G_{\tau} G τ { Y j } \{Y_j\} { Y j }
G τ ( z ) = d − ( z + ∑ j ≥ 1 Y j G τ , j ( z ) ) − 1 . (3) G_{\tau}(z)\mathrel{\overset{\mathrm{d}}{=}}-\left(z+\sum_{j\geq1}Y_jG_{\tau,j}(z)\right)^{-1}. \tag*{(3)} G τ ( z ) = d − ( z + j ≥ 1 ∑ Y j G τ , j ( z ) ) − 1 . (3) Moreover,
∫ R μ τ ( d x ) x − z = E G τ ( z ) . \int_{\mathbb{R}}\frac{\mu_{\tau}(\mathrm{d}x)}{x-z}=\mathbb{E}G_{\tau}(z). ∫ R x − z μ τ ( d x ) = E G τ ( z ) . Proof. Removing the root leaves independent rooted subtrees. The children of each branch root are governed by a fresh Poisson process with intensity L τ L_{\tau} L τ
b N = ∑ j > N Y j δ o j , b_N=\sum_{j>N}\sqrt{Y_j}\,\delta_{o_j}, b N = j > N ∑ Y j δ o j , where o j o_j o j j j j
B N = ∣ δ o ⟩ ⟨ b N ∣ + ∣ b N ⟩ ⟨ δ o ∣ . B_N=|\delta_o\rangle\langle b_N|+|b_N\rangle\langle\delta_o|. B N = ∣ δ o ⟩ ⟨ b N ∣ + ∣ b N ⟩ ⟨ δ o ∣. Unlike the finite principal compressions used above, this truncation removes only connections at the root; each descendant branch remains unchanged. Set A τ ( N ) = A τ − B N A_{\tau}^{(N)}=A_{\tau}-B_N A τ ( N ) = A τ − B N N N N A τ ( N ) A_{\tau}^{(N)} A τ ( N ) B N B_N B N
rank B N ≤ 2 , ∥ B N ∥ ≤ 2 ∥ b N ∥ = 2 ( ∑ j > N Y j ) 1 / 2 ⟶ 0. \operatorname{rank}B_N\leq2,\qquad\lVert B_N\rVert\leq2\lVert b_N\rVert=2\left(\sum_{j>N}Y_j\right)^{1/2}\longrightarrow0. rank B N ≤ 2 , ∥ B N ∥ ≤ 2 ∥ b N ∥ = 2 j > N ∑ Y j 1/2 ⟶ 0. Thus A τ ( N ) → A τ A_{\tau}^{(N)}\to A_{\tau} A τ ( N ) → A τ Q τ Q_{\tau} Q τ F τ \mathcal{F}_{\tau} F τ Theorem 4.10 establish the measurability of z ↦ G τ ( z ) z \mapsto G_\tau(z) z ↦ G τ ( z ) H \mathcal{H} H
For uniqueness, let Q , Q ′ Q,Q' Q , Q ′ W 1 W_1 W 1 z 1 , … , z r ∈ C + z_1,\ldots,z_r \in\mathbb{C}_+ z 1 , … , z r ∈ C + η ℓ = Im z ℓ > 1 \eta_\ell=\operatorname{Im} z_\ell>1 η ℓ = Im z ℓ > 1 ν , ν ′ \nu,\nu' ν , ν ′ C r \mathbb{C}^r C r
c ( u , v ) = ∑ ℓ = 1 r ∣ u ℓ − v ℓ ∣ . c(u,v)=\sum_{\ell=1}^{r}|u_\ell-v_\ell|. c ( u , v ) = ℓ = 1 ∑ r ∣ u ℓ − v ℓ ∣. Take an optimal coupling of the joint evaluation vectors with laws ν , ν ′ \nu,\nu' ν , ν ′ D ℓ , D ℓ ′ D_\ell,D'_\ell D ℓ , D ℓ ′
∣ D ℓ ∣ , ∣ D ℓ ′ ∣ ≥ η ℓ |D_\ell|,|D'_\ell|\geq\eta_\ell ∣ D ℓ ∣ , ∣ D ℓ ′ ∣ ≥ η ℓ and
E ∣ − 1 D ℓ + 1 D ℓ ′ ∣ ≤ 1 η ℓ 2 E ∑ j ≥ 1 Y j ∣ h j ( z ℓ ) − h j ′ ( z ℓ ) ∣ = 1 η ℓ 2 E ∣ h ( z ℓ ) − h ′ ( z ℓ ) ∣ . \mathbb{E}\left|-\frac{1}{D_\ell}+\frac{1}{D'_\ell}\right|
\leq\frac{1}{\eta_\ell^2}\mathbb{E}\sum_{j\geq1}Y_j|h_j(z_\ell)-h'_j(z_\ell)|
=\frac{1}{\eta_\ell^2}\mathbb{E}|h(z_\ell)-h'(z_\ell)|. E − D ℓ 1 + D ℓ ′ 1 ≤ η ℓ 2 1 E j ≥ 1 ∑ Y j ∣ h j ( z ℓ ) − h j ′ ( z ℓ ) ∣ = η ℓ 2 1 E ∣ h ( z ℓ ) − h ′ ( z ℓ ) ∣. because ∫ y L τ ( d y ) = 1 \int y\,L_\tau(\mathrm{d}y)=1 ∫ y L τ ( d y ) = 1 max ℓ η ℓ − 2 < 1 \max_\ell\eta_\ell^{-2}<1 max ℓ η ℓ − 2 < 1 Q Q Q Q ′ Q' Q ′ Im z > 1 \operatorname{Im}z>1 Im z > 1 Lemma 4.12 gives Q = Q ′ Q=Q' Q = Q ′
Finally, Theorem 4.10 and the resolvent bound justify Fubini’s theorem, which gives the Stieltjes-transform identity.
Corollary 4.14 (Structure of the critical moments). For every k ≥ 1 k\geq1 k ≥ 1 M 2 k ( τ ) M_{2k}(\tau) M 2 k ( τ ) τ − 1 \tau^{-1} τ − 1 k − 1 k-1 k − 1 Cat k \operatorname{Cat}_k Cat k τ 1 − k \tau^{1-k} τ 1 − k ( ( k − 1 ) ! ) 2 ((k-1)!)^2 (( k − 1 )! ) 2
M 2 ( τ ) = 1 , M 4 ( τ ) = 2 + τ − 1 , M 6 ( τ ) = 5 + 6 τ − 1 + 4 τ − 2 . M_2(\tau)=1,\qquad M_4(\tau)=2+\tau^{-1},\qquad M_6(\tau)=5+6\tau^{-1}+4\tau^{-2}. M 2 ( τ ) = 1 , M 4 ( τ ) = 2 + τ − 1 , M 6 ( τ ) = 5 + 6 τ − 1 + 4 τ − 2 . Moreover,
μ τ ⟹ μ s c ( τ → ∞ ) . \mu_\tau\Longrightarrow\mu_{\mathrm{sc}}\qquad(\tau\to\infty). μ τ ⟹ μ sc ( τ → ∞ ) . Proof. The power of τ − 1 \tau^{-1} τ − 1 γ \gamma γ k − e ( γ ) k-e(\gamma) k − e ( γ ) e ( γ ) = k e(\gamma)=k e ( γ ) = k Cat k \operatorname{Cat}_k Cat k e ( γ ) = 1 e(\gamma)=1 e ( γ ) = 1 ( ( k − 1 ) ! ) 2 τ 1 − k ((k-1)!)^2\tau^{1-k} (( k − 1 )! ) 2 τ 1 − k k = 1 , 2 , 3 k=1,2,3 k = 1 , 2 , 3
To prove the endpoint convergence, let τ n → ∞ \tau_n\to\infty τ n → ∞ μ τ n \mu_{\tau_n} μ τ n p p p τ n ≥ 1 \tau_n\geq1 τ n ≥ 1
sup n M 2 p + 2 ( τ n ) < ∞ . \sup_n M_{2p+2}(\tau_n)<\infty. n sup M 2 p + 2 ( τ n ) < ∞. Thus powers of order 2 p 2p 2 p e ( γ ) < k e(\gamma)<k e ( γ ) < k τ n → ∞ \tau_n\to\infty τ n → ∞ e ( γ ) = k e(\gamma)=k e ( γ ) = k Cat k \operatorname{Cat}_k Cat k μ τ n ⇒ μ s c \mu_{\tau_n}\Rightarrow\mu_{\mathrm{sc}} μ τ n ⇒ μ sc
Subcritical collapse and the escaped spectral energy
When d α d 2 → 0 d\alpha_d^2 \to0 d α d 2 → 0 d \sqrt{d} d δ 0 \delta_0 δ 0 a > 0 a>0 a > 0 q > 0 q>0 q > 0
( a ) q = Γ ( a + q ) Γ ( a ) . (a)_q = \frac{\Gamma(a+q)}{\Gamma(a)}. ( a ) q = Γ ( a ) Γ ( a + q ) . This is the real-order extension of the rising factorial introduced above and agrees with it when q q q
Lemma 4.15 (Gamma-ratio asymptotics). For every fixed q > 0 q>0 q > 0
( a ) q ∼ Γ ( q ) a ( a ↓ 0 ) , ( a ) q ∼ a q ( a → ∞ ) . (a)_q \sim\Gamma(q)a \quad(a\downarrow0), \qquad(a)_q \sim a^q \quad(a\to\infty). ( a ) q ∼ Γ ( q ) a ( a ↓ 0 ) , ( a ) q ∼ a q ( a → ∞ ) . Proof. The first relation follows from Γ ( a + q ) → Γ ( q ) \Gamma(a+q)\to\Gamma(q) Γ ( a + q ) → Γ ( q ) a Γ ( a ) → 1 a\Gamma(a)\to1 a Γ ( a ) → 1 Γ ( a + q ) / Γ ( a ) ∼ a q \Gamma(a+q)/\Gamma(a)\sim a^q Γ ( a + q ) /Γ ( a ) ∼ a q
Theorem 4.16 (Subcritical collapse and escape of the second moment). Let d → ∞ d\to\infty d → ∞ α d > 0 \alpha_d>0 α d > 0
τ d : = d α d 2 ⟶ 0. \tau_d := d\alpha_d^2 \longrightarrow0. τ d := d α d 2 ⟶ 0. Then
( d ⋅ ) # μ d , α d ⟹ δ 0 . (\sqrt{d}\,\cdot)_{\#}\mu_{d,\alpha_d}\Longrightarrow\delta_0. ( d ⋅ ) # μ d , α d ⟹ δ 0 . More strongly, for every 0 < p < 2 0<p<2 0 < p < 2
∫ R ∣ x ∣ p ( d ⋅ ) # μ d , α d ( d x ) ⟶ 0 , \int_{\mathbb{R}} |x|^p(\sqrt{d}\,\cdot)_{\#}\mu_{d,\alpha_d}(\mathrm{d}x)\longrightarrow0, ∫ R ∣ x ∣ p ( d ⋅ ) # μ d , α d ( d x ) ⟶ 0 , whereas
∫ R x 2 ( d ⋅ ) # μ d , α d ( d x ) = 1 \int_{\mathbb{R}} x^2(\sqrt{d}\,\cdot)_{\#}\mu_{d,\alpha_d}(\mathrm{d}x)=1 ∫ R x 2 ( d ⋅ ) # μ d , α d ( d x ) = 1 for every d d d x 2 x^2 x 2
Proof. Put θ d = d α d \theta_d=d\alpha_d θ d = d α d q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 )
S o : = ⟨ δ o , T ∞ 2 δ o ⟩ = ∑ u ∼ o X o , u X u , o . S_o := \langle\delta_o,T_\infty^2\delta_o\rangle= \sum_{u\sim o} X_{o,u}X_{u,o}. S o := ⟨ δ o , T ∞ 2 δ o ⟩ = u ∼ o ∑ X o , u X u , o . Let ρ o ( 2 ) \rho_o^{(2)} ρ o ( 2 ) T ∞ 2 T_\infty^2 T ∞ 2 δ o \delta_o δ o
⟨ δ o , ( T ∞ 2 ) q δ o ⟩ = ∫ [ 0 , 1 ] x q ρ o ( 2 ) ( d x ) , S o = ∫ [ 0 , 1 ] x ρ o ( 2 ) ( d x ) . \langle\delta_o,(T_\infty^2)^q\delta_o\rangle=\int_{[0,1]}x^q\rho_o^{(2)}(\mathrm{d}x),\qquad S_o=\int_{[0,1]}x\rho_o^{(2)}(\mathrm{d}x). ⟨ δ o , ( T ∞ 2 ) q δ o ⟩ = ∫ [ 0 , 1 ] x q ρ o ( 2 ) ( d x ) , S o = ∫ [ 0 , 1 ] x ρ o ( 2 ) ( d x ) . Jensen’s inequality, applied to the probability measure ρ o ( 2 ) \rho_o^{(2)} ρ o ( 2 ) x ↦ x q x\mapsto x^q x ↦ x q
⟨ δ o , ( T ∞ 2 ) q δ o ⟩ ≤ S o q . \langle\delta_o,(T_\infty^2)^q\delta_o\rangle\le S_o^q. ⟨ δ o , ( T ∞ 2 ) q δ o ⟩ ≤ S o q . Since q < 1 q<1 q < 1
E S o q ≤ ∑ u ∼ o E [ X o , u q X u , o q ] = d ( ( α d ) q ( θ d ) q ) 2 . \mathbb{E}S_o^q \leq\sum_{u\sim o}\mathbb{E}\left[X_{o,u}^qX_{u,o}^q\right]
= d\left(\frac{(\alpha_d)_q}{(\theta_d)_q}\right)^2. E S o q ≤ u ∼ o ∑ E [ X o , u q X u , o q ] = d ( ( θ d ) q ( α d ) q ) 2 . Consequently,
∫ ∣ x ∣ 2 q ( d ⋅ ) # μ d , α ( d x ) ≤ d q + 1 ( ( α d ) q ( θ d ) q ) 2 . \int|x|^{2q}\,(\sqrt{d}\cdot)_{\#}\mu_{d,\alpha}(dx)
\leq d^{q+1}\left(\frac{(\alpha_d)_q}{(\theta_d)_q}\right)^2. ∫ ∣ x ∣ 2 q ( d ⋅ ) # μ d , α ( d x ) ≤ d q + 1 ( ( θ d ) q ( α d ) q ) 2 . We show that the right-hand side tends to zero by considering subsequences. Every subsequence has a further subsequence on which θ d → θ ∈ [ 0 , ∞ ] \theta_d\to\theta\in[0,\infty] θ d → θ ∈ [ 0 , ∞ ] τ d → 0 \tau_d\to0 τ d → 0 α d → 0 \alpha_d\to0 α d → 0 θ = 0 \theta=0 θ = 0 ( α d ) q / ( θ d ) q ∼ α d / θ d = 1 / d (\alpha_d)_q/(\theta_d)_q\sim\alpha_d/\theta_d=1/d ( α d ) q / ( θ d ) q ∼ α d / θ d = 1/ d d q − 1 d^{q-1} d q − 1 θ ∈ ( 0 , ∞ ) \theta\in(0,\infty) θ ∈ ( 0 , ∞ ) O ( d q + 1 α d 2 ) = O ( θ d 2 d q − 1 ) O(d^{q+1}\alpha_d^2)=O(\theta_d^2d^{q-1}) O ( d q + 1 α d 2 ) = O ( θ d 2 d q − 1 ) θ = ∞ \theta=\infty θ = ∞
d q + 1 ( ( α d ) q ( θ d ) q ) 2 ∼ Γ ( q ) 2 τ d 1 − q . d^{q+1}\left(\frac{(\alpha_d)_q}{(\theta_d)_q}\right)^2
\sim\Gamma(q)^2\tau_d^{1-q}. d q + 1 ( ( θ d ) q ( α d ) q ) 2 ∼ Γ ( q ) 2 τ d 1 − q . All three bounds tend to zero. Because every subsequence has a further subsequence with this property, the original bound also tends to zero. Taking p = 2 q p=2q p = 2 q δ 0 \delta_0 δ 0 m 2 ( d , α ) = 1 / d m_2(d,\alpha)=1/d m 2 ( d , α ) = 1/ d
Combining the preceding theorem with the semicircle and critical theorems gives a complete classification on the variance scale. Write
μ ^ d , α = ( d ⋅ ) # μ d , α , R ‾ + = [ 0 , ∞ ] , \widehat{\mu}_{d,\alpha}=(\sqrt{d}\cdot)_{\#}\mu_{d,\alpha},\qquad\overline{\mathbb{R}}_+=[0,\infty], μ d , α = ( d ⋅ ) # μ d , α , R + = [ 0 , ∞ ] , where R ‾ + \overline{\mathbb{R}}_+ R + τ ↦ τ / ( 1 + τ ) \tau\mapsto\tau/(1+\tau) τ ↦ τ / ( 1 + τ ) ∞ \infty ∞ 1 1 1
Φ ( τ ) = { δ 0 , τ = 0 , μ τ , 0 < τ < ∞ , μ s c , τ = ∞ . \Phi(\tau)=
\begin{cases}
\delta_0, & \tau=0,\\
\mu_\tau, & 0<\tau<\infty,\\
\mu_{\mathrm{sc}}, & \tau=\infty.
\end{cases} Φ ( τ ) = ⎩ ⎨ ⎧ δ 0 , μ τ , μ sc , τ = 0 , 0 < τ < ∞ , τ = ∞. Corollary 4.17 (Complete variance-scale classification). Let d → ∞ d\to\infty d → ∞ α d > 0 \alpha_d>0 α d > 0 τ d = d α d 2 \tau_d=d\alpha_d^2 τ d = d α d 2 μ ^ d , α d \widehat{\mu}_{d,\alpha_d} μ d , α d Φ ( τ ) \Phi(\tau) Φ ( τ ) τ ∈ R ‾ + \tau\in\overline{\mathbb{R}}_+ τ ∈ R + τ d \tau_d τ d μ ^ d , α d \widehat{\mu}_{d,\alpha_d} μ d , α d τ d \tau_d τ d R ‾ + \overline{\mathbb{R}}_+ R + τ d → τ \tau_d\to\tau τ d → τ
W 1 ( μ ^ d , α d , Φ ( τ ) ) ⟶ 0. W_1\left(\widehat{\mu}_{d,\alpha_d},\Phi(\tau)\right)\longrightarrow0. W 1 ( μ d , α d , Φ ( τ ) ) ⟶ 0. In particular,
μ ^ d , α d ⟹ μ s c ⟺ d α d 2 ⟶ ∞ . \widehat{\mu}_{d,\alpha_d}\Longrightarrow\mu_{\mathrm{sc}}
\quad\Longleftrightarrow\quad
d\alpha_d^2\longrightarrow\infty. μ d , α d ⟹ μ sc ⟺ d α d 2 ⟶ ∞. At τ = 0 \tau=0 τ = 0
Proof. Along a sequence with τ d → τ \tau_d \to\tau τ d → τ Theorems 4.3 , 4.5 , and 4.16 give the weak limit Φ ( τ ) \Phi(\tau) Φ ( τ ) ∫ x 2 μ ^ d , α ( d x ) = 1 \int x^2\,\widehat{\mu}_{d,\alpha}(\mathrm{d}x)=1 ∫ x 2 μ d , α ( d x ) = 1
∫ ∣ x ∣ > R ∣ x ∣ μ ^ d , α ( d x ) ≤ R − 1 , R > 0 , \int_{\lvert x\rvert>R} \lvert x\rvert\,\widehat{\mu}_{d,\alpha}(\mathrm{d}x)\le R^{-1},\qquad R>0, ∫ ∣ x ∣ > R ∣ x ∣ μ d , α ( d x ) ≤ R − 1 , R > 0 , which gives uniform integrability of first moments and hence W 1 W_1 W 1 R ‾ + \overline{\mathbb{R}}_+ R + τ d \tau_d τ d
The laws Φ ( τ ) \Phi(\tau) Φ ( τ ) τ = 0 \tau=0 τ = 0 0 < τ < ∞ 0<\tau<\infty 0 < τ < ∞ 2 + τ − 1 2+\tau^{-1} 2 + τ − 1 2 2 2 R ‾ + \overline{\mathbb{R}}_+ R +
We next transfer the classification to the quantum walk. The comparison concerns the normalized inherited component, with its two phase branches centered at π / 2 \pi/2 π /2 − π / 2 -\pi/2 − π /2
Corollary 4.18 (Quantitative inherited eigenphase transition). Let ϑ ( x ) = arcsin x \vartheta(x)=\arcsin x ϑ ( x ) = arcsin x − 1 ≤ x ≤ 1 -1\le x\le1 − 1 ≤ x ≤ 1 ϕ ± ( x ) = exp { ± i ( π / 2 − ϑ ( x ) ) } \phi_\pm(x)=\exp\{\pm i(\pi/2-\vartheta(x))\} ϕ ± ( x ) = exp { ± i ( π /2 − ϑ ( x ))} π / 2 \pi/2 π /2 − π / 2 -\pi/2 − π /2 − ϑ ( x ) -\vartheta(x) − ϑ ( x ) ϑ ( x ) \vartheta(x) ϑ ( x ) μ d , α \mu_{d,\alpha} μ d , α ϑ # μ d , α \vartheta_\#\mu_{d,\alpha} ϑ # μ d , α
η ^ d , α = ( d ϑ ) # μ d , α , C p h = π 2 − 1. \widehat{\eta}_{d,\alpha}=(\sqrt{d}\,\vartheta)_\#\mu_{d,\alpha},\qquad C_{\mathrm{ph}}=\frac{\pi}{2}-1. η d , α = ( d ϑ ) # μ d , α , C ph = 2 π − 1. For every d ≥ 3 d\ge3 d ≥ 3 α > 0 \alpha>0 α > 0
W 1 ( η ^ d , α , μ ^ d , α ) ≤ C p h m 4 ( d , α ) ≤ C p h d . (4) W_1(\widehat{\eta}_{d,\alpha},\widehat{\mu}_{d,\alpha})\le C_{\mathrm{ph}}\sqrt{m_4(d,\alpha)}\le\frac{C_{\mathrm{ph}}}{\sqrt{d}}. \tag*{(4)} W 1 ( η d , α , μ d , α ) ≤ C ph m 4 ( d , α ) ≤ d C ph . (4) For every sequence d → ∞ d\to\infty d → ∞ α d > 0 \alpha_d>0 α d > 0 d α d 2 d\alpha_d^2 d α d 2 R ‾ + \overline{\mathbb{R}}_+ R + d α d 2 → τ d\alpha_d^2\to\tau d α d 2 → τ η ^ d , α d → Φ ( τ ) \widehat{\eta}_{d,\alpha_d}\to\Phi(\tau) η d , α d → Φ ( τ ) W 1 W_1 W 1 O ( d − 1 ) O(d^{-1}) O ( d − 1 ) sup d d 2 m 4 ( d , α d ) < ∞ \sup_d d^2m_4(d,\alpha_d)<\infty sup d d 2 m 4 ( d , α d ) < ∞ ( δ 1 + δ − 1 ) / 2 (\delta_1+\delta_{-1})/2 ( δ 1 + δ − 1 ) /2
Proof. The expansion
arcsin t − t = ∑ j ≥ 1 a j t 2 j + 1 , a j = ( 2 j j ) 4 j ( 2 j + 1 ) > 0 , ∑ j ≥ 1 a j = π 2 − 1 , \arcsin t-t=\sum_{j\ge1}a_jt^{2j+1},\qquad a_j=\frac{\binom{2j}{j}}{4^j(2j+1)}>0,\qquad\sum_{j\ge1}a_j=\frac{\pi}{2}-1, arcsin t − t = j ≥ 1 ∑ a j t 2 j + 1 , a j = 4 j ( 2 j + 1 ) ( j 2 j ) > 0 , j ≥ 1 ∑ a j = 2 π − 1 , valid on [ 0 , 1 ] [0,1] [ 0 , 1 ] ∣ arcsin x − x ∣ ≤ C p h ∣ x ∣ 3 \lvert\arcsin x-x\rvert\le C_{\mathrm{ph}}\lvert x\rvert^3 ∣ arcsin x − x ∣ ≤ C ph ∣ x ∣ 3 ∣ x ∣ ≤ 1 \lvert x\rvert\le1 ∣ x ∣ ≤ 1 X ∼ μ d , α X\sim\mu_{d,\alpha} X ∼ μ d , α E X 2 = m 2 ( d , α ) = 1 / d \mathbb{E}X^2=m_2(d,\alpha)=1/d E X 2 = m 2 ( d , α ) = 1/ d ∣ X ∣ ≤ 1 \lvert X\rvert\le1 ∣ X ∣ ≤ 1 m 4 ( d , α ) ≤ m 2 ( d , α ) = 1 / d m_4(d,\alpha)\le m_2(d,\alpha)=1/d m 4 ( d , α ) ≤ m 2 ( d , α ) = 1/ d E ∣ X ∣ 3 ≤ ( E X 2 E X 4 ) 1 / 2 \mathbb{E}\lvert X\rvert^3\le(\mathbb{E}X^2\,\mathbb{E}X^4)^{1/2} E ∣ X ∣ 3 ≤ ( E X 2 E X 4 ) 1/2
W 1 ( η ^ d , α , μ ^ d , α ) ≤ C p h d E ∣ X ∣ 3 ≤ C p h m 4 ( d , α ) ≤ C p h d . W_1(\widehat{\eta}_{d,\alpha},\widehat{\mu}_{d,\alpha})\le C_{\mathrm{ph}}\sqrt{d}\,\mathbb{E}\lvert X\rvert^3\le C_{\mathrm{ph}}\sqrt{m_4(d,\alpha)}\le\frac{C_{\mathrm{ph}}}{\sqrt{d}}. W 1 ( η d , α , μ d , α ) ≤ C ph d E ∣ X ∣ 3 ≤ C ph m 4 ( d , α ) ≤ d C ph . The two families therefore have the same weak subsequential limits. Corollary 4.17 gives the classification and the W 1 W_1 W 1 Theorems 4.3 and 4.5 gives the O ( d − 1 ) O(d^{-1}) O ( d − 1 )
We next describe the spectral energy that escapes the weak-limit scale. Set
θ d = d α d , s d = 1 + θ d . \theta_d=d\alpha_d,\qquad s_d=1+\theta_d. θ d = d α d , s d = 1 + θ d . Because d m 2 ( d , α d ) = 1 dm_2(d,\alpha_d)=1 d m 2 ( d , α d ) = 1
π d e n : = ( s d ⋅ ) # [ d x 2 μ d , α d ( d x ) ] \pi_d^{\mathrm{en}}:=(s_d\cdot)_{\#}\left[d x^2\mu_{d,\alpha_d}(\mathrm{d}x)\right] π d en := ( s d ⋅ ) # [ d x 2 μ d , α d ( d x ) ] is a probability measure. The factor d x 2 d x^2 d x 2 x ↦ s d x x\mapsto s_d x x ↦ s d x
Theorem 4.19 (Subcritical spectral-energy profile). Assume that
d α d 2 ⟶ 0 , d α d ⟶ θ ∈ R ‾ + . d\alpha_d^2\longrightarrow0,\qquad d\alpha_d\longrightarrow\theta\in\overline{\mathbb{R}}_+. d α d 2 ⟶ 0 , d α d ⟶ θ ∈ R + . Let χ \chi χ 0 < θ < ∞ 0<\theta<\infty 0 < θ < ∞ B θ , B θ ′ B_\theta,B'_\theta B θ , B θ ′ ( 1 , θ ) (1,\theta) ( 1 , θ ) B 0 = B 0 ′ = 1 B_0=B'_0=1 B 0 = B 0 ′ = 1 θ ↓ 0 \theta\downarrow0 θ ↓ 0 θ = ∞ \theta=\infty θ = ∞ E , E ′ E,E' E , E ′
Z θ = { χ ( 1 + θ ) B θ B θ ′ , 0 ≤ θ < ∞ , χ E E ′ , θ = ∞ . Z_\theta=
\begin{cases}
\chi(1+\theta)\sqrt{B_\theta B'_\theta}, & 0\leq\theta<\infty,\\
\chi\sqrt{EE'}, & \theta=\infty.
\end{cases} Z θ = { χ ( 1 + θ ) B θ B θ ′ , χ E E ′ , 0 ≤ θ < ∞ , θ = ∞. Then
π d e n ⟹ π θ : = L ( Z θ ) , \pi_d^{\mathrm{en}}\Longrightarrow\pi_\theta:=\mathcal{L}(Z_\theta), π d en ⟹ π θ := L ( Z θ ) , and all moments converge. The odd moments vanish, and
∫ x 2 k π θ ( d x ) = { ( 1 + θ ) 2 k [ k ! ( 1 + θ ) k ] 2 , 0 ≤ θ < ∞ , ( k ! ) 2 , θ = ∞ . \int x^{2k}\,\pi_\theta(\mathrm{d}x)=
\begin{cases}
(1+\theta)^{2k}\left[\dfrac{k!}{(1+\theta)_k}\right]^2, & 0\leq\theta<\infty,\\
(k!)^2, & \theta=\infty.
\end{cases} ∫ x 2 k π θ ( d x ) = ⎩ ⎨ ⎧ ( 1 + θ ) 2 k [ ( 1 + θ ) k k ! ] 2 , ( k ! ) 2 , 0 ≤ θ < ∞ , θ = ∞. In particular, π 0 = 1 2 ( δ − 1 + δ 1 ) \pi_0=\frac{1}{2}(\delta_{-1}+\delta_1) π 0 = 2 1 ( δ − 1 + δ 1 ) π ∞ \pi_\infty π ∞
Proof. The total mass of π d e n \pi_d^{\mathrm{en}} π d en d m 2 ( d , α d ) = 1 dm_2(d,\alpha_d)=1 d m 2 ( d , α d ) = 1 μ d , α d \mu_{d,\alpha_d} μ d , α d π d e n \pi_d^{\mathrm{en}} π d en k ≥ 0 k\geq0 k ≥ 0
∫ x 2 k π d e n ( d x ) = d d 2 k m 2 k + 2 ( d , α d ) . \int x^{2k}\,\pi_d^{\mathrm{en}}(\mathrm{d}x)=d_d^{2k}m_{2k+2}(d,\alpha_d). ∫ x 2 k π d en ( d x ) = d d 2 k m 2 k + 2 ( d , α d ) . Set K = k + 1 K=k+1 K = k + 1 Proposition 4.1 . The unique class with one traced edge contributes
L d , k = d d 2 s d 2 k ( α d ) k + 1 2 ( θ d ) k + 1 2 = s d 2 k [ ( α d ) k + 1 / α d ( 1 + θ d ) k ] 2 . L_{d,k}=d_d^2s_d^{2k}\frac{(\alpha_d)_{k+1}^2}{(\theta_d)_{k+1}^2}
=s_d^{2k}\left[\frac{(\alpha_d)_{k+1}/\alpha_d}{(1+\theta_d)_k}\right]^2. L d , k = d d 2 s d 2 k ( θ d ) k + 1 2 ( α d ) k + 1 2 = s d 2 k [ ( 1 + θ d ) k ( α d ) k + 1 / α d ] 2 . Since α d → 0 \alpha_d \to0 α d → 0
( α d ) k + 1 α d ⟶ k ! . \frac{(\alpha_d)_{k+1}}{\alpha_d} \longrightarrow k!. α d ( α d ) k + 1 ⟶ k ! . If θ < ∞ \theta< \infty θ < ∞
L d , k ⟶ ( 1 + θ ) 2 k [ k ! ( 1 + θ ) k ] 2 . L_{d,k} \longrightarrow(1+\theta)^{2k}\left[\frac{k!}{(1+\theta)_k}\right]^2. L d , k ⟶ ( 1 + θ ) 2 k [ ( 1 + θ ) k k ! ] 2 . If θ = ∞ \theta= \infty θ = ∞ s d k / ( 1 + θ d ) k → 1 s_d^k/(1+\theta_d)_k \to1 s d k / ( 1 + θ d ) k → 1 ( k ! ) 2 (k!)^2 ( k ! ) 2
We now bound the contribution of every class with at least two traced edges. Fix K K K d d d d ≥ 2 K d \ge2K d ≥ 2 K α d ≤ 1 \alpha_d \le1 α d ≤ 1 1 ≤ r ≤ K 1 \le r \le K 1 ≤ r ≤ K
( α d ) r ≤ α d r ! . (\alpha_d)_r \le\alpha_d r!. ( α d ) r ≤ α d r ! . Moreover, for every positive integer R R R
( θ d ) R ≥ θ d ( 1 + θ d ) R − 1 = θ d s d R − 1 . (\theta_d)_R \ge\theta_d(1+\theta_d)^{R-1} = \theta_d s_d^{R-1}. ( θ d ) R ≥ θ d ( 1 + θ d ) R − 1 = θ d s d R − 1 . A canonical class with e ≥ 2 e \ge2 e ≥ 2 e + 1 e+1 e + 1 N d ( γ ) ≤ d e N_d(\gamma) \le d^e N d ( γ ) ≤ d e ∑ v R v = 2 K \sum_v R_v = 2K ∑ v R v = 2 K d s d 2 K − 2 m 2 K ( d , α d ) d s_d^{2K-2}m_{2K}(d,\alpha_d) d s d 2 K − 2 m 2 K ( d , α d )
C K d e + 1 s d 2 K − 2 α d 2 e θ d e + 1 s d 2 K − e − 1 = C K [ α d s d ] e − 1 = C K ( α d + d α d 2 ) e − 1 . \begin{aligned}
C_K\frac{d^{e+1}s_d^{2K-2}\alpha_d^{2e}}{\theta_d^{e+1}s_d^{2K-e-1}} = C_K[\alpha_d s_d]^{e-1} \\
&= C_K(\alpha_d+d\alpha_d^2)^{e-1}.
\end{aligned} C K θ d e + 1 s d 2 K − e − 1 d e + 1 s d 2 K − 2 α d 2 e = C K [ α d s d ] e − 1 = C K ( α d + d α d 2 ) e − 1 . Here
α d s d = α d + d α d 2 ⟶ 0. \alpha_d s_d = \alpha_d+d\alpha_d^2 \longrightarrow0. α d s d = α d + d α d 2 ⟶ 0. The bound therefore tends to zero. Since W K \mathcal{W}_K W K
For finite θ \theta θ
E [ B θ k ] = k ! ( 1 + θ ) k , \mathbb{E}[B_\theta^k] = \frac{k!}{(1+\theta)_k}, E [ B θ k ] = ( 1 + θ ) k k ! , with the same formula under the convention B 0 = 1 B_0=1 B 0 = 1 E [ E k ] = k ! \mathbb{E}[E^k]=k! E [ E k ] = k ! Z θ Z_\theta Z θ k k k 2 k + 2 2k+2 2 k + 2 ∣ x ∣ 2 k |x|^{2k} ∣ x ∣ 2 k θ \theta θ θ = ∞ \theta=\infty θ = ∞ ( k ! ) 2 (k!)^2 ( k ! ) 2
( ( k ! ) 2 ) − 1 / ( 2 k ) = ( k ! ) − 1 / k ∼ e k , ((k!)^2)^{-1/(2k)}=(k!)^{-1/k}\sim\frac{e}{k}, (( k ! ) 2 ) − 1/ ( 2 k ) = ( k ! ) − 1/ k ∼ k e , so Carleman’s condition holds. Every weak subsequential limit has the moments of Z θ Z_\theta Z θ
Corollary 4.20 (Classification of the escaped-energy limits). The map θ ↦ π θ \theta\mapsto\pi_\theta θ ↦ π θ R ‾ + \overline{\mathbb{R}}_+ R + W 1 W_1 W 1 d → ∞ d \to\infty d → ∞ d α d 2 → 0 d\alpha_d^2 \to0 d α d 2 → 0 θ d = d α d \theta_d=d\alpha_d θ d = d α d π d e n \pi_d^{\mathrm{en}} π d en π θ \pi_\theta π θ θ ∈ R ‾ + \theta\in\overline{\mathbb{R}}_+ θ ∈ R + θ d \theta_d θ d π d e n \pi_d^{\mathrm{en}} π d en θ d \theta_d θ d R ‾ + \overline{\mathbb{R}}_+ R + W 1 W_1 W 1
Proof. For each k ≥ 0 k \ge0 k ≥ 0 Theorem 4.19 is continuous in θ ∈ R ‾ + \theta\in\overline{\mathbb{R}}_{+} θ ∈ R + ( k ! ) 2 (k!)^2 ( k ! ) 2 ( 1 + θ ) k ≤ ( 1 + θ ) k (1+\theta)^k \le(1+\theta)_k ( 1 + θ ) k ≤ ( 1 + θ ) k θ ≥ 0 \theta\ge0 θ ≥ 0 θ = ∞ \theta=\infty θ = ∞ π θ \pi_\theta π θ W 1 W_1 W 1
∫ x 4 π θ ( d x ) = 4 ( 1 + θ 2 + θ ) 2 ( 0 ≤ θ < ∞ ) , ∫ x 4 π ∞ ( d x ) = 4. \int x^4 \pi_\theta(\mathrm{d}x)=4\left(\frac{1+\theta}{2+\theta}\right)^2 \quad(0\le\theta<\infty), \qquad\int x^4\pi_\infty(\mathrm{d}x)=4. ∫ x 4 π θ ( d x ) = 4 ( 2 + θ 1 + θ ) 2 ( 0 ≤ θ < ∞ ) , ∫ x 4 π ∞ ( d x ) = 4. The fourth moment is strictly increasing on R ‾ + \overline{\mathbb{R}}_{+} R + Theorem 3.3 gives the exact second moment
∫ y 2 π d e n ( d y ) = d ( 1 + d α d ) 2 m 4 ( d , α d ) = 1 + ( 4 − 2 d ) α d + ( 2 d − 1 ) α d 2 ⟶ 1 , \begin{aligned}
\int y^2\pi_d^{\mathrm{en}}(\mathrm{d}y)=d(1+d\alpha_d)^2m_4(d,\alpha_d) \\
&=1+\left(4-\frac{2}{d}\right)\alpha_d+(2d-1)\alpha_d^2\longrightarrow1,
\end{aligned} ∫ y 2 π d en ( d y ) = d ( 1 + d α d ) 2 m 4 ( d , α d ) = 1 + ( 4 − d 2 ) α d + ( 2 d − 1 ) α d 2 ⟶ 1 , since d α d 2 → 0 d\alpha_d^2\to0 d α d 2 → 0 α d → 0 \alpha_d\to0 α d → 0 R ‾ + \overline{\mathbb{R}}_{+} R + Theorem 4.19 . This identifies all weak subsequential limits. Injectivity proves the converse, and uniform integrability gives W 1 W_1 W 1
The energy-biased law can also be converted back to a spectral intensity by dividing out the quadratic weight. This measure is naturally considered away from the origin, on R ∗ = R ∖ { 0 } \mathbb{R}^*=\mathbb{R}\setminus\{0\} R ∗ = R ∖ { 0 }
Λ d = d s d 2 ( s d ⋅ ) # μ d , α d . \Lambda_d=\frac{d}{s_d^2}(s_d\cdot)_{\#}\mu_{d,\alpha_d}. Λ d = s d 2 d ( s d ⋅ ) # μ d , α d . Then π d e n ( d x ) = x 2 Λ d ( d x ) \pi_d^{\mathrm{en}}(\mathrm{d}x)=x^2\Lambda_d(\mathrm{d}x) π d en ( d x ) = x 2 Λ d ( d x )
Corollary 4.21 (Subcritical spectral-intensity limit). Under the assumptions of Theorem 4.19 ,
Λ d → v Λ θ , Λ θ ( d x ) = x − 2 π θ ( d x ) , \Lambda_d\xrightarrow{\mathrm{v}}\Lambda_\theta,\qquad\Lambda_\theta(\mathrm{d}x)=x^{-2}\pi_\theta(\mathrm{d}x), Λ d v Λ θ , Λ θ ( d x ) = x − 2 π θ ( d x ) , vaguely on R ∗ \mathbb{R}^* R ∗ R ∗ \mathbb{R}^* R ∗
∫ R ∗ x 2 Λ θ ( d x ) = 1. \int_{\mathbb{R}^*}x^2\Lambda_\theta(\mathrm{d}x)=1. ∫ R ∗ x 2 Λ θ ( d x ) = 1. At the two endpoints,
Λ 0 = 1 2 ( δ − 1 + δ 1 ) , \Lambda_0=\frac{1}{2}(\delta_{-1}+\delta_1), Λ 0 = 2 1 ( δ − 1 + δ 1 ) , whereas
π ∞ ( d x ) = 2 ∣ x ∣ K 0 ( 2 ∣ x ∣ ) d x , Λ ∞ ( d x ) = 2 K 0 ( 2 ∣ x ∣ ) ∣ x ∣ d x , x ≠ 0 , \pi_\infty(\mathrm{d}x)=2|x|K_0(2|x|)\,\mathrm{d}x,\qquad\Lambda_\infty(\mathrm{d}x)=\frac{2K_0(2|x|)}{|x|}\,\mathrm{d}x,\qquad x\ne0, π ∞ ( d x ) = 2∣ x ∣ K 0 ( 2∣ x ∣ ) d x , Λ ∞ ( d x ) = ∣ x ∣ 2 K 0 ( 2∣ x ∣ ) d x , x = 0 , where K 0 K_0 K 0
Proof. For f ∈ C c ( R ∗ ) f \in C_{\mathrm{c}}(\mathbb{R}^{*}) f ∈ C c ( R ∗ ) f f f
g f ( x ) = { f ( x ) / x 2 , x ≠ 0 , 0 , x = 0 g_f(x)=
\begin{cases}
f(x)/x^2, & x\ne0,\\
0, & x=0
\end{cases} g f ( x ) = { f ( x ) / x 2 , 0 , x = 0 , x = 0 is a bounded continuous function on R \mathbb{R} R
∫ f ( x ) Λ d ( d x ) = ∫ g f ( x ) π d e n ( d x ) . \int f(x)\,\Lambda_d(\mathrm{d}x)=\int g_f(x)\,\pi_d^{\mathrm{en}}(\mathrm{d}x). ∫ f ( x ) Λ d ( d x ) = ∫ g f ( x ) π d en ( d x ) . Theorem 4.19 gives the vague limit. The second-moment identity follows because π θ \pi_\theta π θ π 0 \pi_0 π 0 { ± 1 } \{\pm1\} { ± 1 } θ = ∞ \theta=\infty θ = ∞ Y = E E ′ Y=EE' Y = E E ′
y ⟼ 2 K 0 ( 2 y ) , y > 0 , y\longmapsto2K_0(2\sqrt{y}),\qquad y>0, y ⟼ 2 K 0 ( 2 y ) , y > 0 , by the integral representation of K 0 K_0 K 0 [23 Eq. 10.32.10] . Passing from Y Y Y χ Y \chi\sqrt{Y} χ Y
The two families meet at different parameter endpoints: as the critical parameter τ \tau τ θ = ∞ \theta=\infty θ = ∞
Corollary 4.23 (The lower endpoint of the critical family). For every q ∈ ( 0 , 1 ) q\in(0,1) q ∈ ( 0 , 1 )
∫ R ∣ x ∣ 2 q μ τ ( d x ) ≤ Γ ( q ) 2 τ 1 − q , τ > 0. \int_{\mathbb{R}} |x|^{2q}\,\mu_\tau(\mathrm{d}x)\leq\Gamma(q)^2\tau^{1-q},\qquad\tau>0. ∫ R ∣ x ∣ 2 q μ τ ( d x ) ≤ Γ ( q ) 2 τ 1 − q , τ > 0. Consequently,
μ τ ⟹ δ 0 ( τ ↓ 0 ) . \mu_\tau\Longrightarrow\delta_0\qquad(\tau\downarrow0). μ τ ⟹ δ 0 ( τ ↓ 0 ) . Moreover, if
Π τ : = ( τ ⋅ ) # [ x 2 μ τ ( d x ) ] , \Pi_\tau:=(\sqrt{\tau}\mathbin{\cdot})_{\#}[x^2\mu_\tau(\mathrm{d}x)], Π τ := ( τ ⋅ ) # [ x 2 μ τ ( d x )] , then
Π τ ⟹ π ∞ ( τ ↓ 0 ) , \Pi_\tau\Longrightarrow\pi_\infty\qquad(\tau\downarrow0), Π τ ⟹ π ∞ ( τ ↓ 0 ) , and equivalently
τ − 1 ( τ ⋅ ) # μ τ → v Λ ∞ \tau^{-1}(\sqrt{\tau}\mathbin{\cdot})_{\#}\mu_\tau\mathrel{\xrightarrow{\mathrm{v}}}\Lambda_\infty τ − 1 ( τ ⋅ ) # μ τ v Λ ∞ vaguely on R ∗ \mathbb{R}^{*} R ∗
Proof. For fixed τ > 0 \tau> 0 τ > 0 d α d 2 → τ d\alpha_d^2 \to\tau d α d 2 → τ d α d → ∞ d\alpha_d \to\infty d α d → ∞ Theorem 4.16 gives
lim sup d → ∞ ∫ ∣ x ∣ 2 q ( d ⋅ ) # μ d , α d ( d x ) ≤ Γ ( q ) 2 τ 1 − q . \limsup_{d\to\infty} \int|x|^{2q}(\sqrt{d}\cdot)_{\#}\mu_{d,\alpha_d}(\mathrm{d}x) \le\Gamma(q)^2\tau^{1-q}. d → ∞ lim sup ∫ ∣ x ∣ 2 q ( d ⋅ ) # μ d , α d ( d x ) ≤ Γ ( q ) 2 τ 1 − q . Theorem 4.5 and the Portmanteau theorem, applied to the nonnegative lower semicontinuous function ∣ x ∣ 2 q |x|^{2q} ∣ x ∣ 2 q μ τ \mu_\tau μ τ ε > 0 \varepsilon> 0 ε > 0
μ τ ( ∣ x ∣ > ε ) ≤ ε − 2 q Γ ( q ) 2 τ 1 − q ⟶ 0 , \mu_\tau(|x| > \varepsilon) \le\varepsilon^{-2q}\Gamma(q)^2\tau^{1-q} \longrightarrow0, μ τ ( ∣ x ∣ > ε ) ≤ ε − 2 q Γ ( q ) 2 τ 1 − q ⟶ 0 , which proves weak convergence to δ 0 \delta_0 δ 0
For the energy-biased law, note first that Π τ \Pi_\tau Π τ M 2 ( τ ) = 1 M_2(\tau)=1 M 2 ( τ ) = 1
∫ x 2 k Π τ ( d x ) = τ k M 2 k + 2 ( τ ) . \int x^{2k}\Pi_\tau(\mathrm{d}x)=\tau^k M_{2k+2}(\tau). ∫ x 2 k Π τ ( d x ) = τ k M 2 k + 2 ( τ ) . In the closed-walk formula for M 2 k + 2 ( τ ) M_{2k+2}(\tau) M 2 k + 2 ( τ ) ( k ! ) 2 τ − k (k!)^2\tau^{-k} ( k ! ) 2 τ − k τ k \tau^k τ k ( k ! ) 2 (k!)^2 ( k ! ) 2 π ∞ \pi_\infty π ∞
∫ x 2 Π τ ( d x ) = τ M 4 ( τ ) = 1 + 2 τ , \int x^2\Pi_\tau(\mathrm{d}x)=\tau M_4(\tau)=1+2\tau, ∫ x 2 Π τ ( d x ) = τ M 4 ( τ ) = 1 + 2 τ , so the family is tight as τ ↓ 0 \tau\downarrow0 τ ↓ 0 k k k
τ k + 1 M 2 k + 4 ( τ ) = ∑ γ ∈ W k + 2 c γ τ e ( γ ) − 1 , c γ = ∏ e ∈ E ( H γ ) ( r e ( γ ) − 1 ) ! 2 ≥ 0 , \tau^{k+1}M_{2k+4}(\tau)=\sum_{\gamma\in\mathcal{W}_{k+2}}c_\gamma\tau^{e(\gamma)-1},\qquad c_\gamma=\prod_{e\in E(H_\gamma)}(r_e(\gamma)-1)!^2\ge0, τ k + 1 M 2 k + 4 ( τ ) = γ ∈ W k + 2 ∑ c γ τ e ( γ ) − 1 , c γ = e ∈ E ( H γ ) ∏ ( r e ( γ ) − 1 ) ! 2 ≥ 0 , and is bounded on ( 0 , 1 ] (0,1] ( 0 , 1 ] π ∞ \pi_\infty π ∞ Π τ ⇒ π ∞ \Pi_\tau\Rightarrow\pi_\infty Π τ ⇒ π ∞
Finally, for f ∈ C c ( R ∗ ) f\in C_c(\mathbb{R}^*) f ∈ C c ( R ∗ ) f ( x ) / x 2 f(x)/x^2 f ( x ) / x 2
Π τ ( d x ) = x 2 τ − 1 ( τ ⋅ ) # μ τ ( d x ) \Pi_\tau(\mathrm{d}x)=x^2\tau^{-1}(\sqrt{\tau}\cdot)_{\#}\mu_\tau(\mathrm{d}x) Π τ ( d x ) = x 2 τ − 1 ( τ ⋅ ) # μ τ ( d x ) then gives the asserted vague convergence.
Uniform approximation across the phase diagram
The three regimes combine into a single approximation, uniform over all positive Dirichlet parameters. The approximating law is indexed by the current value of d α 2 d\alpha^2 d α 2
Proposition 4.24 (Uniform Wasserstein approximation). The map Φ : R + → P 1 ( R ) \Phi:\mathbb{R}_+ \to\mathcal{P}_1(\mathbb{R}) Φ : R + → P 1 ( R ) is continuous and injective, where P 1 ( R ) \mathcal{P}_1(\mathbb{R}) P 1 ( R ) denotes the probability measures with finite first moment, equipped with W 1 W_1 W 1 Moreover,
lim sup d → ∞ α > 0 W 1 ( μ ^ d , α , Φ ( d α 2 ) ) = 0 , (5) \limsup_{\substack{d\to\infty\\ \alpha>0}}W_1\left(\widehat{\mu}_{d,\alpha},\Phi(d\alpha^2)\right)=0, \tag*{(5)} d → ∞ α > 0 lim sup W 1 ( μ d , α , Φ ( d α 2 ) ) = 0 , (5) lim sup d → ∞ α > 0 W 1 ( η ^ d , α , Φ ( d α 2 ) ) = 0. (6) \limsup_{\substack{d\to\infty\\ \alpha>0}}W_1\left(\widehat{\eta}_{d,\alpha},\Phi(d\alpha^2)\right)=0. \tag*{(6)} d → ∞ α > 0 lim sup W 1 ( η d , α , Φ ( d α 2 ) ) = 0. (6) Proof. Suppose τ j → τ ∈ ( 0 , ∞ ) \tau_j \to\tau\in(0,\infty) τ j → τ ∈ ( 0 , ∞ ) τ \tau τ τ − 1 \tau^{-1} τ − 1 Theorem 4.5 therefore gives μ τ j ⇒ μ τ \mu_{\tau_j} \Rightarrow\mu_\tau μ τ j ⇒ μ τ W 1 W_1 W 1 W 1 ( μ τ , μ s c ) → 0 W_1(\mu_\tau,\mu_{\mathrm{sc}}) \to0 W 1 ( μ τ , μ sc ) → 0 q = 1 / 2 q=1/2 q = 1/2
W 1 ( μ τ , δ 0 ) = ∫ ∣ x ∣ μ τ ( d x ) ≤ π τ . W_1(\mu_\tau,\delta_0)=\int|x|\,\mu_\tau(\mathrm{d}x)\leq\pi\sqrt{\tau}. W 1 ( μ τ , δ 0 ) = ∫ ∣ x ∣ μ τ ( d x ) ≤ π τ . This proves continuity on R ‾ + \overline{\mathbb{R}}_+ R +
Suppose (5) fails. Then there exist ε > 0 \varepsilon>0 ε > 0 d j → ∞ d_j\to\infty d j → ∞ α j > 0 \alpha_j>0 α j > 0 ε \varepsilon ε d j α j 2 → τ ∈ R ‾ + d_j\alpha_j^2\to\tau\in\overline{\mathbb{R}}_+ d j α j 2 → τ ∈ R + μ ^ d j , α j → Φ ( τ ) \widehat{\mu}_{d_j,\alpha_j}\to\Phi(\tau) μ d j , α j → Φ ( τ ) W 1 W_1 W 1 Φ ( d j α j 2 ) → Φ ( τ ) \Phi(d_j\alpha_j^2)\to\Phi(\tau) Φ ( d j α j 2 ) → Φ ( τ ) W 1 W_1 W 1 (4) imply (6) .
Conclusions and further directions
We have studied a sparse self-adjoint random operator in which independent Dirichlet vectors on vertex stars couple the incident weights. The operator is the discriminant of a quenched reflection quantum walk, giving an exact spectral relation between the two. At fixed degree, empirical spectral measures along locally tree-like graph sequences converge almost surely to a deterministic tree law with an all-order closed-walk formula. Rare paths give spectrum and essential spectrum [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
For expected root spectral laws on the infinite tree, variance-scale convergence is equivalent to convergence of d α d 2 d\alpha_d^2 d α d 2 [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] W 1 W_1 W 1 α > 0 \alpha>0 α > 0 d α d d\alpha_d d α d ± 1 \pm1 ± 1
These limits lead to further questions about the underlying operators. A simultaneous finite-graph limit with n , d → ∞ n,d\to\infty n , d → ∞ Theorem 4.10 also motivates studying convergence of the rescaled edge-weight point processes and their associated operators or cavity maps. At finite α \alpha α
Finite-degree cavity recursion and its analytic fixed point
Removing an edge from a tree separates its two branches. This gives a recursion for the Green function, but the Dirichlet constraint creates a dependence that must remain in the recursion. We record it by weighting the branch Green function with the mass assigned to the deleted edge.
We use the Herglotz resolvent convention
R H ( z ) = ( H − z ) − 1 , z ∈ C + . R_H(z) = (H-z)^{-1}, \qquad z \in\mathbb{C}_+. R H ( z ) = ( H − z ) − 1 , z ∈ C + . With this convention, diagonal resolvent entries have positive imaginary part. The finite-degree recursion below is the bounded-operator counterpart of the critical Poisson recursion in Theorem 4.13 .
We work in the positive gauge on the tree, so that
a x y = X x , y X y , x . a_{xy} = \sqrt{X_{x,y}X_{y,x}}. a x y = X x , y X y , x . For an oriented edge x → p x \to p x → p { x , p } \{x,p\} { x , p } T x → p \mathbb{T}_{x\to p} T x → p x x x T x → p T_{x\to p} T x → p T ∞ T_\infty T ∞ ℓ 2 ( T x → p ) \ell^2(\mathbb{T}_{x\to p}) ℓ 2 ( T x → p )
Γ x → p ( z ) = ⟨ δ x , ( T x → p − z ) − 1 δ x ⟩ . \Gamma_{x\to p}(z) = \left\langle\delta_x, (T_{x\to p}-z)^{-1}\delta_x \right\rangle. Γ x → p ( z ) = ⟨ δ x , ( T x → p − z ) − 1 δ x ⟩ . The coordinate X x , p X_{x,p} X x , p x x x p p p
Lemma A.1 (Herglotz convention and resolvent bounds). For every oriented edge x → p x \to p x → p z ∈ C + z \in\mathbb{C}_+ z ∈ C +
Γ x → p ( z ) ∈ C + , ∣ Γ x → p ( z ) ∣ ≤ 1 Im z . \Gamma_{x\to p}(z) \in\mathbb{C}_+, \qquad\left|\Gamma_{x\to p}(z)\right| \le\frac{1}{\operatorname{Im} z}. Γ x → p ( z ) ∈ C + , ∣ Γ x → p ( z ) ∣ ≤ Im z 1 . Moreover,
Im Γ x → p ( z ) = ( Im z ) ∥ ( T x → p − z ) − 1 δ x ∥ 2 , \operatorname{Im}\Gamma_{x\to p}(z) = (\operatorname{Im}z)\left\|(T_{x\to p}-z)^{-1}\delta_x\right\|^2, Im Γ x → p ( z ) = ( Im z ) ( T x → p − z ) − 1 δ x 2 , and
− i y Γ x → p ( i y ) ⟶ 1 ( y → ∞ ) . -\mathrm{i}y\Gamma_{x\to p}(\mathrm{i}y) \longrightarrow1 \qquad(y \to\infty). − i y Γ x → p ( i y ) ⟶ 1 ( y → ∞ ) . Proof. The resolvent identity and self-adjointness give
( T x → p − z ) − 1 − ( T x → p − z ‾ ) − 1 2 i = ( Im z ) ( T x → p − z ) − 1 ( T x → p − z ‾ ) − 1 . \frac{(T_{x\to p}-z)^{-1}-(T_{x\to p}-\overline{z})^{-1}}{2\mathrm{i}}
=(\operatorname{Im}z)(T_{x\to p}-z)^{-1}(T_{x\to p}-\overline{z})^{-1}. 2 i ( T x → p − z ) − 1 − ( T x → p − z ) − 1 = ( Im z ) ( T x → p − z ) − 1 ( T x → p − z ) − 1 . Taking the matrix element at δ x \delta_x δ x ∥ ( T x → p − z ) − 1 ∥ ≤ ( Im z ) − 1 \left\|(T_{x\to p}-z)^{-1}\right\| \le(\operatorname{Im}z)^{-1} ( T x → p − z ) − 1 ≤ ( Im z ) − 1
( T x → p − i y ) − 1 = − 1 i y ( I − T x → p i y ) − 1 , (T_{x\to p}-\mathrm{i}y)^{-1}=-\frac{1}{\mathrm{i}y}\left(I-\frac{T_{x\to p}}{\mathrm{i}y}\right)^{-1}, ( T x → p − i y ) − 1 = − i y 1 ( I − i y T x → p ) − 1 , and ∥ T x → p ∥ ≤ 1 \left\|T_{x\to p}\right\| \le1 ∥ T x → p ∥ ≤ 1
The masses assigned to the child directions sum to 1 − X x , p 1-X_{x,p} 1 − X x , p Γ x → p \Gamma_{x\to p} Γ x → p
m x → p ( z ) = X x , p Γ x → p ( z ) . \mathbf{m}_{x\to p}(z)=X_{x,p}\Gamma_{x\to p}(z). m x → p ( z ) = X x , p Γ x → p ( z ) . Since X x , p > 0 X_{x,p}>0 X x , p > 0 α > 0 \alpha>0 α > 0
m x → p ( z ) ∈ C + , ∣ m x → p ( z ) ∣ ≤ 1 Im z \mathbf{m}_{x\to p}(z)\in\mathbb{C}_{+},\qquad\lvert\mathbf{m}_{x\to p}(z)\rvert\leq\frac{1}{\operatorname{Im}z} m x → p ( z ) ∈ C + , ∣ m x → p ( z )∣ ≤ Im z 1 and
− i y m x → p ( i y ) ⟶ X x , p ( y → ∞ ) . -\mathrm{i}y\mathbf{m}_{x\to p}(\mathrm{i}y)\longrightarrow X_{x,p}\qquad(y\to\infty). − i y m x → p ( i y ) ⟶ X x , p ( y → ∞ ) . Proposition A.2 (Closed weighted-message recursion). Let m \mathbf{m} m have the law of the canonical weighted message m x → p \mathbf{m}_{x\to p} m x → p and let
( X 0 , X 1 , … , X d − 1 ) ∼ Dirichlet ( α , … , α ) , (X_0,X_1,\ldots,X_{d-1})\sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha), ( X 0 , X 1 , … , X d − 1 ) ∼ Dirichlet ( α , … , α ) , where X 0 X_0 X 0 x x x X 1 , … , X d − 1 X_1,\ldots,X_{d-1} X 1 , … , X d − 1 m 1 , … , m d − 1 \mathbf{m}_1,\ldots,\mathbf{m}_{d-1} m 1 , … , m d − 1 m \mathbf{m} m
m ( z ) = d − X 0 z + ∑ j = 1 d − 1 X j m j ( z ) , z ∈ C + . \mathbf{m}(z)\mathrel{\overset{\mathrm{d}}{=}}-\frac{X_0}{z+\sum_{j=1}^{d-1}X_j\mathbf{m}_j(z)},\qquad z\in\mathbb{C}_{+}. m ( z ) = d − z + ∑ j = 1 d − 1 X j m j ( z ) X 0 , z ∈ C + . For the root formula, take d d d m 1 , … , m d \mathbf{m}_1,\ldots,\mathbf{m}_d m 1 , … , m d m \mathbf{m} m ( X 1 , … , X d ) ∼ Dirichlet ( α , … , α ) (X_1,\ldots,X_d)\sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha) ( X 1 , … , X d ) ∼ Dirichlet ( α , … , α )
G o ( z ) : = ⟨ δ o , ( T ∞ − z ) − 1 δ o ⟩ = d − 1 z + ∑ j = 1 d X j m j ( z ) . G_o(z):=\langle\delta_o,(T_\infty-z)^{-1}\delta_o\rangle\mathrel{\overset{\mathrm{d}}{=}}-\frac{1}{z+\sum_{j=1}^{d}X_j\mathbf{m}_j(z)}. G o ( z ) := ⟨ δ o , ( T ∞ − z ) − 1 δ o ⟩ = d − z + ∑ j = 1 d X j m j ( z ) 1 . Consequently,
s d , α ( z ) : = ∫ R μ d , α ( d x ) x − z = E G o ( z ) . s_{d,\alpha}(z):=\int_{\mathbb{R}}\frac{\mu_{d,\alpha}(\mathrm{d}x)}{x-z}=\mathbb{E}G_o(z). s d , α ( z ) := ∫ R x − z μ d , α ( d x ) = E G o ( z ) . Proof. Let y 1 , … , y d − 1 y_1,\ldots,y_{d-1} y 1 , … , y d − 1 x x x T x → p \mathcal{T}_{x\to p} T x → p
ℓ 2 ( T x → p ) = C δ x ⊕ ⨁ j = 1 d − 1 ℓ 2 ( T y j → x ) , \ell^2(\mathcal{T}_{x\to p})=\mathbb{C}\delta_x\oplus\bigoplus_{j=1}^{d-1}\ell^2(\mathcal{T}_{y_j\to x}), ℓ 2 ( T x → p ) = C δ x ⊕ j = 1 ⨁ d − 1 ℓ 2 ( T y j → x ) , the operator T x → p − z T_{x\to p}-z T x → p − z
( − z B ∗ B ⨁ j = 1 d − 1 ( T y j → x − z ) ) , \begin{pmatrix}
-z & B^* \\
B & \bigoplus_{j=1}^{d-1}(T_{y_j\to x}-z)
\end{pmatrix}, ( − z B B ∗ ⨁ j = 1 d − 1 ( T y j → x − z ) ) , where
B δ x = ⨁ j = 1 d − 1 a x , y j δ y j . B\delta_x = \bigoplus_{j=1}^{d-1} a_{x,y_j}\delta_{y_j}. B δ x = j = 1 ⨁ d − 1 a x , y j δ y j . The lower-right block is invertible for z ∈ C + z \in\mathbb{C}_+ z ∈ C +
Γ x → p ( z ) = − [ z + ∑ j = 1 d − 1 a x , y j 2 Γ y j → x ( z ) ] − 1 . \Gamma_{x\to p}(z) = -\left[z+\sum_{j=1}^{d-1}a_{x,y_j}^{2}\Gamma_{y_j\to x}(z)\right]^{-1}. Γ x → p ( z ) = − [ z + j = 1 ∑ d − 1 a x , y j 2 Γ y j → x ( z ) ] − 1 . Since
a x , y j 2 Γ y j → x ( z ) = X x , y j m y j → x ( z ) , a_{x,y_j}^{2}\Gamma_{y_j\to x}(z)=X_{x,y_j}m_{y_j\to x}(z), a x , y j 2 Γ y j → x ( z ) = X x , y j m y j → x ( z ) , multiplying by X x , p X_{x,p} X x , p x x x d d d ρ o \rho_o ρ o
G o ( z ) = ∫ R ρ o ( d x ) x − z , μ d , α = E ρ o . G_o(z)=\int_{\mathbb{R}}\frac{\rho_o(dx)}{x-z},\qquad\mu_{d,\alpha}=\mathbb{E}\rho_o. G o ( z ) = ∫ R x − z ρ o ( d x ) , μ d , α = E ρ o . The resolvent bound justifies interchanging expectation and the spectral integral, giving the final identity. □
Remark A.3 (Dirichlet neutrality). Since X 0 < 1 X_0<1 X 0 < 1
Y = X 0 , Z j = X j 1 − X 0 , 1 ≤ j ≤ d − 1. Y=X_0,\qquad Z_j=\frac{X_j}{1-X_0},\qquad1\leq j\leq d-1. Y = X 0 , Z j = 1 − X 0 X j , 1 ≤ j ≤ d − 1. Then
Y ∼ Beta ( α , ( d − 1 ) α ) , ( Z 1 , … , Z d − 1 ) ∼ Dirichlet ( α , … , α ) , Y\sim\operatorname{Beta}(\alpha,(d-1)\alpha),\qquad(Z_1,\ldots,Z_{d-1})\sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha), Y ∼ Beta ( α , ( d − 1 ) α ) , ( Z 1 , … , Z d − 1 ) ∼ Dirichlet ( α , … , α ) , and Y Y Y Z Z Z
m ( z ) = d − Y z + ( 1 − Y ) ∑ j = 1 d − 1 Z j m j ( z ) . m(z)\mathrel{\overset{\mathrm{d}}{=}}-\frac{Y}{z+(1-Y)\sum_{j=1}^{d-1}Z_jm_j(z)}. m ( z ) = d − z + ( 1 − Y ) ∑ j = 1 d − 1 Z j m j ( z ) Y . We now formulate the recursion as an equation for the probability law of a random analytic function. Recall the standard Borel space H \mathcal{H} H (2) . For a probability law Q Q Q H \mathcal{H} H F d , α ( Q ) \mathcal{F}_{d,\alpha}(Q) F d , α ( Q )
Φ X ( h 1 , … , h d − 1 ) ( z ) = − X 0 z + ∑ j = 1 d − 1 X j h j ( z ) . \Phi_X(h_1,\ldots,h_{d-1})(z)=-\frac{X_0}{z+\sum_{j=1}^{d-1}X_jh_j(z)}. Φ X ( h 1 , … , h d − 1 ) ( z ) = − z + ∑ j = 1 d − 1 X j h j ( z ) X 0 . where X = ( X 0 , … , X d − 1 ) ∼ Dirichlet ( α , … , α ) X=(X_0,\ldots,X_{d-1})\sim\operatorname{Dirichlet}(\alpha,\ldots,\alpha) X = ( X 0 , … , X d − 1 ) ∼ Dirichlet ( α , … , α ) h 1 , … , h d − 1 h_1,\ldots,h_{d-1} h 1 , … , h d − 1 Q Q Q X X X
( X , h 1 , … , h d − 1 ) ⟼ Φ X ( h 1 , … , h d − 1 ) (X,h_1,\ldots,h_{d-1})\longmapsto\Phi_X(h_1,\ldots,h_{d-1}) ( X , h 1 , … , h d − 1 ) ⟼ Φ X ( h 1 , … , h d − 1 ) is Borel measurable in the compact-open topology. The denominator has positive imaginary part, so Φ X ( h 1 , … , h d − 1 ) ∈ H \Phi_X(h_1,\ldots,h_{d-1})\in\mathcal{H} Φ X ( h 1 , … , h d − 1 ) ∈ H
Theorem A.4 (Unique analytic cavity fixed point). The law Q d , α Q_{d,\alpha} Q d , α of the canonical weighted message m x → p \mathbf{m}_{x\to p} m x → p is a fixed point of F d , α \mathcal{F}_{d,\alpha} F d , α It is the unique probability law on H \mathcal{H} H satisfying
Q = F d , α ( Q ) . Q=\mathcal{F}_{d,\alpha}(Q). Q = F d , α ( Q ) . Proof. Existence follows from Proposition A.2. For uniqueness, set
κ d , α = E [ X 0 ( 1 − X 0 ) ] = α ( d − 1 ) d ( d α + 1 ) . \kappa_{d,\alpha}=\mathbb{E}[X_0(1-X_0)]=\frac{\alpha(d-1)}{d(d\alpha+1)}. κ d , α = E [ X 0 ( 1 − X 0 )] = d ( d α + 1 ) α ( d − 1 ) . Let Q Q Q Q ′ Q' Q ′ z 1 , … , z r ∈ C + z_1,\ldots,z_r\in\mathbb{C}_+ z 1 , … , z r ∈ C +
η ℓ : = Im z ℓ > κ d , α ( 1 ≤ ℓ ≤ r ) . \eta_\ell:=\operatorname{Im}z_\ell>\sqrt{\kappa_{d,\alpha}}\qquad(1\leq\ell\leq r). η ℓ := Im z ℓ > κ d , α ( 1 ≤ ℓ ≤ r ) . Let ν \nu ν ν ′ \nu' ν ′ Q Q Q Q ′ Q' Q ′ z 1 , … , z r z_1,\ldots,z_r z 1 , … , z r C r \mathbb{C}^r C r
c ( u , v ) = ∑ ℓ = 1 r ∣ u ℓ − v ℓ ∣ . c(u,v)=\sum_{\ell=1}^{r}|u_\ell-v_\ell|. c ( u , v ) = ℓ = 1 ∑ r ∣ u ℓ − v ℓ ∣. and let W 1 W_1 W 1 W 1 W_1 W 1 ν \nu ν ν ′ \nu' ν ′ d − 1 d-1 d − 1 D ℓ D_\ell D ℓ D ℓ ′ D'_\ell D ℓ ′
∣ D ℓ ∣ , ∣ D ℓ ′ ∣ ≥ η ℓ |D_\ell|,|D'_\ell|\geq\eta_\ell ∣ D ℓ ∣ , ∣ D ℓ ′ ∣ ≥ η ℓ and therefore
∣ − X 0 D ℓ + X 0 D ℓ ′ ∣ ≤ X 0 η ℓ 2 ∑ j = 1 d − 1 X j ∣ h j ( z ℓ ) − h j ′ ( z ℓ ) ∣ . \left|-\frac{X_0}{D_\ell}+\frac{X_0}{D'_\ell}\right|
\leq\frac{X_0}{\eta_\ell^2}\sum_{j=1}^{d-1}X_j|h_j(z_\ell)-h'_j(z_\ell)|. − D ℓ X 0 + D ℓ ′ X 0 ≤ η ℓ 2 X 0 j = 1 ∑ d − 1 X j ∣ h j ( z ℓ ) − h j ′ ( z ℓ ) ∣. Taking expectations and using independence gives
E ∣ − X 0 D ℓ + X 0 D ℓ ′ ∣ ≤ κ d , α η ℓ 2 E ∣ h ( z ℓ ) − h ′ ( z ℓ ) ∣ . \mathbb{E}\left|-\frac{X_0}{D_\ell}+\frac{X_0}{D'_\ell}\right|
\leq\frac{\kappa_{d,\alpha}}{\eta_\ell^2}\mathbb{E}|h(z_\ell)-h'(z_\ell)|. E − D ℓ X 0 + D ℓ ′ X 0 ≤ η ℓ 2 κ d , α E ∣ h ( z ℓ ) − h ′ ( z ℓ ) ∣. Consequently, if F z \mathcal{F}_{\mathbf{z}} F z
W 1 ( F z ( ν ) , F z ( ν ′ ) ) ≤ q W 1 ( ν , ν ′ ) , q = max 1 ≤ ℓ ≤ r κ d , α η ℓ 2 < 1. W_1(\mathcal{F}_{\mathbf{z}}(\nu),\mathcal{F}_{\mathbf{z}}(\nu'))\leq qW_1(\nu,\nu'),\qquad q=\max_{1\leq\ell\leq r}\frac{\kappa_{d,\alpha}}{\eta_\ell^2}<1. W 1 ( F z ( ν ) , F z ( ν ′ )) ≤ q W 1 ( ν , ν ′ ) , q = 1 ≤ ℓ ≤ r max η ℓ 2 κ d , α < 1. Since Q Q Q Q ′ Q' Q ′ ν \nu ν ν ′ \nu' ν ′ F z \mathcal{F}_{\mathbf{z}} F z W 1 ( ν , ν ′ ) ≤ q W 1 ( ν , ν ′ ) W_1(\nu,\nu')\leq qW_1(\nu,\nu') W 1 ( ν , ν ′ ) ≤ q W 1 ( ν , ν ′ ) ν = ν ′ \nu=\nu' ν = ν ′
Thus all finite-dimensional evaluation laws of Q Q Q Q ′ Q' Q ′
S = { z ∈ C + : Im z > κ d , α } . S=\{z\in\mathbb{C}_+:\operatorname{Im}z>\sqrt{\kappa_{d,\alpha}}\}. S = { z ∈ C + : Im z > κ d , α } . Choose a countable dense set D 0 ⊂ S D_0\subset S D 0 ⊂ S D 0 D_0 D 0 C + \mathbb{C}_+ C + Q = Q ′ Q=Q' Q = Q ′ □ \square □
Remark A.5 (A uniform contraction half-plane). For d ≥ 3 d\geq3 d ≥ 3
κ d , α < d − 1 d 2 ≤ 2 9 < 1 4 . \kappa_{d,\alpha}<\frac{d-1}{d^2}\leq\frac{2}{9}<\frac{1}{4}. κ d , α < d 2 d − 1 ≤ 9 2 < 4 1 . Thus the uniqueness argument works uniformly in d d d α \alpha α Im z > 1 / 2 \operatorname{Im}z>1/2 Im z > 1/2
The fixed-point law can also be obtained from finite-depth branches. Starting at the leaves gives the following sequence of laws, which converges to the infinite-branch message law.
Proposition A.6 (Finite-depth construction). Let Q 0 Q_0 Q 0
z ⟼ − Y z , Y ∼ Beta ( α , ( d − 1 ) α ) , z \longmapsto-\frac{Y}{z}, \qquad Y \sim\operatorname{Beta}(\alpha,(d-1)\alpha), z ⟼ − z Y , Y ∼ Beta ( α , ( d − 1 ) α ) , and define recursively
Q R + 1 = F d , α ( Q R ) . Q_{R+1} = \mathcal{F}_{d,\alpha}(Q_R). Q R + 1 = F d , α ( Q R ) . Then Q R Q_R Q R m R \mathbf{m}_R m R R R R
Q R ⟹ Q d , α Q_R \Longrightarrow Q_{d,\alpha} Q R ⟹ Q d , α in the topology of locally uniform convergence on C + \mathbb{C}_+ C +
Proof. The same block Schur complement gives the finite-depth recursion, with leaf resolvent − 1 / z -1/z − 1/ z P R P_R P R R R R
T R = P R T x → p P R , T_R = P_R T_{x\to p} P_R, T R = P R T x → p P R , extended by zero to the full branch space. On each finitely supported vector, T R T_R T R T x → p T_{x\to p} T x → p R R R ∥ T R ∥ ≤ 1 \lVert T_R\rVert\le1 ∥ T R ∥ ≤ 1 T R → T x → p T_R \to T_{x\to p} T R → T x → p
The cavity messages therefore converge pointwise for every z ∈ C + z \in\mathbb{C}_+ z ∈ C + C + \mathbb{C}_+ C + ∣ m R ( z ) ∣ ≤ ( Im z ) − 1 \lvert m_R(z)\rvert\le(\operatorname{Im} z)^{-1} ∣ m R ( z )∣ ≤ ( Im z ) − 1 R R R
Statements and Declarations
Funding. This work was supported by JSPS KAKENHI Grant Number 23K12507.
Competing interests. The authors have no relevant financial or non-financial interests to disclose.
Author contributions. Both authors contributed to the conception and development of the study, the mathematical analysis, and the preparation of the manuscript.
Data availability. No datasets were generated or analyzed during the current study.
Code availability. No custom code is required to obtain the results reported in this article.
