Introduction
Overview
Consider a basic problem for a system of N N N R \mathbb{R} R N → ∞ N \to\infty N → ∞
This question has two natural extensions. The first is to vary the strength of the interaction between particles. The logarithmic repulsion in the classical invariant matrix ensembles leads to the notion of a β \beta β β > 0 \beta> 0 β > 0 β = 1 , 2 , 4 \beta= 1, 2, 4 β = 1 , 2 , 4 R N \mathbb{R}^{N} R N
1 Z N , β , V exp ( − β N 2 ∑ i = 1 N V ( a i ) ) ∏ 1 ≤ i < j ≤ N ∣ a i − a j ∣ β \frac{1}{Z_{N,\beta,V}} \exp\left(-\frac{\beta N}{2}\sum_{i=1}^{N} V(a_i)\right) \prod_{1 \le i < j \le N} \lvert a_i-a_j\rvert^{\beta} Z N , β , V 1 exp ( − 2 β N i = 1 ∑ N V ( a i ) ) 1 ≤ i < j ≤ N ∏ ∣ a i − a j ∣ β where V V V Z N , β , V Z_{N,\beta,V} Z N , β , V [29 ] for a systematic account. Starting with Johansson [37 ] , a substantial theory of global fluctuations has developed; see, among many others, [24 , 15 , 57 , 4 , 41 ] . Fluctuations at mesoscopic scales have also been studied extensively, including for general and circular β \beta β [5 , 40 ] ; see also [16 ] for optimal local laws and logarithmically correlated Gaussian fluctuations.
The second extension is to consider addition: how do the global limit and the fluctuations behave when two independent random matrices are added? The classical result of Voiculescu [63 ] identifies free additive convolution as the operation governing the limiting spectra of broad classes of independent matrices, establishing a fundamental connection between random matrices and free probability. First-order free cumulants linearize this operation and provide the free-probabilistic analogue of classical cumulants; see [50 ] for a systematic account. At the fluctuation level, Mingo and Speicher developed second-order freeness in [48 ] , with Gaussian and Wishart matrices as fundamental examples. To characterize the covariances of the Gaussian fluctuation under the addition, Collins, Mingo, Śniady, and Speicher [20 ] introduced the second-order free cumulants and gave the explicit covariance–cumulant relations; see also [47 , 49 ] . In this way, second-order free probability extends the role of free convolution from limiting spectral measures to their fluctuations, and makes it possible to study the latter through quantities that are additive under matrix addition.
For β \beta β β = 0 \beta= 0 β = 0 high-temperature regime , where θ = β / 2 \theta= \beta/2 θ = β /2 N θ → γ > 0 N\theta\to\gamma> 0 N θ → γ > 0 γ \gamma γ [1 ] . At the level of fluctuations, the change is equally substantial: normalized global statistics of the classical ensembles fluctuate on the N − 1 / 2 N^{-1/2} N − 1/2 N − 1 N^{-1} N − 1 [51 , 26 , 46 ] . From the additive viewpoint, Benaych-Georges, Cuenca, and Gorin [7 ] introduced γ \gamma γ γ \gamma γ
The present paper brings together these perspectives by giving a characterization of the CLT for general continuous N N N β \beta β N N N β \beta β [30 , 7 ] . We prove that if both inputs satisfy a CLT in either regime, then their β \beta β
An important feature of this characterization is that it does not assume a log-gas density, which makes it suitable for studying operations such as β \beta β β \beta β [37 , 15 , 4 , 41 , 26 ] . Here, by working with the Bessel generating function, we obtain a criterion that applies beyond prescribed log-gas densities and is compatible with addition, while also giving a combinatorial description of the covariance through cumulants and weighted lattice paths.
From the free-probabilistic viewpoint, our starting point is symmetric functions and Łukasiewicz paths, rather than an operator-algebraic model or annular non-crossing partitions, used in the classical literature. The resulting combinatorics gives a different description of the same second-order structure at fixed temperature: we show that our covariance formula agrees, up to the factor 1 / θ 1/\theta 1/ θ [20 ] . Its usefulness becomes more apparent at high temperature, where it provides new second-order free γ \gamma γ [7 ] . While a high-temperature analogue is not immediate from the classical descriptions, it arises naturally from the same symmetric-function and lattice-path formulation.
These results also connect free cumulants with the idea of topological expansions . [13 , 12 ] interpreted the first- and second-order free moment–cumulant relations through ordinary and fully simple maps, relating their generating series by monotone Hurwitz numbers, which count the number of branched coverings of orientable closed surfaces. In a different direction, [32 ] obtained an expansion of the HCIZ integral in terms of monotone Hurwitz numbers. This is directly related to our setting: the multivariate Bessel function we use in this text specializes to the HCIZ integral at θ = 1 \theta=1 θ = 1 θ > 0 \theta>0 θ > 0 [22 ] , gives a symmetric-function description of free cumulants through the Bessel topological expansion, derived from a combinatorial expansion of ribbon graphs on surfaces in [18 ] . It also reveals a distinction between the two temperature regimes: genus zero governs the fixed-temperature limits, whereas contributions of arbitrary genus enter at high temperature. We discuss this perspective and the role of the expansion in § 2.2 .
Main results
The main object of this work has a natural representation-theoretic origin. For θ = 1 / 2 , 1 , 2 \theta=1/2,1,2 θ = 1/2 , 1 , 2 G θ ( N ) G_\theta(N) G θ ( N ) O ( N ) O(N) O ( N ) U ( N ) U(N) U ( N ) S p ( N ) Sp(N) S p ( N ) N N N a a a x ⃗ \vec{x} x A = diag ( a ) A=\operatorname{diag}(a) A = diag ( a ) X = diag ( x ⃗ ) X=\operatorname{diag}(\vec{x}) X = diag ( x )
B a ( x ⃗ ; θ ) = ∫ G θ ( N ) exp ( tr ( A U X U ∗ ) ) d U . B_a(\vec{x};\theta)=\int_{G_\theta(N)}\exp(\operatorname{tr}(AUXU^*))\,\mathrm{d}U. B a ( x ; θ ) = ∫ G θ ( N ) exp ( tr ( A U X U ∗ )) d U . where d U \mathrm{d}U d U 1 θ = 1 \theta=1 θ = 1 [33 , 36 ] ; the other two cases are its orthogonal and symplectic analogues. Replacing x ⃗ \vec{x} x i x ⃗ i\vec{x} i x [35 ] , and play an important role in matrix models of mathematical physics [36 ] .
The integral is symmetric separately in a a a x ⃗ \vec{x} x θ \theta θ [52 ] , the so-called Jack polynomial, which naturally extends it to every θ > 0 \theta>0 θ > 0 multivariate Bessel function B a ( x ⃗ ; θ ) B_a(\vec{x};\theta) B a ( x ; θ ) ( a , x ⃗ ) ∈ C 2 N (a,\vec{x})\in\mathbb{C}^{2N} ( a , x ) ∈ C 2 N B a ( 0 N ; θ ) = 1 B_a(0^N;\theta)=1 B a ( 0 N ; θ ) = 1 [56 , 2 ] and § 5.2 below.
These functions provide a powerful characterization of a class of continuous particle systems in integrable probability, notably for the β \beta β [21 ] and β \beta β [7 ] . The Bessel-function approach has also been used to establish global scaling limits [7 , 64 , 65 , 22 ] and local edge limits [38 , 31 ] . Their large-N N N [54 ] and its extension to the orbital beta process [3 ] .
We consider the following class of N N N N N N μ N \mu_N μ N N N N R N \mathbb{R}^N R N
⟨ μ N , 1 ⟩ = N ! . \langle\mu_N,1\rangle=N!. ⟨ μ N , 1 ⟩ = N ! . In the first case, E μ N \mathbb{E}_{\mu_N} E μ N
E μ N [ f ] : = 1 N ! ⟨ μ N , f ⟩ . \mathbb{E}_{\mu_N}[f]:=\frac{1}{N!}\langle\mu_N,f\rangle. E μ N [ f ] := N ! 1 ⟨ μ N , f ⟩ . Thus E μ N [ 1 ] = 1 \mathbb{E}_{\mu_N}[1]=1 E μ N [ 1 ] = 1 μ N \mu_N μ N Bessel generating function as
G N , θ ( x ⃗ ) : = G θ ( x ⃗ ; μ N ) : = E μ N [ B a ( x ⃗ ; θ ) ] . G_{N,\theta}(\vec{x}):=G_\theta(\vec{x};\mu_N):=\mathbb{E}_{\mu_N}[B_a(\vec{x};\theta)]. G N , θ ( x ) := G θ ( x ; μ N ) := E μ N [ B a ( x ; θ )] . The BGF is a natural analog of the characteristic function/Laplace transform for the N N N R N \mathbb{R}^N R N μ N \mu_N μ N exp ( t x ) \exp(tx) exp ( t x ) d d t \frac{d}{dt} d t d § 5.1 . Moreover, there is a collection of additive quantities of a random variable, namely the (classical) cumulants, which are bijective to the moments and are encoded by the Taylor coefficients of the logarithm of its Laplace transform. This motivates our study of the logarithmic BGF. Since G N , θ G_{N,\theta} G N , θ 0 N 0^N 0 N G N , θ ( 0 N ) = 1 G_{N,\theta}(0^N)=1 G N , θ ( 0 N ) = 1 0 N 0^N 0 N
Definition 1.1. For any partition λ \lambda λ ∣ λ ∣ ≤ N |\lambda|\leq N ∣ λ ∣ ≤ N c λ = c λ ( N , θ ) c_\lambda=c_\lambda(N,\theta) c λ = c λ ( N , θ ) p λ ( x ⃗ ) p_\lambda(\vec{x}) p λ ( x ) log G N , θ ( x ⃗ ) \log G_{N,\theta}(\vec{x}) log G N , θ ( x ) x ⃗ = 0 N \vec{x}=0^N x = 0 N
log G N , θ ( x ⃗ ) = ∑ ∣ λ ∣ ≤ N c λ p λ ( x ⃗ ) + O ( ∥ x ∥ N + 1 ) . \log G_{N,\theta}(\vec{x})=\sum_{|\lambda|\leq N}c_\lambda p_\lambda(\vec{x})+O(\lVert x\rVert^{N+1}). log G N , θ ( x ) = ∣ λ ∣ ≤ N ∑ c λ p λ ( x ) + O (∥ x ∥ N + 1 ) . For the coordinate tuple a = ( a 1 , … , a N ) a=(a_1,\ldots,a_N) a = ( a 1 , … , a N )
p k N : = 1 N ∑ i = 1 N a i k , p ~ k N : = 1 N ∑ i = 1 N ( a i N ) k . p_k^N:=\frac{1}{N}\sum_{i=1}^N a_i^k,\qquad\widetilde{p}_k^N:=\frac{1}{N}\sum_{i=1}^N\left(\frac{a_i}{N}\right)^k. p k N := N 1 i = 1 ∑ N a i k , p k N := N 1 i = 1 ∑ N ( N a i ) k . The second normalization is the natural one when the particles themselves have size N N N
All expectations below mean normalized moment pairings with μ N \mu_N μ N
Cov μ N ( f , g ) = E μ N [ f g ] − E μ N [ f ] E μ N [ g ] . \operatorname{Cov}_{\mu_N}(f,g)=\mathbb{E}_{\mu_N}[fg]-\mathbb{E}_{\mu_N}[f]\mathbb{E}_{\mu_N}[g]. Cov μ N ( f , g ) = E μ N [ f g ] − E μ N [ f ] E μ N [ g ] . No positivity is imposed on this bilinear form.
Fixed temperature.
Definition 1.2 (Fixed-temperature CLT-satisfaction). Fix θ > 0 \theta>0 θ > 0 { μ N } N ≥ 1 \{\mu_N\}_{N\geq1} { μ N } N ≥ 1 { m k } k ≥ 1 \{m_k\}_{k\geq1} { m k } k ≥ 1 { α k , l } k , l ≥ 1 \{\alpha_{k,l}\}_{k,l\geq1} { α k , l } k , l ≥ 1
For every k ≥ 1 k\geq1 k ≥ 1
lim N → ∞ E μ N [ p ~ k N ] = m k . \lim_{N\to\infty}\mathbb{E}_{\mu_N}[\widetilde{p}_k^N]=m_k. N → ∞ lim E μ N [ p k N ] = m k . For every s ≥ 1 s\geq1 s ≥ 1 k 1 , … , k s ≥ 1 k_1,\ldots,k_s\geq1 k 1 , … , k s ≥ 1
lim N → ∞ N s E μ N [ ∏ j = 1 s ( p ~ k j N − E μ N [ p ~ k j N ] ) ] = ∑ π ∏ { i , j } ∈ π α k i , k j . \lim_{N\to\infty}N^s\mathbb{E}_{\mu_N}\left[\prod_{j=1}^s\left(\widetilde{p}_{k_j}^N-\mathbb{E}_{\mu_N}[\widetilde{p}_{k_j}^N]\right)\right]=\sum_{\pi}\prod_{\{i,j\}\in\pi}\alpha_{k_i,k_j}. N → ∞ lim N s E μ N [ j = 1 ∏ s ( p k j N − E μ N [ p k j N ] ) ] = π ∑ { i , j } ∈ π ∏ α k i , k j . Here the sum is over all pair partitions π \pi π { 1 , … , s } \{1,\ldots,s\} { 1 , … , s } s s s
The two directions give the concise result below. The covariance formula appears later in Theorem 3.12 .
Theorem 1.3 (Fixed-temperature characterization). Under the standing assumptions above, fix θ > 0 \theta> 0 θ > 0 { μ N } \{\mu_N\} { μ N } Definition 1.2 if and only if there exist constants κ d \kappa_d κ d κ r , s = κ s , r \kappa_{r,s} = \kappa_{s,r} κ r , s = κ s , r d , r , s ≥ 1 d,r,s \ge1 d , r , s ≥ 1 λ \lambda λ ℓ ( λ ) ≥ 3 \ell(\lambda) \ge3 ℓ ( λ ) ≥ 3
d N c ( d ) ⟶ θ 1 − d κ d , ( 1 + 1 r = s ) r s c ( r , s ) ⟶ θ 1 − r − s κ r , s , c λ ⟶ 0. \frac{d}{N}c_{(d)} \longrightarrow\theta^{1-d}\kappa_d,\qquad(1+\mathbf{1}_{r=s})rs\,c_{(r,s)} \longrightarrow\theta^{1-r-s}\kappa_{r,s},\qquad c_\lambda\longrightarrow0. N d c ( d ) ⟶ θ 1 − d κ d , ( 1 + 1 r = s ) r s c ( r , s ) ⟶ θ 1 − r − s κ r , s , c λ ⟶ 0. The limiting means are determined by the first-order free cumulants { κ d } \{\kappa_d\} { κ d } { κ d } \{\kappa_d\} { κ d } { κ r , s } \{\kappa_{r,s}\} { κ r , s } (14) .
The first limit is the fixed-temperature LLN criterion of [22 ] . The new content at second order is that the two-row BGF coefficients are exactly the second-order free cumulants, while the decay of all longer coefficients is equivalent to the vanishing of higher classical cumulants. In particular, the condition is intrinsic to the BGF and does not require the particles to be eigenvalues of an actual random matrix.
High temperature.
Definition 1.4 (High-temperature CLT-satisfaction). Let θ = θ N → 0 \theta= \theta_N \to0 θ = θ N → 0 N θ N → γ > 0 N\theta_N \to\gamma> 0 N θ N → γ > 0 { μ N } N ≥ 1 \{\mu_N\}_{N\ge1} { μ N } N ≥ 1 { m k } k ≥ 1 \{m_k\}_{k\ge1} { m k } k ≥ 1 { α k , l } k , l ≥ 1 \{\alpha_{k,l}\}_{k,l\ge1} { α k , l } k , l ≥ 1
lim N → ∞ E μ N [ p k N ] = m k . \lim_{N\to\infty}\mathbb{E}_{\mu_N}[p_k^N] = m_k. N → ∞ lim E μ N [ p k N ] = m k . For every s ≥ 1 s \ge1 s ≥ 1 k 1 , … , k s ≥ 1 k_1,\ldots,k_s \ge1 k 1 , … , k s ≥ 1
lim N → ∞ N s / 2 E μ N [ ∏ j = 1 s ( p k j N − E μ N [ p k j N ] ) ] = ∑ π ∏ { i , j } ∈ π α k i , k j , \lim_{N\to\infty}N^{s/2}\mathbb{E}_{\mu_N}\left[\prod_{j=1}^{s}\left(p_{k_j}^N-\mathbb{E}_{\mu_N}[p_{k_j}^N]\right)\right]=\sum_{\pi}\prod_{\{i,j\}\in\pi}\alpha_{k_i,k_j}, N → ∞ lim N s /2 E μ N [ j = 1 ∏ s ( p k j N − E μ N [ p k j N ] ) ] = π ∑ { i , j } ∈ π ∏ α k i , k j , with the same pair-partition convention as in Definition 1.2 .
The different fluctuation scale is reflected exactly in the scaling of the BGF coefficients.
Theorem 1.5 (High-temperature characterization). Under the standing assumptions above, let θ = θ N → 0 \theta= \theta_N \to0 θ = θ N → 0 N θ N → γ > 0 N\theta_N \to\gamma> 0 N θ N → γ > 0 { μ N } \{\mu_N\} { μ N } κ d γ \kappa_d^\gamma κ d γ κ r , s γ = κ s , r γ \kappa_{r,s}^\gamma= \kappa_{s,r}^\gamma κ r , s γ = κ s , r γ d , r , s ≥ 1 d,r,s \ge1 d , r , s ≥ 1 λ \lambda λ ℓ ( λ ) ≥ 3 \ell(\lambda) \ge3 ℓ ( λ ) ≥ 3
d c ( d ) ⟶ κ d γ , N ( 1 + 1 r = s ) r s c ( r , s ) ⟶ κ r , s γ , N ℓ ( λ ) / 2 c λ ⟶ 0. d\,c_{(d)} \longrightarrow\kappa_d^\gamma,\qquad N(1+\mathbf{1}_{r=s})rs\,c_{(r,s)} \longrightarrow\kappa_{r,s}^\gamma,\qquad N^{\ell(\lambda)/2}c_\lambda\longrightarrow0. d c ( d ) ⟶ κ d γ , N ( 1 + 1 r = s ) r s c ( r , s ) ⟶ κ r , s γ , N ℓ ( λ ) /2 c λ ⟶ 0. The limiting means are determined by the first-order free γ \gamma γ { κ d γ } \{\kappa_d^\gamma\} { κ d γ } { κ d γ } \{\kappa_d^\gamma\} { κ d γ } γ \gamma γ { κ r , s γ } \{\kappa_{r,s}^\gamma\} { κ r , s γ } (16) .
Remark 1.6. Definitions 1.2 and 1.4 are entirely in terms of moments and do not require ( α k , l ) k , l ≥ 1 (\alpha_{k,l})_{k,l\ge1} ( α k , l ) k , l ≥ 1 α k , l \alpha_{k,l} α k , l
If each normalized N N N E μ N \mathbb{E}_{\mu_N} E μ N ( α k , l ) (\alpha_{k,l}) ( α k , l )
This theorem is the fluctuation-level counterpart of the high-temperature LLN criterion of [7 ] . The first-order free γ \gamma γ γ \gamma γ β \beta β
Theorems 1.3 and 1.5 have the same three-level architecture, but their scalings are not formal substitutions for one another. At fixed temperature the one-row coefficients have order N N N N − 1 N^{-1} N − 1 N − 1 N^{-1} N − 1 N − 1 / 2 N^{-1/2} N − 1/2
Applications
Our if-and-only-if characterizations reduce fixed- and high-temperature CLTs to the corresponding coefficient asymptotics of logarithmic BGFs. Whenever these transforms can be described explicitly, the criteria give direct access to fluctuations and their limiting covariance. We give three applications of such characterization.
First, the classical Gaussian and Laguerre β \beta β [37 , 24 ] . Applying our high-temperature criterion to these explicit transforms gives CLTs and explicit limiting covariances directly; see Section 8.1.
Second, consider N N N a ( N ) a(N) a ( N ) b ( N ) b(N) b ( N ) [7 Section 2.4] . We regard them as the eigenvalues of virtual random matrices. Their β \beta β c ( N ) = a ( N ) + θ b ( N ) c(N)=a(N)+_\theta b(N) c ( N ) = a ( N ) + θ b ( N ) β = 2 θ \beta=2\theta β = 2 θ
G N , θ c ( x ⃗ ) = G N , θ a ( x ⃗ ) G N , θ b ( x ⃗ ) . G^c_{N,\theta}(\vec{x})=G^a_{N,\theta}(\vec{x})G^b_{N,\theta}(\vec{x}). G N , θ c ( x ) = G N , θ a ( x ) G N , θ b ( x ) . When the inputs are probability measures and θ = 1 / 2 , 1 , 2 \theta=1/2,1,2 θ = 1/2 , 1 , 2 [30 , 7 ] . For general θ > 0 \theta>0 θ > 0
Third, the high-temperature theory interpolates between the classical theory at θ = 0 \theta=0 θ = 0 γ → ∞ \gamma\to\infty γ → ∞ [7 Section 8.2] . We obtain the corresponding transition at the fluctuation level: after appropriately rescaling the first- and second-order free γ \gamma γ
Beyond these applications, one can extend the Gaussian fluctuation to two-dimensional particle systems, and study the limiting Gaussian fields. For a broad class of two-dimensional random surface models, global height fluctuations are expected to be described by the Gaussian free field (GFF). This picture is supported by results for random tilings and their discrete log-gas marginals [39 , 11 , 17 , 14 ] , for the six-vertex model in certain parameter regimes [23 ] , and for anisotropic ( 2 + 1 ) (2+1) ( 2 + 1 ) [9 ] . On the random matrix side, the single-level fluctuation of log-gases at fixed-temperature can be viewed as a one-dimensional projection of a GFF, and the corners process of Wigner, Wishart matrices and the Jacobi ensembles in two dimension are also shown to be GFF [37 , 8 , 25 , 10 ] . A natural multilevel extension of a one-dimensional β \beta β β \beta β [21 ] . By [21 Proposition 4.1] , the BGF formalism is compatible with the interlacing projections of these processes, suggesting that our characterization can be extended to fixed- and high-temperature CLTs for general two-dimensional corners processes. We will pursue this direction in subsequent work.
Methodology and topological expansion
The covariance in the if direction is obtained by applying Dunkl operators to the BGF and expanding their action into weighted sums of Lukasiewicz paths with associated block functions. Such a moment-method approach using Dunkl operators was used for LLNs in [7 , 64 , 65 , 22 ] and for edge limits in [38 , 31 ] . At the fluctuation scale, the leading moment terms cancel, and the covariance is determined by contributions that are invisible to the LLN. We organize the terms into configurations and track their cancellations to isolate the surviving weights. Unlike the edge calculations, here the moment degrees remain fixed, and we evaluate these weights directly, without passing to a Brownian limit of the paths. The surviving configurations give the explicit covariance formulas in Eqns. (14) and (16) . A separate combinatorial argument in § 10 identifies the fixed-temperature formula with the second-order free-cumulant formula of Collins–Mingo–Śniady–Speicher [20 ] , up to a factor 1 / θ 1/\theta 1/ θ
The only if direction and the vanishing of higher classical cumulants in the if direction use a different tool: a topological expansion of the multivariate Bessel function. To recover the coefficients of log G N , θ \log G_{N,\theta} log G N , θ
log G N , θ ( x ⃗ ) = log E [ exp ( log B a ( x ⃗ ; θ ) ) ] \log G_{N,\theta}(\vec{x}) = \log\mathbb{E}[\exp(\log B_a(\vec{x};\theta))] log G N , θ ( x ) = log E [ exp ( log B a ( x ; θ ))] and expand the right-hand side in classical joint cumulants. The required input is an explicit expansion of log B a \log B_a log B a [22 ] from the Jack-series identities of Chapuy–Dołęga [18 ] and used there to study LLNs; we recall it in Theorem 2.8 . Here we use it to relate BGF coefficients to joint cumulants at the fluctuation scale, in both directions of the CLT characterization. It is indexed by connected constellations , a class of ribbon graphs embedded in surfaces, allowing twisted ribbons and hence non-orientable surfaces. Each term contains a weight
b ν ρ , b = θ − 1 − 1 = 2 / β − 1 , b^{\nu_\rho}, \qquad b = \theta^{-1} - 1 = 2/\beta- 1, b ν ρ , b = θ − 1 − 1 = 2/ β − 1 , where ν ρ \nu_\rho ν ρ N N N
Combining this formula with the classical cumulant expansion expresses the BGF coefficients in terms of joint cumulants of the power sums and gives the estimates needed for necessity. For sufficiency, we use the same relation in reverse. At each fixed total degree d d d
The idea of topological expansion has its root in gauge theory [34 , 19 ] , and has a long tradition in random matrix theory. For GUE and GOE, the expansion comes directly from the matrix structure: expanding traces into a polynomial of matrix entries and applying Wick’s formula turns moments into sums over polygon gluings, equivalently maps on orientable surfaces; non-orientable surfaces occur in the real case. At finite N N N N N N
The picture changes under the edge scaling, which captures the local limit of the largest eigenvalues: the powers in the moment method grow with N N N [53 ] for the edge calculation of GUE, and [58 , 59 , 60 , 28 , 61 ] for GOE and more general real symmetric matrices.
A distinct source of topological expansion is the group structure appearing in the corresponding random matrix object. For Haar-distributed compact groups, Haar integration and Weingarten calculus replace Wick pairing, see [44 , 45 ] . A symmetric-group counterpart appears in Okounkov’s study of random partitions [53 ] , where moments under the Plancherel measure are expressed as weighted sums over branched coverings of surfaces.
For general β \beta β Jack series . [43 ] related the Gaussian β \beta β § 5.4 , and [18 ] generalizes the expansion by introducing the more involved notion of constellations. In this setting, the expansion is extracted from the combinatorics of symmetric functions, without requiring a realization by concrete matrix or group structure. Recently, such perspective was used in [66 ] to study the soft edge limits of the Jack–Plancherel process and their relation to asymptotic β \beta β
All the above approaches treat β > 0 \beta>0 β > 0 β = 1 , 2 , 4 \beta=1,2,4 β = 1 , 2 , 4 θ ∼ γ / N \theta\sim\gamma/N θ ∼ γ / N b ∼ N / γ b\sim N/\gamma b ∼ N / γ N − 2 g b ν ρ N^{-2g}b^{\nu_\rho} N − 2 g b ν ρ N − 2 g + ν ρ N^{-2g+\nu_\rho} N − 2 g + ν ρ 0 ≤ ν ρ ≤ 2 g 0\le\nu_\rho\le2g 0 ≤ ν ρ ≤ 2 g ν ρ = 2 g \nu_\rho=2g ν ρ = 2 g g = 0 g=0 g = 0 γ > 0 \gamma>0 γ > 0
The paper is organized as follows. § 5 recalls the backgrounds in symmetric functions and topological expansion. Sections 3 and § 7 state and prove the covariance formulas. § 8 proves the only if direction, and § 9 completes the if direction by proving the vanishing of higher classical cumulants. § 10 gives the second-order free-probabilistic identification. § 11 presents the applications.
Notation. Throughout the paper, N N N N N N x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N x ⃗ = ( x 1 , … , x N ) \vec{x}=(x_1,\ldots,x_N) x = ( x 1 , … , x N )
∂ i : = ∂ ∂ x i , i = 1 , … , N , \partial_i:=\frac{\partial}{\partial x_i},\qquad i=1,\ldots,N, ∂ i := ∂ x i ∂ , i = 1 , … , N , and σ i , j \sigma_{i,j} σ i , j x i x_i x i x j x_j x j fixed-temperature regime means that θ > 0 \theta>0 θ > 0 N → ∞ N\to\infty N → ∞ high-temperature regime means that θ = θ N → 0 \theta=\theta_N\to0 θ = θ N → 0 N θ N → γ N\theta_N\to\gamma N θ N → γ γ > 0 \gamma>0 γ > 0 u N u_N u N N N N s ∈ Z ∪ { − ∞ } s\in\mathbb{Z}\cup\{-\infty\} s ∈ Z ∪ { − ∞ } lim sup N → ∞ N − s ∣ u N ∣ < ∞ \limsup_{N\to\infty}N^{-s}|u_N|<\infty lim sup N → ∞ N − s ∣ u N ∣ < ∞
Table 1. Sources and limiting regimes of the topological expansions discussed above. “All genera” means that the leading asymptotics are not restricted to genus zero.
Acknowledgements
We thank Maciej Dołęga, Cesar Cuenca and Vadim Gorin for helpful discussions. Some of the discussions took place during our stay in the Institut Mittag-Leffler in 2024 fall, and we are grateful to the institute and the organizers of the program “Random Matrices and Scaling Limits".
Statement of AI use
Most of the math ingredients in this text were figured out before 2025 spring, and the rest were also generated later by the human author. During the preparation of this text, we used ChatGPT 5.5, 5.6 and 6 for literature search, error detection and improving the exposition. In particular, LLM pointed out the useful reference for the Leonov–Shiryaev formula in Lemma 5.2.
Preliminaries
In this section, we collect the necessary background on multivariate Bessel functions and Bessel generating functions. Sections 2.1–2.3 contain classical results, while Section 2.4 recalls a more recent expansion from [18 ] and [22 ] .
Dunkl operators
The Dunkl operators, introduced in [27 ] , are defined by
D i N , θ : = ∂ i + θ ∑ j : j ≠ i 1 x i − x j ( 1 − σ i , j ) , i = 1 , … , N . \mathcal{D}_{i}^{N,\theta} := \partial_i + \theta\sum_{j:j\ne i} \frac{1}{x_i-x_j}(1-\sigma_{i,j}), \qquad i=1,\ldots,N. D i N , θ := ∂ i + θ j : j = i ∑ x i − x j 1 ( 1 − σ i , j ) , i = 1 , … , N . They pairwise commute, D i N , θ D j N , θ = D j N , θ D i N , θ \mathcal{D}^{N,\theta}_i\mathcal{D}^{N,\theta}_j=\mathcal{D}^{N,\theta}_j\mathcal{D}^{N,\theta}_i D i N , θ D j N , θ = D j N , θ D i N , θ i , j i,j i , j D i N , θ \mathcal{D}^{N,\theta}_i D i N , θ C N \mathbb{C}^N C N
P k N , θ : = ∑ i = 1 N ( D i N , θ ) k , k = 1 , 2 , … . \mathcal{P}^{N,\theta}_k:=\sum_{i=1}^{N}\left(\mathcal{D}^{N,\theta}_i\right)^k,\qquad k=1,2,\ldots. P k N , θ := i = 1 ∑ N ( D i N , θ ) k , k = 1 , 2 , … . For simplicity, we drop the superscripts N , θ N,\theta N , θ D i N , θ , P k N , θ \mathcal{D}^{N,\theta}_i,\mathcal{P}^{N,\theta}_k D i N , θ , P k N , θ D i , P k \mathcal{D}_i,\mathcal{P}_k D i , P k
Multivariate Bessel functions
Theorem 2.1 ([55 ] ). Let a 1 , … , a N ∈ C a_1,\ldots,a_N\in\mathbb{C} a 1 , … , a N ∈ C F ( x 1 , … , x N ) F(x_1,\ldots,x_N) F ( x 1 , … , x N ) F ( 0 N ) = 1 F(0^N)=1 F ( 0 N ) = 1
P k F ( x 1 , … , x N ) = ( ∑ i = 1 N a i k ) F ( x 1 , … , x N ) , for all k ≥ 1. \mathcal{P}_kF(x_1,\ldots,x_N)=\left(\sum_{i=1}^{N}a_i^k\right)F(x_1,\ldots,x_N),\qquad\text{for all } k\ge1. P k F ( x 1 , … , x N ) = ( i = 1 ∑ N a i k ) F ( x 1 , … , x N ) , for all k ≥ 1. If we denote this solution by B ( a 1 , … , a N ) ( x 1 , … , x N ; θ ) B_{(a_1,\ldots,a_N)}(x_1,\ldots,x_N;\theta) B ( a 1 , … , a N ) ( x 1 , … , x N ; θ ) ( a , x ) ↦ B a ( x ; θ ) (a,x)\mapsto B_a(x;\theta) ( a , x ) ↦ B a ( x ; θ ) 2 N 2N 2 N
Indeed, B a ( x ⃗ ; θ ) B_a(\vec{x};\theta) B a ( x ; θ ) [52 Sec. 4] ):
B a ( x ⃗ ; θ ) = ∑ ℓ ( λ ) ≤ N ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + θ ) ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + 1 ) ( N θ + j − 1 − θ ( i − 1 ) ) J λ ( x ⃗ ; θ ) J λ ( a ; θ ) . (1) B_a(\vec{x};\theta)=\sum_{\ell(\lambda)\le N}\frac{\prod_{(i,j)\in\lambda}\left((\lambda_i-j)+\theta(\lambda'_j-i)+\theta\right)}{\prod_{(i,j)\in\lambda}\left((\lambda_i-j)+\theta(\lambda'_j-i)+1\right)(N\theta+j-1-\theta(i-1))}J_\lambda(\vec{x};\theta)J_\lambda(a;\theta). \tag*{(1)} B a ( x ; θ ) = ℓ ( λ ) ≤ N ∑ ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + 1 ) ( N θ + j − 1 − θ ( i − 1 )) ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + θ ) J λ ( x ; θ ) J λ ( a ; θ ) . (1) Bessel generating functions
Following [7 ] , let M N \mathcal{M}_N M N N N N a 1 ≤ ⋯ ≤ a N a_1\le\cdots\le a_N a 1 ≤ ⋯ ≤ a N R > 0 R>0 R > 0 M N R ⊂ M N \mathcal{M}^R_N\subset\mathcal{M}_N M N R ⊂ M N μ \mu μ
∫ a 1 ≤ ⋯ ≤ a N e N R max i ∣ a i ∣ μ ( d a 1 , … , d a N ) < ∞ . (2) \int_{a_1\le\cdots\le a_N}e^{N R\max_i|a_i|}\,\mu(da_1,\ldots,da_N)<\infty. \tag*{(2)} ∫ a 1 ≤ ⋯ ≤ a N e N R m a x i ∣ a i ∣ μ ( d a 1 , … , d a N ) < ∞. (2) Proposition-Definition 2.2 (Lemma 2.9 of [7 ] ). If μ ∈ M N R \mu\in\mathcal{M}^R_N μ ∈ M N R R > 0 R>0 R > 0
G θ ( x 1 , … , x N ; μ ) : = ∫ a 1 ≤ ⋯ ≤ a N B ( a 1 , … , a N ) ( x 1 , … , x N ; θ ) μ ( d a 1 , … , d a N ) G_\theta(x_1,\ldots,x_N;\mu):=\int_{a_1\le\cdots\le a_N}B_{(a_1,\ldots,a_N)}(x_1,\ldots,x_N;\theta)\mu(da_1,\ldots,da_N) G θ ( x 1 , … , x N ; μ ) := ∫ a 1 ≤ ⋯ ≤ a N B ( a 1 , … , a N ) ( x 1 , … , x N ; θ ) μ ( d a 1 , … , d a N ) defines a holomorphic function in the domain
{ ( x 1 , … , x N ) ∈ C N : ∣ ℜ x i ∣ < R , i = 1 , 2 , … , N } , \{(x_1,\ldots,x_N)\in\mathbb{C}^N:|\Re x_i|<R,\ i=1,2,\ldots,N\}, {( x 1 , … , x N ) ∈ C N : ∣ℜ x i ∣ < R , i = 1 , 2 , … , N } , that is called the Bessel generating function of μ \mu μ
The measures in M N R \mathcal{M}^R_N M N R R R R M N > 0 : = ⋃ R > 0 M N R \mathcal{M}^{>0}_N:=\bigcup_{R>0}\mathcal{M}^R_N M N > 0 := ⋃ R > 0 M N R M N > 0 \mathcal{M}^{>0}_N M N > 0 μ ∈ M N > 0 \mu\in\mathcal{M}^{>0}_N μ ∈ M N > 0 G θ ( x 1 , … , x N ; μ ) G_\theta(x_1,\ldots,x_N;\mu) G θ ( x 1 , … , x N ; μ ) [7 Section 2.4] , we also allow μ \mu μ R N \mathbb{R}^{N} R N ⟨ μ , 1 ⟩ = N ! \langle\mu,1\rangle=N! ⟨ μ , 1 ⟩ = N !
E μ [ f ] : = 1 N ! ⟨ μ , f ⟩ , G θ ( x ⃗ ; μ ) : = 1 N ! ⟨ μ , B ∙ ( x ⃗ ; θ ) ⟩ . \mathbb{E}_{\mu}[f] := \frac{1}{N!}\langle\mu,f\rangle,\qquad G_{\theta}(\vec{x};\mu) := \frac{1}{N!}\langle\mu,B_{\bullet}(\vec{x};\theta)\rangle. E μ [ f ] := N ! 1 ⟨ μ , f ⟩ , G θ ( x ; μ ) := N ! 1 ⟨ μ , B ∙ ( x ; θ )⟩ . The BGF is entire by [7 Proposition 2.13] . Together with exponentially decaying probability measures, these are the N N N E μ [ 1 ] = 1 \mathbb{E}_{\mu}[1]=1 E μ [ 1 ] = 1
In either case, the normalization and symmetry of the Bessel function give
G θ ( 0 N ; μ ) = 1 , G_{\theta}(0^{N};\mu)=1, G θ ( 0 N ; μ ) = 1 , and G θ ( x 1 , … , x N ; μ ) G_{\theta}(x_{1},\ldots,x_{N};\mu) G θ ( x 1 , … , x N ; μ ) x 1 , … , x N x_{1},\ldots,x_{N} x 1 , … , x N
Proposition 2.3 (Propositions 2.11 and 2.13 of [7 ] ). Let s ≥ 1 s\geq1 s ≥ 1 k 1 , … , k s ≥ 1 k_{1},\ldots,k_{s}\geq1 k 1 , … , k s ≥ 1 μ \mu μ
( ∏ i = 1 s P k i ) G θ ( x ⃗ ; μ ) ∣ x ⃗ = 0 = E μ [ ∏ i = 1 s p k i ( a ) ] , \left.\left(\prod_{i=1}^{s}P_{k_i}\right)G_{\theta}(\vec{x};\mu)\right|_{\vec{x}=0}=\mathbb{E}_{\mu}\left[\prod_{i=1}^{s}p_{k_i}(a)\right], ( i = 1 ∏ s P k i ) G θ ( x ; μ ) x = 0 = E μ [ i = 1 ∏ s p k i ( a ) ] , where the right-hand side is the corresponding normalized moment pairing.
Maps on closed surfaces and a topological expansion
In this section, we recall a finite-N N N ln B a ( x ⃗ ; θ ) \ln B_a(\vec{x};\theta) ln B a ( x ; θ ) [18 ] in [22 Section 5] . It is used in both fixed- and high-temperature regimes in § 8 . We give the needed definitions below and reproduce the proof of Theorem 2.8 in Appendix § 12 .
A closed surface is a compact surface without boundary; all surfaces here are connected. For a cellularly embedded graph (that is, an embedding whose complementary regions are open disks), the Euler characteristic is
χ = V − E + F = 2 − 2 g . \chi=V-E+F=2-2g. χ = V − E + F = 2 − 2 g . Here V V V E E E F F F g = 0 , 1 , 2 , … g=0,1,2,\ldots g = 0 , 1 , 2 , … h ≥ 1 h\geq1 h ≥ 1 g = h / 2 g=h/2 g = h /2 g g g
Definition 2.4. A map M \mathbf{M} M
Definition 2.5. Let k ≥ 1 k\geq1 k ≥ 1 k k k M \mathbf{M} M { 0 , 1 , … , k } \{0,1,\ldots,k\} { 0 , 1 , … , k }
Each vertex of color 0 0 0 1 1 1
Each vertex of color k k k k − 1 k-1 k − 1
For 1 ≤ i ≤ k − 1 1\leq i\leq k-1 1 ≤ i ≤ k − 1 i i i i − 1 i-1 i − 1 i + 1 i+1 i + 1
Finally, we also color all corners with the same colors as their corresponding vertices.
The size of M M M ∣ M ∣ |M| ∣ M ∣ M M M k k k M M M k ⋅ ∣ M ∣ k \cdot|M| k ⋅ ∣ M ∣ ∣ M ∣ |M| ∣ M ∣
We say that M M M connected if the underlying graph is connected, and M M M orientable/non-orientable if its underlying surface is orientable/non-orientable. The genus of M M M g ( M ) g(M) g ( M )
An oriented corner is a corner together with one of the two local directions around its incident vertex; we may encode this choice by a sign ϵ ∈ { − 1 , + 1 } \epsilon\in\{-1,+1\} ϵ ∈ { − 1 , + 1 } M M M rooted if it has a distinguished oriented corner of color 0, which is called the root of M M M
By definition, in any k k k M M M k k k k − 1 k-1 k − 1 k + 1 k+1 k + 1 k k k k k k ( k + 1 ) (k+1) ( k + 1 ) k k k ( k + 1 ) (k+1) ( k + 1 ) C d , ∙ ( k ) C_{d,\bullet}^{(k)} C d , ∙ ( k ) k k k d d d
i k : C d , ∙ ( k ) ↪ C d , ∙ ( k + 1 ) , k ≥ 1. i_k:C_{d,\bullet}^{(k)} \hookrightarrow C_{d,\bullet}^{(k+1)}, \qquad k \ge1. i k : C d , ∙ ( k ) ↪ C d , ∙ ( k + 1 ) , k ≥ 1. A rooted, connected, infinite constellation of size d d d k 0 k_0 k 0 k 0 ≥ 1 k_0 \ge1 k 0 ≥ 1 i k 0 , i k 0 + 1 , … i_{k_0},i_{k_0+1},\ldots i k 0 , i k 0 + 1 , …
C d , ∙ ( ∞ ) = lim k C d , ∙ ( k ) . C_{d,\bullet}^{(\infty)}=\lim_{k} C_{d,\bullet}^{(k)}. C d , ∙ ( ∞ ) = k lim C d , ∙ ( k ) . Note that the initial level k 0 k_0 k 0 d d d d d d
Definition 2.6. For M ∈ C d , ∙ ( ∞ ) M \in C_{d,\bullet}^{(\infty)} M ∈ C d , ∙ ( ∞ ) F ( M ) F(M) F ( M )
For any f ∈ F ( M ) f \in F(M) f ∈ F ( M ) deg ( f ) \deg(f) deg ( f ) f f f μ 1 ( M ) \mu_1(M) μ 1 ( M ) deg ( f ) \deg(f) deg ( f ) f ∈ F ( M ) f \in F(M) f ∈ F ( M )
Denote by V 0 ( M ) V_0(M) V 0 ( M ) M M M v 0 ( M ) : = ∣ V 0 ( M ) ∣ v_0(M):=|V_0(M)| v 0 ( M ) := ∣ V 0 ( M ) ∣
For any v ∈ V 0 ( M ) v \in V_0(M) v ∈ V 0 ( M ) deg ( v ) \deg(v) deg ( v ) 2 μ 2 ( M ) \mu_2(M) μ 2 ( M ) deg ( v ) \deg(v) deg ( v ) v ∈ V 0 ( M ) v \in V_0(M) v ∈ V 0 ( M )
For any i ≥ 1 i \ge1 i ≥ 1 v i ( M ) v_i(M) v i ( M ) M M M i i i
η ( M ) : = ( ∣ M ∣ − v i ( M ) ) i ≥ 1 . \eta(M):=\left(|M|-v_i(M)\right)_{i\ge1}. η ( M ) := ( ∣ M ∣ − v i ( M ) ) i ≥ 1 . We say that M M M normal if v 1 ( M ) ≤ v 2 ( M ) ≤ ⋯ v_1(M) \le v_2(M) \le\cdots v 1 ( M ) ≤ v 2 ( M ) ≤ ⋯ v i ( M ) ≤ ∣ M ∣ v_i(M) \le|M| v i ( M ) ≤ ∣ M ∣ i i i η ( M ) \eta(M) η ( M )
By definition, ∣ μ 1 ( M ) ∣ = ∣ μ 2 ( M ) ∣ = ∣ M ∣ |\mu_1(M)|=|\mu_2(M)|=|M| ∣ μ 1 ( M ) ∣ = ∣ μ 2 ( M ) ∣ = ∣ M ∣
The parts of each partition are listed in decreasing order. In particular, ℓ ( μ 1 ) \ell(\mu_1) ℓ ( μ 1 ) ℓ ( μ 2 ) = v 0 \ell(\mu_2)=v_0 ℓ ( μ 2 ) = v 0 η \eta η ∣ η ∣ = ∑ i ≥ 1 ( d − v i ) |\eta|=\sum_{i\ge1}(d-v_i) ∣ η ∣ = ∑ i ≥ 1 ( d − v i )
Proposition 2.7. [18 Eqn. (5)] For any M ∈ C d , ∙ ( ∞ ) \mathbf{M} \in\mathcal{C}^{(\infty)}_{d,\bullet} M ∈ C d , ∙ ( ∞ ) g g g
∣ η ( M ) ∣ − ℓ ( μ 1 ( M ) ) − ℓ ( μ 2 ( M ) ) = 2 g − 2. (3) |\eta(\mathbf{M})|-\ell(\boldsymbol{\mu}_1(\mathbf{M}))-\ell(\boldsymbol{\mu}_2(\mathbf{M}))=2g-2. \tag*{(3)} ∣ η ( M ) ∣ − ℓ ( μ 1 ( M )) − ℓ ( μ 2 ( M )) = 2 g − 2. (3) Proof. Take a large enough k k k η i ( M ) = 0 \eta_i(\mathbf{M})=0 η i ( M ) = 0 i ≥ k i \geq k i ≥ k k k k M \mathbf{M} M M \mathbf{M} M k ⋅ ∣ M ∣ k\cdot|\mathbf{M}| k ⋅ ∣ M ∣ M \mathbf{M} M ∑ i = 0 k v i ( M ) = ∑ i = 1 k v i ( M ) + ℓ ( μ 2 ( M ) ) \sum_{i=0}^{k}v_i(\mathbf{M})=\sum_{i=1}^{k}v_i(\mathbf{M})+\ell(\boldsymbol{\mu}_2(\mathbf{M})) ∑ i = 0 k v i ( M ) = ∑ i = 1 k v i ( M ) + ℓ ( μ 2 ( M )) M \mathbf{M} M ℓ ( μ 1 ( M ) ) \ell(\boldsymbol{\mu}_1(\mathbf{M})) ℓ ( μ 1 ( M )) E − V − F = 2 g − 2 E-V-F=2g-2 E − V − F = 2 g − 2
The exponent ν ρ ( M , c ) \nu_\rho(\mathbf{M},c) ν ρ ( M , c ) ( M , c ) (\mathbf{M},c) ( M , c ) [18 Sections 3 and 6.1] for details. It satisfies the following two properties:
0 ≤ ν ρ ( M , c ) ≤ 2 g ( M ) , ν ρ ( M , c ) = 0 ⟺ M is orientable . 0\leq\nu_\rho(\mathbf{M},c)\leq2g(\mathbf{M}),\qquad\nu_\rho(\mathbf{M},c)=0\Longleftrightarrow\mathbf{M}\text{ is orientable}. 0 ≤ ν ρ ( M , c ) ≤ 2 g ( M ) , ν ρ ( M , c ) = 0 ⟺ M is orientable . Theorem 2.8. [22 Theorem 5.6] Let n ∈ N n\in\mathbb{N} n ∈ N θ > 0 \theta>0 θ > 0 α = θ − 1 \alpha=\theta^{-1} α = θ − 1 N > max ( α , 1 ) n N>\max(\alpha,1)n N > max ( α , 1 ) n N > n N>n N > n N θ > n N\theta>n N θ > n a ∈ C N \mathbf{a}\in\mathbb{C}^{N} a ∈ C N ln B a ( x ⃗ ; θ ) \ln B_{\mathbf{a}}(\vec{x};\theta) ln B a ( x ; θ ) ln ( 1 + x ) = ∑ r = 1 ∞ ( − 1 ) r − 1 r x r \ln(1+x)=\sum_{r=1}^{\infty}\frac{(-1)^{r-1}}{r}x^r ln ( 1 + x ) = ∑ r = 1 ∞ r ( − 1 ) r − 1 x r n n n x ⃗ \vec{x} x
ln ( B a ( x ⃗ ; θ ) ) = ∑ d = 1 n 1 d ∑ g ∈ 1 2 Z ≥ 0 ( M , c ) ∈ C d , ∙ ( ∞ ) , M n o r m a l , g ( M ) = g α d − 1 ( α − 1 ) ν ρ ( M , c ) ∏ f ∈ F ( M ) p deg ( f ) ( a / N ) N × ∏ v ∈ V 0 ( M ) p deg ( v ) ( x ⃗ ) N ⋅ N 2 − 2 g ⋅ f η ( M ) ( { ( − 1 ) k } k ≥ 1 ) + O ( ∥ x ⃗ ∥ n + 1 ) . (4) \begin{aligned}
\ln\left(B_{\mathbf{a}}(\vec{x};\theta)\right)
={}&\sum_{d=1}^{n}\frac{1}{d}\sum_{\substack{g\in\frac{1}{2}\mathbb{Z}_{\geq0}\\( \mathbf{M},c)\in\mathcal{C}^{(\infty)}_{d,\bullet},\,\mathbf{M}\ \mathrm{normal},\,g(\mathbf{M})=g}}
\alpha^{d-1}(\alpha-1)^{\nu_\rho(\mathbf{M},c)}
\prod_{f\in F(\mathbf{M})}\frac{p_{\deg(f)}(a/N)}{N}\\
&\qquad\times\prod_{v\in V_0(\mathbf{M})}\frac{p_{\deg(v)}(\vec{x})}{N}\cdot N^{2-2g}\cdot f_{\eta(\mathbf{M})}\left(\left\{(-1)^k\right\}_{k\geq1}\right)+O(\lVert\vec{x}\rVert^{n+1}). \tag*{(4)}
\end{aligned} ln ( B a ( x ; θ ) ) = d = 1 ∑ n d 1 g ∈ 2 1 Z ≥ 0 ( M , c ) ∈ C d , ∙ ( ∞ ) , M normal , g ( M ) = g ∑ α d − 1 ( α − 1 ) ν ρ ( M , c ) f ∈ F ( M ) ∏ N p d e g ( f ) ( a / N ) × v ∈ V 0 ( M ) ∏ N p d e g ( v ) ( x ) ⋅ N 2 − 2 g ⋅ f η ( M ) ( { ( − 1 ) k } k ≥ 1 ) + O (∥ x ∥ n + 1 ) . (4) where f η ( e 1 , e 2 , … ) f_\eta(e_1,e_2,\ldots) f η ( e 1 , e 2 , … ) m η m_\eta m η e 1 , e 2 , … e_1,e_2,\ldots e 1 , e 2 , … M \mathbf{M} M d d d g g g
We reproduce the proof of Theorem 2.8 in Appendix § 12 using the same argument3 [22 Theorem 5.6] .
We divide the proofs of Theorems 3.12 and 3.13 into the if and only if directions. Sections 3, § 7 , and § 9 establish the if direction. The present section develops the Lukasiewicz-path framework and states the limiting covariance formulas; § 7 proves these formulas, while § 9 proves the vanishing of the higher joint cumulants. The only if direction is proved in § 8 .
Recall from Definition 1.1 that
ln G θ , N ( x ⃗ ) = ∑ λ ∈ Y c λ p λ ( x ⃗ ) . \ln G_{\theta,N}(\vec{x})=\sum_{\lambda\in\mathcal{Y}}c_\lambda p_\lambda(\vec{x}). ln G θ , N ( x ) = λ ∈ Y ∑ c λ p λ ( x ) . We start by recalling the moment LLN results in the fixed- and high-temperature regimes. Throughout the statements below, μ N \mu_N μ N § 5.3 , and all convergence is understood in terms of its normalized moment functional. The moment–cumulant formulas in the two regimes, together with the two covariance–cumulant formulas in Theorems 3.12 and 3.13 , fall into a unified combinatorial framework in terms of Lukasiewicz paths.
Definition 3.1. A Lukasiewicz path of length k k k E : { 0 , 1 , 2 , … , k } → Z ≥ 0 E:\{0,1,2,\ldots,k\}\to\mathbb{Z}_{\geq0} E : { 0 , 1 , 2 , … , k } → Z ≥ 0 E ( 0 ) = E ( k ) = 0 E(0)=E(k)=0 E ( 0 ) = E ( k ) = 0 t = 1 , 2 , … , k t=1,2,\ldots,k t = 1 , 2 , … , k E ( t ) − E ( t − 1 ) ∈ Z ≥ − 1 E(t)-E(t-1)\in\mathbb{Z}_{\geq-1} E ( t ) − E ( t − 1 ) ∈ Z ≥ − 1 k ∈ Z ≥ 1 k\in\mathbb{Z}_{\geq1} k ∈ Z ≥ 1 k k k L ( k ) L(k) L ( k )
Theorem 3.2. [22 Theorem 1.1] In the fixed-temperature regime, the LLN holds, i.e.,
lim N → ∞ E μ N [ ∏ j = 1 s p ~ k j N ] = ∏ j = 1 s m k j , \lim_{N\to\infty}\mathbb{E}_{\mu_N}\left[\prod_{j=1}^{s}\widetilde{p}_{k_j}^{N}\right]=\prod_{j=1}^{s}m_{k_j}, N → ∞ lim E μ N [ j = 1 ∏ s p k j N ] = j = 1 ∏ s m k j , if and only if for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\geq1} d ∈ Z ≥ 1
d N c ( d ) → θ 1 − d κ d ; 1 N ℓ ( λ ) c λ → 0 for ℓ ( λ ) ≥ 2 (5) \frac{d}{N}c^{(d)}\to\theta^{1-d}\kappa_d;
\qquad
\frac{1}{N^{\ell(\lambda)}}c_\lambda\to0\quad\text{for }\ell(\lambda)\geq2 \tag*{(5)} N d c ( d ) → θ 1 − d κ d ; N ℓ ( λ ) 1 c λ → 0 for ℓ ( λ ) ≥ 2 (5) when N → ∞ N\to\infty N → ∞ m k m_k m k k ≥ 1 k\geq1 k ≥ 1 κ l \kappa_l κ l
m k = ∑ E ∈ L ( k ) ∏ B i ∈ E κ ∣ B i ∣ (6) m_k=\sum_{E\in L(k)}\prod_{B_i\in E}\kappa_{\lvert B_i\rvert} \tag*{(6)} m k = E ∈ L ( k ) ∑ B i ∈ E ∏ κ ∣ B i ∣ (6) where ∣ B i ∣ \lvert B_i\rvert ∣ B i ∣ i i i E E E
Theorem 3.3. [7 Theorem 3.8 and 3.10] In the high-temperature regime, the LLN holds, i.e.,
lim N → ∞ E μ N [ ∏ j = 1 s p k j N ] = ∏ j = 1 s m k j , \lim_{N\to\infty}\mathbb{E}_{\mu_N}\left[\prod_{j=1}^{s}p_{k_j}^{N}\right]=\prod_{j=1}^{s}m_{k_j}, N → ∞ lim E μ N [ j = 1 ∏ s p k j N ] = j = 1 ∏ s m k j , if and only if for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\geq1} d ∈ Z ≥ 1
d ⋅ c ( d ) → κ d γ ; c λ → 0 for ℓ ( λ ) ≥ 2 (7) d\cdot c_{(d)}\to\kappa_d^\gamma;
\qquad
c_\lambda\to0\quad\text{for }\ell(\lambda)\geq2 \tag*{(7)} d ⋅ c ( d ) → κ d γ ; c λ → 0 for ℓ ( λ ) ≥ 2 (7) when N → ∞ N\to\infty N → ∞ m k m_k m k k ≥ 1 k\geq1 k ≥ 1 γ \gamma γ κ l γ \kappa_l^\gamma κ l γ
m k = ∑ E ∈ L ( k ) t : E ( t ) − E ( t − 1 ) = − 1 ( γ + E ( t − 1 ) ) ∏ B i ∈ E κ ∣ B i ∣ γ (8) m_k=\sum_{\substack{E\in L(k)\\ t:E(t)-E(t-1)=-1}}(\gamma+E(t-1))\prod_{B_i\in E}\kappa_{\lvert B_i\rvert}^\gamma \tag*{(8)} m k = E ∈ L ( k ) t : E ( t ) − E ( t − 1 ) = − 1 ∑ ( γ + E ( t − 1 )) B i ∈ E ∏ κ ∣ B i ∣ γ (8) where ∣ B i ∣ \lvert B_i\rvert ∣ B i ∣
Remark 3.4. The condition given in Eqn. (7) is a reformulation of [7 Theorem 3.8] . The combinatorial formula in [7 ] is a weighted sum over partitions which are not necessarily non-crossing, but [65 Proposition 5.12] identifies it with a weighted sum over non-crossing partitions, which is equivalent to Eqn. (7) by the well-known bijection between non-crossing partitions and Lukasiewicz paths, see e.g., [50 Proposition 9.8] .
To formulate our next results, we introduce some further notions related to the configurations of Lukasiewicz paths.
For m , n ∈ Z ≥ 1 m,n \in\mathbb{Z}_{\ge1} m , n ∈ Z ≥ 1 m + n m+n m + n m m m E ( t ) : { 0 , 1 , 2 , … , m + n } → Z ≥ 0 E(t):\{0,1,2,\ldots,m+n\}\to\mathbb{Z}_{\ge0} E ( t ) : { 0 , 1 , 2 , … , m + n } → Z ≥ 0 E ( 0 ) = E ( m + n ) = 0 E(0)=E(m+n)=0 E ( 0 ) = E ( m + n ) = 0 L ~ ( m , n ) \widetilde{L}(m,n) L ( m , n ) E ( m ) > 0 E(m)>0 E ( m ) > 0 L ( m , n ) L(m,n) L ( m , n ) E E E [ 0 , m + n ] [0,m+n] [ 0 , m + n ] E ( t ) = E ( ⌊ t ⌋ ) E(t)=E(\lfloor t\rfloor) E ( t ) = E (⌊ t ⌋) E ( t ) E(t) E ( t ) { 1 , 2 , … , m + n } \{1,2,\ldots,m+n\} { 1 , 2 , … , m + n }
For E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) { q i ( t ) } i = 1 ∞ : [ 0 , m + n ] → Z ≥ 0 \{q_i(t)\}_{i=1}^{\infty}:[0,m+n]\to\mathbb{Z}_{\ge0} { q i ( t ) } i = 1 ∞ : [ 0 , m + n ] → Z ≥ 0 i = 1 , 2 i=1,2 i = 1 , 2 t = m t=m t = m i i i q ( t ) = ∑ i = 1 ∞ q i ( t ) q(t)=\sum_{i=1}^{\infty}q_i(t) q ( t ) = ∑ i = 1 ∞ q i ( t ) t t t { q i } i = 1 , 2 , … \{q_i\}_{i=1,2,\ldots} { q i } i = 1 , 2 , … block functions of E E E
Each q i ( t ) q_i(t) q i ( t ) { 1 , 2 , … , m + n } \{1,2,\ldots,m+n\} { 1 , 2 , … , m + n } q i q_i q i
q ( t ) ≤ E ( t ) q(t)\le E(t) q ( t ) ≤ E ( t ) t ∈ [ 0 , m + n ] t\in[0,m+n] t ∈ [ 0 , m + n ]
q 1 ( t ) = 0 q_1(t)=0 q 1 ( t ) = 0 t ∈ [ 0 , m ] t\in[0,m] t ∈ [ 0 , m ] q 1 ( m + ) = E ( m ) − q ( m ) q_1(m+)=E(m)-q(m) q 1 ( m + ) = E ( m ) − q ( m ) q 2 ( t ) = 0 q_2(t)=0 q 2 ( t ) = 0 t ∈ ( m , m + n ] t\in(m,m+n] t ∈ ( m , m + n ]
At each left-discontinuity point t t t q i q_i q i i i i
(1). If E ( t ) − E ( t − 1 ) = − 1 E(t)-E(t-1)=-1 E ( t ) − E ( t − 1 ) = − 1 q i q_i q i jump at t t t h = q i ( t ) − q i ( t − ) h=q_i(t)-q_i(t-) h = q i ( t ) − q i ( t − ) jump size of q i q_i q i t t t
∣ h ∣ ≤ ∣ E ( t − ) − q ( t − ) − q i ( t − ) − 1 2 ( sgn ( h ) + 1 ) ∣ , |h|\le\left|E(t-)-q(t-)-q_i(t-)-\frac{1}{2}(\operatorname{sgn}(h)+1)\right|, ∣ h ∣ ≤ E ( t − ) − q ( t − ) − q i ( t − ) − 2 1 ( sgn ( h ) + 1 ) , h ⋅ ( E ( t − ) − q ( t − ) − q i ( t − ) ) > 0. (9) h\cdot\left(E(t-)-q(t-)-q_i(t-)\right)>0. \tag*{(9)} h ⋅ ( E ( t − ) − q ( t − ) − q i ( t − ) ) > 0. (9) (2). If E ( t ) − E ( t − ) ≥ 0 E(t)-E(t-)\ge0 E ( t ) − E ( t − ) ≥ 0
k 2 : = q i ( t ) − q i ( t − ) ∈ Z ≥ 1 , k_2:=q_i(t)-q_i(t-)\in\mathbb{Z}_{\ge1}, k 2 := q i ( t ) − q i ( t − ) ∈ Z ≥ 1 , and let k 1 = E ( t ) − E ( t − ) ∈ Z ≥ 0 k_1=E(t)-E(t-)\in\mathbb{Z}_{\ge0} k 1 = E ( t ) − E ( t − ) ∈ Z ≥ 0
Similarly, for k ∈ Z ≥ 1 k\in\mathbb{Z}_{\ge1} k ∈ Z ≥ 1 E ∈ L ( k ) E\in L(k) E ∈ L ( k ) E E E E ( t ) : [ 0 , k ] → Z ≥ 0 E(t):[0,k]\to\mathbb{Z}_{\ge0} E ( t ) : [ 0 , k ] → Z ≥ 0 { q i ( t ) } i = 1 ∞ : [ 0 , k ] → Z ≥ 0 \{q_i(t)\}_{i=1}^{\infty}:[0,k]\to\mathbb{Z}_{\ge0} { q i ( t ) } i = 1 ∞ : [ 0 , k ] → Z ≥ 0 { q i } i = 1 , 2 , … \{q_i\}_{i=1,2,\ldots} { q i } i = 1 , 2 , … block functions of E E E
Each q i ( t ) q_i(t) q i ( t ) { 1 , 2 , … , k } \{1,2,\ldots,k\} { 1 , 2 , … , k } q i q_i q i
q ( t ) ≤ E ( t ) q(t)\le E(t) q ( t ) ≤ E ( t ) t ∈ [ 0 , k ] t\in[0,k] t ∈ [ 0 , k ]
q 1 ( t ) = 0 q_1(t)=0 q 1 ( t ) = 0 t ∈ [ 0 , k ] t\in[0,k] t ∈ [ 0 , k ]
At each discontinuity point t t t q i q_i q i i ≥ 2 i\ge2 i ≥ 2 E ( t ) − E ( t − ) = − 1 E(t)-E(t-)=-1 E ( t ) − E ( t − ) = − 1 q i q_i q i jump at t t t t t t jump times of q i q_i q i h = q i ( t ) − q i ( t − ) h=q_i(t)-q_i(t-) h = q i ( t ) − q i ( t − ) jump height of q i q_i q i t t t
∣ h ∣ ≤ ∣ E ( t − ) − q ( t − ) − q i ( t − ) − 1 2 ( sgn ( h ) + 1 ) ∣ , |h|\le\left|E(t-)-q(t-)-q_i(t-)-\frac{1}{2}(\operatorname{sgn}(h)+1)\right|, ∣ h ∣ ≤ E ( t − ) − q ( t − ) − q i ( t − ) − 2 1 ( sgn ( h ) + 1 ) , h ⋅ ( E ( t − ) − q ( t − ) − q i ( t − ) ) > 0. (10) h\cdot\left(E(t-)-q(t-)-q_i(t-)\right)>0. \tag*{(10)} h ⋅ ( E ( t − ) − q ( t − ) − q i ( t − ) ) > 0. (10) Definition 3.5. For E ( t ) ∈ L ( m , n ) E(t) \in L(m,n) E ( t ) ∈ L ( m , n ) L ( k ) L(k) L ( k ) D ( E ) D(E) D ( E ) t t t E E E E ( t ) − E ( t − ) = − 1 E(t)-E(t-)=-1 E ( t ) − E ( t − ) = − 1 U ( E ) = { 1 , 2 , … , m + n } ∖ D ( E ) U(E)=\{1,2,\ldots,m+n\}\setminus D(E) U ( E ) = { 1 , 2 , … , m + n } ∖ D ( E ) { 1 , 2 , … , k } ∖ D ( E ) \{1,2,\ldots,k\}\setminus D(E) { 1 , 2 , … , k } ∖ D ( E )
The data consisting of a path E ( t ) ∈ L ( m , n ) E(t)\in L(m,n) E ( t ) ∈ L ( m , n ) L ( k ) L(k) L ( k ) ( E , { q i } i = 1 , 2 , … ) (E,\{q_i\}_{i=1,2,\ldots}) ( E , { q i } i = 1 , 2 , … ) ( E , q ⃗ ) (E,\vec{q}) ( E , q ) ( E , q ⃗ ) (E,\vec{q}) ( E , q ) J ( E , q ⃗ ) ⊂ D ( E ) J(E,\vec{q})\subset D(E) J ( E , q ) ⊂ D ( E )
P ( E , q ⃗ ) = D ( E ) ∖ J ( E , q ⃗ ) . P(E,\vec{q})=D(E)\setminus J(E,\vec{q}). P ( E , q ) = D ( E ) ∖ J ( E , q ) . For E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) C = E ( m ) C=E(m) C = E ( m )
τ = inf { t ≥ m : E ( t ) < C } , τ ˉ = sup { t ∈ [ 0 , m ] : E ( t ) < C } . (11) \begin{aligned}
\tau&=\inf\{t\ge m:E(t)<C\},\\
\bar{\tau}&=\sup\{t\in[0,m]:E(t)<C\}. \tag*{(11)}
\end{aligned} τ τ ˉ = inf { t ≥ m : E ( t ) < C } , = sup { t ∈ [ 0 , m ] : E ( t ) < C } . (11) Furthermore, let
A ( E ) = ∣ { τ ˉ + 1 , … , m } ∩ D ( E ) ∣ + 1 , B ( E ) = ∣ { m + 1 , … , τ − 1 } ∩ D ( E ) ∣ + 1. (12) \begin{aligned}
A(E)&=|\{\bar{\tau}+1,\ldots,m\}\cap D(E)|+1,\\
B(E)&=|\{m+1,\ldots,\tau-1\}\cap D(E)|+1. \tag*{(12)}
\end{aligned} A ( E ) B ( E ) = ∣ { τ ˉ + 1 , … , m } ∩ D ( E ) ∣ + 1 , = ∣ { m + 1 , … , τ − 1 } ∩ D ( E ) ∣ + 1. (12) Example 3.6. Consider E ∈ L ( 7 , 4 ) E\in L(7,4) E ∈ L ( 7 , 4 ) E ( 1 ) = 1 , E ( 2 ) = 2 , E ( 3 ) = 1 , E ( 4 ) = 4 , E ( 5 ) = 3 , E ( 6 ) = 2 , E ( 7 ) = 2 , E ( 8 ) = 3 , E ( 9 ) = 2 , E ( 10 ) = 1 E(1)=1,E(2)=2,E(3)=1,E(4)=4,E(5)=3,E(6)=2,E(7)=2,E(8)=3,E(9)=2,E(10)=1 E ( 1 ) = 1 , E ( 2 ) = 2 , E ( 3 ) = 1 , E ( 4 ) = 4 , E ( 5 ) = 3 , E ( 6 ) = 2 , E ( 7 ) = 2 , E ( 8 ) = 3 , E ( 9 ) = 2 , E ( 10 ) = 1 C = 2 , τ ˉ = 4 , τ = 10 C=2,\bar{\tau}=4,\tau=10 C = 2 , τ ˉ = 4 , τ = 10
{ τ ˉ + 1 , … , m } ∩ D ( E ) = { 5 , 6 } , { m + 1 , … , τ − 1 } ∩ D ( E ) = { 9 } , \{\bar{\tau}+1,\ldots,m\}\cap D(E)=\{5,6\},\qquad\{m+1,\ldots,\tau-1\}\cap D(E)=\{9\}, { τ ˉ + 1 , … , m } ∩ D ( E ) = { 5 , 6 } , { m + 1 , … , τ − 1 } ∩ D ( E ) = { 9 } , so A ( E ) = 3 , B ( E ) = 2 A(E)=3,B(E)=2 A ( E ) = 3 , B ( E ) = 2
Figure 1. The Lukasiewicz path in Example 3.6. Solid dots give the function values; open dots give the distinct left limits. The blue and orange down jumps determine A ( E ) A(E) A ( E ) B ( E ) B(E) B ( E )
Definition 3.7. Let k = m + n k=m+n k = m + n E ∈ L ( k ) E\in L(k) E ∈ L ( k ) C = E ( m ) C=E(m) C = E ( m ) ( E , q ⃗ ) (E,\vec{q}) ( E , q ) C > 0 C>0 C > 0
(a).
q i ( t ) ≡ 0 for all i = 1 , 2 , … q_i(t)\equiv0\quad\text{for all }i=1,2,\ldots q i ( t ) ≡ 0 for all i = 1 , 2 , … For E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) ( E , q ⃗ ) (E,\vec{q}) ( E , q ) C > 0 C>0 C > 0
(b). q i ( t ) ≡ 0 q_i(t)\equiv0 q i ( t ) ≡ 0 i ≥ 2 i\ge2 i ≥ 2 q 1 ( t ) = C q_1(t)=C q 1 ( t ) = C ( m , τ ) (m,\tau) ( m , τ ) q 1 ( t ) = 0 q_1(t)=0 q 1 ( t ) = 0
(c). q i ( t ) ≡ 0 q_i(t) \equiv0 q i ( t ) ≡ 0 i ≠ 2 i \ne2 i = 2 q 2 ( t ) = C q_2(t) = C q 2 ( t ) = C [ s , m ] [s,m] [ s , m ] s ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) s \in\{\bar{\tau}+1,\ldots,m\} \cap D(E) s ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) q 2 ( t ) = 0 q_2(t) = 0 q 2 ( t ) = 0
(d). q i ( t ) ≡ 0 q_i(t) \equiv0 q i ( t ) ≡ 0 i ≠ 3 i \ne3 i = 3 q 3 ( t ) = C q_3(t) = C q 3 ( t ) = C [ s , τ ) [s,\tau) [ s , τ ) s ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) s \in\{\bar{\tau}+1,\ldots,m\} \cap D(E) s ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) q 3 ( t ) = 0 q_3(t) = 0 q 3 ( t ) = 0
For m ≥ n m \ge n m ≥ n m , n ∈ Z ≥ 1 m,n \in\mathbb{Z}_{\ge1} m , n ∈ Z ≥ 1 m + n m+n m + n Conf 1 ( m , n ) \operatorname{Conf}_1(m,n) Conf 1 ( m , n )
One can check by definition that all such configurations satisfy the properties of block functions.
Definition 3.9. For E ∈ L ( m , n ) E \in L(m,n) E ∈ L ( m , n ) ( E , q ⃗ ) (E,\vec{q}) ( E , q ) C > 0 C > 0 C > 0
(e). q i ( t ) ≡ 0 q_i(t) \equiv0 q i ( t ) ≡ 0 i ≠ 2 i \ne2 i = 2 q 2 ( t ) = C q_2(t) = C q 2 ( t ) = C [ s , m ] [s,m] [ s , m ] s ∈ { τ ˉ , … , m } ∩ U ( E ) s \in\{\bar{\tau},\ldots,m\} \cap U(E) s ∈ { τ ˉ , … , m } ∩ U ( E ) q 2 ( t ) = 0 q_2(t) = 0 q 2 ( t ) = 0
(f). q i ( t ) ≡ 0 q_i(t) \equiv0 q i ( t ) ≡ 0 i ≠ 3 i \ne3 i = 3 q 3 ( t ) = C q_3(t) = C q 3 ( t ) = C [ s , τ ) [s,\tau) [ s , τ ) s ∈ { τ ˉ , … , m } ∩ U ( E ) s \in\{\bar{\tau},\ldots,m\} \cap U(E) s ∈ { τ ˉ , … , m } ∩ U ( E ) q 3 ( t ) = 0 q_3(t) = 0 q 3 ( t ) = 0
In both cases, we require E ( s ) − E ( s − ) − q ( s ) ≥ 0 E(s)-E(s-) - q(s) \ge0 E ( s ) − E ( s − ) − q ( s ) ≥ 0
For m ≥ n m \ge n m ≥ n m , n ∈ Z ≥ 1 m,n \in\mathbb{Z}_{\ge1} m , n ∈ Z ≥ 1 m + n m+n m + n Conf 2 ( m , n ) \operatorname{Conf}_2(m,n) Conf 2 ( m , n )
Again by definition, all such configurations satisfy the properties of block functions.
Recall that in classical free probability, the limiting covariance is expressed as a combination of two parts: a polynomial in the κ d \kappa_d κ d κ d \kappa_d κ d κ k 1 , k 2 \kappa_{k_1,k_2} κ k 1 , k 2
Definition 3.10. Given a configuration ( E , q ⃗ ) ∈ Conf 1 ( m , n ) (E,\vec{q}) \in\operatorname{Conf}_1(m,n) ( E , q ) ∈ Conf 1 ( m , n ) l = 1 , 2 , … l=1,2,\ldots l = 1 , 2 , … I l ( 1 ) ( E , q ⃗ ) I_l^{(1)}(E,\vec{q}) I l ( 1 ) ( E , q )
I l ( 1 ) ( E , q ⃗ ) = ( − 1 ) m ( q ⃗ ) ∏ B i ∈ E κ ∣ B i ∣ ∑ i 1 < ⋯ < i l ∏ j = 1 l [ E ( i j − ) − q ( i j − ) ] . I_l^{(1)}(E,\vec{q})=(-1)^{m(\vec{q})}\prod_{B_i\in E}\kappa_{\lvert B_i\rvert}\sum_{i_1<\cdots<i_l}\prod_{j=1}^{l}[E(i_j-) - q(i_j-)]. I l ( 1 ) ( E , q ) = ( − 1 ) m ( q ) B i ∈ E ∏ κ ∣ B i ∣ i 1 < ⋯ < i l ∑ j = 1 ∏ l [ E ( i j − ) − q ( i j − )] . where B i B_i B i U ( E ) U(E) U ( E ) m ( q ⃗ ) m(\vec{q}) m ( q ) q i q_i q i i 1 , … , i l ∈ P ( E , q ⃗ ) i_1,\ldots,i_l \in P(E,\vec{q}) i 1 , … , i l ∈ P ( E , q )
Definition 3.11. Given a configuration ( E , q ⃗ ) ∈ Conf 2 ( m , n ) (E,\vec{q}) \in\operatorname{Conf}_2(m,n) ( E , q ) ∈ Conf 2 ( m , n ) s = s ( E , q ⃗ ) s=s(E,\vec{q}) s = s ( E , q ) t ≤ m t \le m t ≤ m q ( t ) − q ( t − ) = C > 0 q(t)-q(t-) = C > 0 q ( t ) − q ( t − ) = C > 0 Definition 3.9 ), and let k 1 ( E , q ⃗ ) = E ( s ) − E ( s − ) − q ( s ) + 1 k_1(E,\vec{q})=E(s)-E(s-)-q(s)+1 k 1 ( E , q ) = E ( s ) − E ( s − ) − q ( s ) + 1 k 2 ( E ) = C ( E ) k_2(E)=C(E) k 2 ( E ) = C ( E ) l = 1 , 2 , … l=1,2,\ldots l = 1 , 2 , … I l ( 2 ) ( E , q ⃗ ) I_l^{(2)}(E,\vec{q}) I l ( 2 ) ( E , q )
I l ( 2 ) ( E , q ⃗ ) = ( − 1 ) m ( q ⃗ ) κ k 1 ( E , q ⃗ ) , k 2 ( E ) k 2 ( E ) ∏ B i ∈ E , t ( B i ) ≠ s ( E , q ⃗ ) κ ∣ B i ∣ ∑ i 1 < ⋯ < i l ∏ j = 1 l [ E ( i j − ) − q ( i j − ) ] . I_l^{(2)}(E,\vec{q})=(-1)^{m(\vec{q})}\frac{\kappa_{k_1(E,\vec{q}),k_2(E)}}{k_2(E)}
\prod_{\substack{B_i\in E,\ t(B_i)\ne s(E,\vec{q})}}\kappa_{\lvert B_i\rvert}
\sum_{i_1<\cdots<i_l}\prod_{j=1}^{l}[E(i_j-) - q(i_j-)]. I l ( 2 ) ( E , q ) = ( − 1 ) m ( q ) k 2 ( E ) κ k 1 ( E , q ) , k 2 ( E ) B i ∈ E , t ( B i ) = s ( E , q ) ∏ κ ∣ B i ∣ i 1 < ⋯ < i l ∑ j = 1 ∏ l [ E ( i j − ) − q ( i j − )] . where B i B_i B i U ( E ) U(E) U ( E ) m ( q ⃗ ) m(\vec{q}) m ( q ) q i q_i q i q i ( t ) − q i ( t − ) < 0 q_i(t)-q_i(t-)<0 q i ( t ) − q i ( t − ) < 0 i 1 , … , i l ∈ P ( E , q ⃗ ) i_1,\ldots,i_l \in P(E,\vec{q}) i 1 , … , i l ∈ P ( E , q )
After the preparation above, we state the two main theorems of this paper, which constitute the “if” parts of Theorems 1.3 and 1.5 .
Theorem 3.12. In the fixed-temperature regime, if for all d ∈ Z ≥ 1 d \in\mathbb{Z}_{\ge1} d ∈ Z ≥ 1 k 1 , k 2 ∈ Z ≥ 1 k_1,k_2 \in\mathbb{Z}_{\ge1} k 1 , k 2 ∈ Z ≥ 1
d N c ( d ) → θ 1 − d κ d ; \frac{d}{N}c_{(d)} \to\theta^{1-d}\kappa_d; N d c ( d ) → θ 1 − d κ d ; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; k_1k_2(1+\mathbf{1}_{k_1=k_2})c_{(k_1,k_2)} \to\theta^{1-k_1-k_2}\kappa_{k_1,k_2}; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; c λ → 0 for ℓ ( λ ) ≥ 3 , (13) c_\lambda\to0 \qquad\text{for }\ell(\lambda)\ge3, \tag*{(13)} c λ → 0 for ℓ ( λ ) ≥ 3 , (13) when N → ∞ N \to\infty N → ∞ α m , n \alpha_{m,n} α m , n κ l \kappa_l κ l κ k 1 , k 2 \kappa_{k_1,k_2} κ k 1 , k 2
α m , n = 1 θ ∑ ( E , q ⃗ ) ∈ Conf 1 ( m , n ) I 1 ( 1 ) ( E , q ⃗ ) + 1 θ ∑ ( E , q ⃗ ) ∈ Conf 2 ( m , n ) I 1 ( 2 ) ( E , q ⃗ ) \alpha_{m,n}=\frac{1}{\theta}\sum_{(E,\vec{q})\in\operatorname{Conf}_1(m,n)}I_1^{(1)}(E,\vec{q})+\frac{1}{\theta}\sum_{(E,\vec{q})\in\operatorname{Conf}_2(m,n)}I_1^{(2)}(E,\vec{q}) α m , n = θ 1 ( E , q ) ∈ Conf 1 ( m , n ) ∑ I 1 ( 1 ) ( E , q ) + θ 1 ( E , q ) ∈ Conf 2 ( m , n ) ∑ I 1 ( 2 ) ( E , q ) = 1 θ ∑ E ∈ L ( m , n ) A ( E ) B ( E ) C ( E ) ∏ B i ∈ E κ ∣ B i ∣ =\frac{1}{\theta}\sum_{E\in L(m,n)}A(E)B(E)C(E)\prod_{B_i\in E}\kappa_{\lvert B_i\rvert} = θ 1 E ∈ L ( m , n ) ∑ A ( E ) B ( E ) C ( E ) B i ∈ E ∏ κ ∣ B i ∣ + 1 θ ∑ E ∈ L ( m , n ) ∑ s ∈ { τ , … , m ‾ } ∩ U ( E ) B ( E ) κ k 1 ( E ) , k 2 ( E ) ∏ B i ∈ E , t ( B i ) ≠ s κ ∣ B i ∣ , (14) \qquad+\frac{1}{\theta}\sum_{E\in L(m,n)}\sum_{s\in\{\tau,\ldots,\overline{m}\}\cap U(E)}B(E)\kappa_{k_1(E),k_2(E)}\prod_{\substack{B_i\in E,\\ t(B_i)\ne s}}\kappa_{\lvert B_i\rvert}, \tag*{(14)} + θ 1 E ∈ L ( m , n ) ∑ s ∈ { τ , … , m } ∩ U ( E ) ∑ B ( E ) κ k 1 ( E ) , k 2 ( E ) B i ∈ E , t ( B i ) = s ∏ κ ∣ B i ∣ , (14) where k 1 ( E ) : = E ( s ) − E ( s − ) − C ( E ) + 1 k_1(E):=E(s)-E(s-)-C(E)+1 k 1 ( E ) := E ( s ) − E ( s − ) − C ( E ) + 1 κ k 1 ( E ) , k 2 ( E ) : = 0 \kappa_{k_1(E),k_2(E)}:=0 κ k 1 ( E ) , k 2 ( E ) := 0 k 1 ( E ) ≤ 0 k_1(E)\le0 k 1 ( E ) ≤ 0
For θ = 1 2 , 1 \theta=\frac{1}{2},1 θ = 2 1 , 1 [20 ] ; see also [49 Chapter 5] . We will show in § 10 that our Eqn. (14) is equivalent to the classical formula when θ = 1 2 , 1 \theta=\frac{1}{2},1 θ = 2 1 , 1 θ > 0 \theta>0 θ > 0 θ \theta θ [49 Section 5.5] also apply.
Using a similar approach, we prove the CLT for the high-temperature regime and provide an analogous combinatorial covariance–cumulant formula.
Theorem 3.13. In the high-temperature regime, if for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\ge1} d ∈ Z ≥ 1 k 1 , k 2 ∈ Z ≥ 1 k_1,k_2\in\mathbb{Z}_{\ge1} k 1 , k 2 ∈ Z ≥ 1
d ⋅ c ( d ) → κ d γ ; d\cdot c_{(d)}\to\kappa_d^\gamma; d ⋅ c ( d ) → κ d γ ; N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → κ k 1 , k 2 γ ; Nk_1k_2(1+\mathbf{1}_{k_1=k_2})c_{(k_1,k_2)}\to\kappa_{k_1,k_2}^\gamma; N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → κ k 1 , k 2 γ ; N ℓ ( λ ) / 2 c λ → 0 for ℓ ( λ ) ≥ 3 , (15) N^{\ell(\lambda)/2}c_\lambda\to0 \qquad\text{for }\ell(\lambda)\ge3, \tag*{(15)} N ℓ ( λ ) /2 c λ → 0 for ℓ ( λ ) ≥ 3 , (15) when N → ∞ N\to\infty N → ∞ θ → 0 \theta\to0 θ → 0 N θ → γ > 0 N\theta\to\gamma>0 N θ → γ > 0 α m , n \alpha_{m,n} α m , n γ \gamma γ κ l γ \kappa_l^\gamma κ l γ γ \gamma γ κ k 1 , k 2 γ \kappa_{k_1,k_2}^\gamma κ k 1 , k 2 γ
α m , n = ∑ ( E , q ⃗ ) ∈ Conf 1 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − k I k ( 1 ) ( E , q ⃗ ; γ ) \alpha_{m,n}=
\sum_{(E,\vec{q})\in\operatorname{Conf}_1(m,n)}
\sum_{k=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-k}I_k^{(1)}(E,\vec{q};\gamma) α m , n = ( E , q ) ∈ Conf 1 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − k I k ( 1 ) ( E , q ; γ ) + ∑ ( E , q ⃗ ) ∈ Conf 2 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − k I k ( 2 ) ( E , q ⃗ ; γ ) . (16) \qquad+
\sum_{(E,\vec{q})\in\operatorname{Conf}_2(m,n)}
\sum_{k=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-k}I_k^{(2)}(E,\vec{q};\gamma). \tag*{(16)} + ( E , q ) ∈ Conf 2 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − k I k ( 2 ) ( E , q ; γ ) . (16) where I k ( 1 ) ( E , q ⃗ ; γ ) I_k^{(1)}(E,\vec{q};\gamma) I k ( 1 ) ( E , q ; γ ) I k ( 2 ) ( E , q ⃗ ; γ ) I_k^{(2)}(E,\vec{q};\gamma) I k ( 2 ) ( E , q ; γ ) Definitions 3.10 and 3.11 , except that we replace κ l \kappa_l κ l κ k 1 , k 2 \kappa_{k_1,k_2} κ k 1 , k 2 κ l γ \kappa_l^\gamma κ l γ κ k 1 , k 2 γ \kappa_{k_1,k_2}^\gamma κ k 1 , k 2 γ
Remark 3.14. From the above two statements, one can see that
κ k 1 , k 2 = κ k 2 , k 1 , κ k 1 , k 2 γ = κ k 2 , k 1 γ . \kappa_{k_1,k_2}=\kappa_{k_2,k_1},\qquad\kappa^\gamma_{k_1,k_2}=\kappa^\gamma_{k_2,k_1}. κ k 1 , k 2 = κ k 2 , k 1 , κ k 1 , k 2 γ = κ k 2 , k 1 γ . Hence it suffices to consider k 1 ≥ k 2 k_1\ge k_2 k 1 ≥ k 2 k 1 , k 2 k_1,k_2 k 1 , k 2
Action of Dunkl operators
We split the proofs of Theorems 3.12 and 3.13 into two parts. The corresponding laws of large numbers already give the first-cumulant limits. Thus, by the moment–cumulant relations and Definitions 1.2 and 1.4, it remains to establish two assertions for the power sums under the respective scalings: their second joint cumulants, equivalently their covariances, converge to the limits stated in the theorems, and every joint cumulant of order at least three converges to zero. We prove the former in this section and the latter in § 9 . Throughout these two parts, c l ( X 1 , … , X l ) c_l(X_1,\ldots,X_l) c l ( X 1 , … , X l ) X 1 , … , X l X_1,\ldots,X_l X 1 , … , X l c λ c_\lambda c λ
Recall the type A Dunkl operators
D i : = ∂ i + θ ⋅ ∑ j : j ≠ i 1 x i − x j ( 1 − σ i j ) , for i = 1 , 2 , … , N . (17) D_i:=\partial_i+\theta\cdot\sum_{j:j\ne i}\frac{1}{x_i-x_j}(1-\sigma_{ij}),\qquad\text{for }i=1,2,\ldots,N. \tag*{(17)} D i := ∂ i + θ ⋅ j : j = i ∑ x i − x j 1 ( 1 − σ ij ) , for i = 1 , 2 , … , N . (17) defined in [27 ] . They pairwise commute, i.e. D i D j = D j D i D_iD_j=D_jD_i D i D j = D j D i
Consider the operators
P k : = ∑ i = 1 N ( D i ) k , P ~ k : = P k N k , k ≥ 1. P_k:=\sum_{i=1}^{N}(D_i)^k,\qquad\widetilde{P}_k:=\frac{P_k}{N^k},\qquad k\ge1. P k := i = 1 ∑ N ( D i ) k , P k := N k P k , k ≥ 1. The symbols with tildes will be used for the fixed-temperature regime and those without tildes for the high-temperature regime.
For μ N \mu_N μ N § 5.3 , consider the polynomial observables
p k ( a ⃗ ) : = ∑ i = 1 N ( a i ) k , p ~ k ( a ⃗ ) : = p k ( a ⃗ ) N k , k ≥ 1. p_k(\vec a):=\sum_{i=1}^{N}(a_i)^k,\qquad\widetilde{p}_k(\vec a):=\frac{p_k(\vec a)}{N^k},\qquad k\ge1. p k ( a ) := i = 1 ∑ N ( a i ) k , p k ( a ) := N k p k ( a ) , k ≥ 1. Throughout this section we use
F N , θ ( x ⃗ ) : = log G N , θ ( x ⃗ ) = ∑ λ ∈ Y c λ p λ ( x ⃗ ) . F_{N,\theta}(\vec x):=\log G_{N,\theta}(\vec x)=\sum_{\lambda\in\mathbb{Y}}c_\lambda p_\lambda(\vec x). F N , θ ( x ) := log G N , θ ( x ) = λ ∈ Y ∑ c λ p λ ( x ) . Thus every coefficient below is the BGF coefficient c λ c_\lambda c λ N N N θ \theta θ
For fixed-temperature scaling, we have
Cov ( p ~ m ( a ⃗ ) , p ~ n ( a ⃗ ) ) = [ P ~ m P ~ n exp ( ln G θ , N ) − P ~ m exp ( ln G θ , N ) ⋅ P ~ n exp ( ln G θ , N ) ] x ⃗ = 0 = N ( N − 1 ) ( D 2 N ) n ( D 1 N ) m exp ( ln G θ , N ) ∣ x ⃗ = 0 + N ( D 1 N ) m + n exp ( ln G θ , N ) ∣ x ⃗ = 0 − N 2 ( D 1 N ) m exp ( ln G θ , N ) ∣ x ⃗ = 0 ( D 1 N ) n exp ( ln G θ , N ) ∣ x ⃗ = 0 . (18) \begin{aligned}
\operatorname{Cov}\bigl(\widetilde p_m(\vec a),\widetilde p_n(\vec a)\bigr)
&=\left[\widetilde P_m\widetilde P_n\exp(\ln G_{\theta,N})-\widetilde P_m\exp(\ln G_{\theta,N})\cdot\widetilde P_n\exp(\ln G_{\theta,N})\right]_{\vec x=0}\\
&=N(N-1)\left.\left(\frac{D_2}{N}\right)^n\left(\frac{D_1}{N}\right)^m\exp(\ln G_{\theta,N})\right|_{\vec x=0}
+N\left.\left(\frac{D_1}{N}\right)^{m+n}\exp(\ln G_{\theta,N})\right|_{\vec x=0}\\
&\quad-N^2\left.\left(\frac{D_1}{N}\right)^m\exp(\ln G_{\theta,N})\right|_{\vec x=0}
\left.\left(\frac{D_1}{N}\right)^n\exp(\ln G_{\theta,N})\right|_{\vec x=0}. \tag*{(18)}
\end{aligned} Cov ( p m ( a ) , p n ( a ) ) = [ P m P n exp ( ln G θ , N ) − P m exp ( ln G θ , N ) ⋅ P n exp ( ln G θ , N ) ] x = 0 = N ( N − 1 ) ( N D 2 ) n ( N D 1 ) m exp ( ln G θ , N ) x = 0 + N ( N D 1 ) m + n exp ( ln G θ , N ) x = 0 − N 2 ( N D 1 ) m exp ( ln G θ , N ) x = 0 ( N D 1 ) n exp ( ln G θ , N ) x = 0 . (18) For high-temperature scaling, we have
N − 1 Cov ( p m ( a ⃗ ) , p n ( a ⃗ ) ) = N − 1 [ P m P n exp ( ln G θ , N ) − P m exp ( ln G θ , N ) ⋅ P n exp ( ln G θ , N ) ] ∣ x ⃗ = 0 = ( N − 1 ) D 2 n D 1 m exp ( ln G θ , N ) ∣ x ⃗ = 0 + D 1 m + n exp ( ln G θ , N ) ∣ x ⃗ = 0 − N D 1 m exp ( ln G θ , N ) ∣ x ⃗ = 0 D 1 n exp ( ln G θ , N ) ∣ x ⃗ = 0 . (19) \begin{aligned}
N^{-1}\operatorname{Cov}\left(p_m(\vec{a}),p_n(\vec{a})\right)
&=N^{-1}\left[P_mP_n\exp(\ln G_{\theta,N})-P_m\exp(\ln G_{\theta,N})\cdot P_n\exp(\ln G_{\theta,N})\right]\bigg|_{\vec{x}=0}\\
&=(N-1)\mathcal{D}_2^n\mathcal{D}_1^m\exp(\ln G_{\theta,N})\bigg|_{\vec{x}=0}
+\mathcal{D}_1^{m+n}\exp(\ln G_{\theta,N})\bigg|_{\vec{x}=0}\\
&\quad-N\mathcal{D}_1^m\exp(\ln G_{\theta,N})\bigg|_{\vec{x}=0}\mathcal{D}_1^n\exp(\ln G_{\theta,N})\bigg|_{\vec{x}=0}. \tag*{(19)}
\end{aligned} N − 1 Cov ( p m ( a ) , p n ( a ) ) = N − 1 [ P m P n exp ( ln G θ , N ) − P m exp ( ln G θ , N ) ⋅ P n exp ( ln G θ , N ) ] x = 0 = ( N − 1 ) D 2 n D 1 m exp ( ln G θ , N ) x = 0 + D 1 m + n exp ( ln G θ , N ) x = 0 − N D 1 m exp ( ln G θ , N ) x = 0 D 1 n exp ( ln G θ , N ) x = 0 . (19) Let h ( x ⃗ ) h(\vec{x}) h ( x ) x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N κ \kappa κ p κ ( x ⃗ ) = ∏ i = 1 ℓ ( κ ) p κ i ( x ⃗ ) p_\kappa(\vec{x})=\prod_{i=1}^{\ell(\kappa)}p_{\kappa_i}(\vec{x}) p κ ( x ) = ∏ i = 1 ℓ ( κ ) p κ i ( x )
∂ i ( h ( x ⃗ ) p κ ( x ⃗ ) e F N , θ ( x ⃗ ) ) = [ ∂ i h ( x ⃗ ) p κ ( x ⃗ ) + h ( x ⃗ ) ∑ s = 1 ℓ ( κ ) κ s x i κ s − 1 ∏ j ≠ s p κ j ( x ⃗ ) ] ⋅ e F N , θ ( x ⃗ ) + ( ∑ d = 1 ∞ d c ( d ) x i d − 1 ) h ( x ⃗ ) p κ ( x ⃗ ) e F N , θ ( x ⃗ ) + ( ∑ λ : ℓ ( λ ) ≥ 2 c λ ∑ j = 1 ℓ ( λ ) λ j x i λ j − 1 ∏ k ≠ j p λ k ( x ⃗ ) ) h ( x ⃗ ) p κ ( x ⃗ ) e F N , θ ( x ⃗ ) , \begin{aligned}
\partial_i\left(h(\vec{x})p_\kappa(\vec{x})e^{F_{N,\theta}(\vec{x})}\right)
={}&\left[\partial_i h(\vec{x})p_\kappa(\vec{x})+h(\vec{x})\sum_{s=1}^{\ell(\kappa)}\kappa_sx_i^{\kappa_s-1}\prod_{j\ne s}p_{\kappa_j}(\vec{x})\right]\cdot e^{F_{N,\theta}(\vec{x})}\\
&+\left(\sum_{d=1}^{\infty}d c_{(d)}x_i^{d-1}\right)h(\vec{x})p_\kappa(\vec{x})e^{F_{N,\theta}(\vec{x})}\\
&+\left(\sum_{\lambda:\ell(\lambda)\ge2}c_\lambda\sum_{j=1}^{\ell(\lambda)}\lambda_jx_i^{\lambda_j-1}\prod_{k\ne j}p_{\lambda_k}(\vec{x})\right)h(\vec{x})p_\kappa(\vec{x})e^{F_{N,\theta}(\vec{x})},
\end{aligned} ∂ i ( h ( x ) p κ ( x ) e F N , θ ( x ) ) = ∂ i h ( x ) p κ ( x ) + h ( x ) s = 1 ∑ ℓ ( κ ) κ s x i κ s − 1 j = s ∏ p κ j ( x ) ⋅ e F N , θ ( x ) + ( d = 1 ∑ ∞ d c ( d ) x i d − 1 ) h ( x ) p κ ( x ) e F N , θ ( x ) + λ : ℓ ( λ ) ≥ 2 ∑ c λ j = 1 ∑ ℓ ( λ ) λ j x i λ j − 1 k = j ∏ p λ k ( x ) h ( x ) p κ ( x ) e F N , θ ( x ) , 1 − σ i j x i − x j ( h ( x ⃗ ) p κ ( x ⃗ ) e F N , θ ( x ⃗ ) ) = 1 − σ i j x i − x j ( h ( x ⃗ ) p κ ( x ⃗ ) ) e F N , θ ( x ⃗ ) , \frac{1-\sigma_{ij}}{x_i-x_j}\left(h(\vec{x})p_\kappa(\vec{x})e^{F_{N,\theta}(\vec{x})}\right)
=\frac{1-\sigma_{ij}}{x_i-x_j}\left(h(\vec{x})p_\kappa(\vec{x})\right)e^{F_{N,\theta}(\vec{x})}, x i − x j 1 − σ ij ( h ( x ) p κ ( x ) e F N , θ ( x ) ) = x i − x j 1 − σ ij ( h ( x ) p κ ( x ) ) e F N , θ ( x ) , 1 − σ i j x i − x j ( x i m i x j m j ) = { − ∑ a = m i m j − 1 x i a x j m i + m j − 1 − a , if m i < m j , ∑ a = m j m i − 1 x i a x j m i + m j − 1 − a , if m i ≥ m j . (20) \frac{1-\sigma_{ij}}{x_i-x_j}\left(x_i^{m_i}x_j^{m_j}\right)
=
\begin{cases}
-\sum_{a=m_i}^{m_j-1}x_i^a x_j^{m_i+m_j-1-a},&\text{if }m_i<m_j,\\[6pt]
\sum_{a=m_j}^{m_i-1}x_i^a x_j^{m_i+m_j-1-a},&\text{if }m_i\ge m_j.
\end{cases} \tag*{(20)} x i − x j 1 − σ ij ( x i m i x j m j ) = ⎩ ⎨ ⎧ − ∑ a = m i m j − 1 x i a x j m i + m j − 1 − a , ∑ a = m j m i − 1 x i a x j m i + m j − 1 − a , if m i < m j , if m i ≥ m j . (20) Based on the above calculation, we then introduce the following linear operators on polynomials, uniquely defined by their actions on monomials x 1 m 1 ⋯ x N m N x_1^{m_1}\cdots x_N^{m_N} x 1 m 1 ⋯ x N m N m 1 , … , m N ∈ Z ≥ 0 m_1,\ldots,m_N\in\mathbb{Z}_{\ge0} m 1 , … , m N ∈ Z ≥ 0
U i ( d ) ( x 1 m 1 ⋯ x N m N ) : = d c ( d ) x i d − 1 ⋅ x 1 m 1 ⋯ x N m N . U_i^{(d)}(x_1^{m_1}\cdots x_N^{m_N}):=d c_{(d)}x_i^{d-1}\cdot x_1^{m_1}\cdots x_N^{m_N}. U i ( d ) ( x 1 m 1 ⋯ x N m N ) := d c ( d ) x i d − 1 ⋅ x 1 m 1 ⋯ x N m N . U i ( λ ) ( x 1 m 1 ⋯ x N m N ) : = c λ ∑ j = 1 ℓ ( λ ) λ j x i λ j − 1 ∏ k ≠ j p λ k ( x ⃗ ) ⋅ x 1 m 1 ⋯ x N m N . U_i^{(\lambda)}(x_1^{m_1}\cdots x_N^{m_N}):=c_\lambda\sum_{j=1}^{\ell(\lambda)}\lambda_jx_i^{\lambda_j-1}\prod_{k\ne j}p_{\lambda_k}(\vec{x})\cdot x_1^{m_1}\cdots x_N^{m_N}. U i ( λ ) ( x 1 m 1 ⋯ x N m N ) := c λ j = 1 ∑ ℓ ( λ ) λ j x i λ j − 1 k = j ∏ p λ k ( x ) ⋅ x 1 m 1 ⋯ x N m N . In particular, if λ = ( λ 1 , λ 2 ) \lambda=(\lambda_1,\lambda_2) λ = ( λ 1 , λ 2 )
U i ( λ , 1 , k ) ( x 1 m 1 ⋯ x N m N ) : = c λ λ 1 x i λ 1 − 1 x k λ 2 ⋅ x 1 m 1 ⋯ x N m N , U_i^{(\lambda,1,k)}(x_1^{m_1}\cdots x_N^{m_N}):=c_\lambda\lambda_1x_i^{\lambda_1-1}x_k^{\lambda_2}\cdot x_1^{m_1}\cdots x_N^{m_N}, U i ( λ , 1 , k ) ( x 1 m 1 ⋯ x N m N ) := c λ λ 1 x i λ 1 − 1 x k λ 2 ⋅ x 1 m 1 ⋯ x N m N , U i ( λ , 2 , k ) ( x 1 m 1 ⋯ x N m N ) : = c λ λ 2 x i λ 2 − 1 x k λ 1 ⋅ x 1 m 1 ⋯ x N m N , U_i^{(\lambda,2,k)}(x_1^{m_1}\cdots x_N^{m_N}):=c_\lambda\lambda_2x_i^{\lambda_2-1}x_k^{\lambda_1}\cdot x_1^{m_1}\cdots x_N^{m_N}, U i ( λ , 2 , k ) ( x 1 m 1 ⋯ x N m N ) := c λ λ 2 x i λ 2 − 1 x k λ 1 ⋅ x 1 m 1 ⋯ x N m N , for k = 1 , 2 , … , N k=1,2,\ldots,N k = 1 , 2 , … , N U i ( λ ) = ∑ k = 1 N ( U i ( λ , 1 , k ) + U i ( λ , 2 , k ) ) U_i^{(\lambda)}=\sum_{k=1}^N\left(U_i^{(\lambda,1,k)}+U_i^{(\lambda,2,k)}\right) U i ( λ ) = ∑ k = 1 N ( U i ( λ , 1 , k ) + U i ( λ , 2 , k ) )
d i , j ( x 1 m 1 ⋯ x N m N ) : = { x 1 m 1 ⋯ x i − 1 m i − 1 x i m i − 1 x j m j − 1 x i + 1 m i + 1 ⋯ x N m N , m i > m j , 0 , m i ≤ m j . d_{i,j}(x_1^{m_1}\cdots x_N^{m_N})
:=
\begin{cases}
x_1^{m_1}\cdots x_{i-1}^{m_{i-1}}x_i^{m_i-1}x_j^{m_j-1}x_{i+1}^{m_{i+1}}\cdots x_N^{m_N},&m_i>m_j,\\
0,&m_i\le m_j.
\end{cases} d i , j ( x 1 m 1 ⋯ x N m N ) := { x 1 m 1 ⋯ x i − 1 m i − 1 x i m i − 1 x j m j − 1 x i + 1 m i + 1 ⋯ x N m N , 0 , m i > m j , m i ≤ m j . For 1 ≤ i ≤ N 1 \le i \le N 1 ≤ i ≤ N 1 ≤ j ≤ N 1 \le j \le N 1 ≤ j ≤ N j ≠ i j \ne i j = i a ∈ Z ≥ 1 a \in\mathbb{Z}_{\ge1} a ∈ Z ≥ 1
d ~ i , j ( a ) ( x 1 n 1 ⋯ x N n N ) : = { x 1 n 1 ⋯ x N n N x i − a x j a − 1 , if n i > n j , 1 < a ≤ n i − n j , − x 1 n 1 ⋯ x N n N x i a − 1 x j − a , if n i < n j , 1 ≤ a ≤ n j − n i , 0 , otherwise. (21) \tilde{d}_{i,j}^{(a)}(x_1^{n_1}\cdots x_N^{n_N}) :=
\begin{cases}
x_1^{n_1}\cdots x_N^{n_N}x_i^{-a}x_j^{a-1}, & \text{if } n_i>n_j,\ 1<a\le n_i-n_j,\\
-x_1^{n_1}\cdots x_N^{n_N}x_i^{a-1}x_j^{-a}, & \text{if } n_i<n_j,\ 1\le a\le n_j-n_i,\\
0, & \text{otherwise.}
\end{cases} \tag*{(21)} d ~ i , j ( a ) ( x 1 n 1 ⋯ x N n N ) := ⎩ ⎨ ⎧ x 1 n 1 ⋯ x N n N x i − a x j a − 1 , − x 1 n 1 ⋯ x N n N x i a − 1 x j − a , 0 , if n i > n j , 1 < a ≤ n i − n j , if n i < n j , 1 ≤ a ≤ n j − n i , otherwise. (21) ∂ i ( x 1 m 1 ⋯ x N m N ) = m i ⋅ d i ( x 1 m 1 ⋯ x N m N ) , \partial_i(x_1^{m_1}\cdots x_N^{m_N})=m_i\cdot d_i(x_1^{m_1}\cdots x_N^{m_N}), ∂ i ( x 1 m 1 ⋯ x N m N ) = m i ⋅ d i ( x 1 m 1 ⋯ x N m N ) , where
d i ( x 1 m 1 ⋯ x N m N ) : = 1 { m i ≥ 1 } ⋅ x 1 m 1 ⋯ x i − 1 m i − 1 x i m i − 1 x i + 1 m i + 1 ⋯ x N m N . d_i(x_1^{m_1}\cdots x_N^{m_N}):=\mathbf{1}_{\{m_i\ge1\}}\cdot x_1^{m_1}\cdots x_{i-1}^{m_{i-1}}x_i^{m_i-1}x_{i+1}^{m_{i+1}}\cdots x_N^{m_N}. d i ( x 1 m 1 ⋯ x N m N ) := 1 { m i ≥ 1 } ⋅ x 1 m 1 ⋯ x i − 1 m i − 1 x i m i − 1 x i + 1 m i + 1 ⋯ x N m N . Then for a partition4 μ = ( μ 1 , … , μ s ) \mu=(\mu_1,\ldots,\mu_s) μ = ( μ 1 , … , μ s )
[ D s μ s ⋯ D 2 μ 2 D 1 μ 1 ] e F N , θ ( x ⃗ ) ∣ x ⃗ = ( 0 N ) = ( ∑ d U s ( d ) + ∑ λ = ( λ 1 , λ 2 ) ∑ k = 1 N ( U s ( λ , 1 , k ) + U s ( λ , 2 , k ) ) + θ ∑ j ≠ s d s , j + θ ∑ j ≠ s ∑ a ≥ 1 d ~ s , j ( a ) + ∂ s + ∑ ℓ ( λ ) ≥ 3 U s ( λ ) ) μ s \left[D_s^{\mu_s}\cdots D_2^{\mu_2}D_1^{\mu_1}\right]e^{F_{N,\theta}(\vec{x})}\bigg|_{\vec{x}=(0^N)}
=
\left(\sum_d U_s^{(d)}+\sum_{\lambda=(\lambda_1,\lambda_2)}\sum_{k=1}^{N}\left(U_s^{(\lambda,1,k)}+U_s^{(\lambda,2,k)}\right)+\theta\sum_{j\ne s}d_{s,j}+\theta\sum_{j\ne s}\sum_{a\ge1}\tilde{d}_{s,j}^{(a)}+\partial_s+\sum_{\ell(\lambda)\ge3}U_s^{(\lambda)}\right)^{\mu_s} [ D s μ s ⋯ D 2 μ 2 D 1 μ 1 ] e F N , θ ( x ) x = ( 0 N ) = d ∑ U s ( d ) + λ = ( λ 1 , λ 2 ) ∑ k = 1 ∑ N ( U s ( λ , 1 , k ) + U s ( λ , 2 , k ) ) + θ j = s ∑ d s , j + θ j = s ∑ a ≥ 1 ∑ d ~ s , j ( a ) + ∂ s + ℓ ( λ ) ≥ 3 ∑ U s ( λ ) μ s ( ∑ d U 1 ( d ) + ∑ λ = ( λ 1 , λ 2 ) ∑ k = 1 N ( U 1 ( λ , 1 , k ) + U 1 ( λ , 2 , k ) ) + θ ∑ j ≠ 1 d 1 , j + θ ∑ j ≠ 1 ∑ a ≥ 1 d ~ 1 , j ( a ) + ∂ 1 + ∑ ℓ ( λ ) ≥ 3 U 1 ( λ ) ) μ 1 ∣ x ⃗ = ( 0 N ) (22) \qquad(\sum_d U_1^{(d)}+\sum_{\lambda=(\lambda_1,\lambda_2)}\sum_{k=1}^{N}(U_1^{(\lambda,1,k)}+U_1^{(\lambda,2,k)})+\theta\sum_{j\ne1}d_{1,j}+\theta\sum_{j\ne1}\sum_{a\ge1}\tilde{d}_{1,j}^{(a)}+\partial_1+\sum_{\ell(\lambda)\ge3}U_1^{(\lambda)})^{\mu_1}|_{\vec{x}=(0^N)} \tag*{(22)} ( d ∑ U 1 ( d ) + λ = ( λ 1 , λ 2 ) ∑ k = 1 ∑ N ( U 1 ( λ , 1 , k ) + U 1 ( λ , 2 , k ) ) + θ j = 1 ∑ d 1 , j + θ j = 1 ∑ a ≥ 1 ∑ d ~ 1 , j ( a ) + ∂ 1 + ℓ ( λ ) ≥ 3 ∑ U 1 ( λ ) ) μ 1 ∣ x = ( 0 N ) (22) = ∑ T ( t ) : 1 ≤ t ≤ ∣ μ ∣ T ( ∣ μ ∣ ) ⋯ T ( 1 ) ( 1 ) ∣ x ⃗ = ( 0 N ) , (23) =\left.\sum_{T(t):1\le t\le|\mu|}T(|\mu|)\cdots T(1)(1)\right|_{\vec{x}=(0^N)}, \tag*{(23)} = T ( t ) : 1 ≤ t ≤ ∣ μ ∣ ∑ T ( ∣ μ ∣ ) ⋯ T ( 1 ) ( 1 ) x = ( 0 N ) , (23) where the summation in the last line is over the finitely many choices of the operators T ( t ) T(t) T ( t ) T ( t ) T(t) T ( t ) U i ( d ) U_i^{(d)} U i ( d ) U i ( λ , j , k ) U_i^{(\lambda,j,k)} U i ( λ , j , k ) ℓ ( λ ) = 2 \ell(\lambda)=2 ℓ ( λ ) = 2 θ ⋅ d i , j \theta\cdot d_{i,j} θ ⋅ d i , j θ ⋅ d ~ i , j ( a ) \theta\cdot\tilde{d}_{i,j}^{(a)} θ ⋅ d ~ i , j ( a ) ∂ i \partial_i ∂ i U i ( λ ) U_i^{(\lambda)} U i ( λ ) ℓ ( λ ) ≥ 3 \ell(\lambda)\ge3 ℓ ( λ ) ≥ 3 ∑ j = 1 i − 1 μ j + 1 ≤ t ≤ ∑ j = 1 i μ j \sum_{j=1}^{i-1}\mu_j+1\le t\le\sum_{j=1}^{i}\mu_j ∑ j = 1 i − 1 μ j + 1 ≤ t ≤ ∑ j = 1 i μ j
In the order displayed, these are the following six types:
I U i ( d ) I I U i ( λ , j , k ) , ℓ ( λ ) = 2 I I I θ d i , j I V θ d ~ i , j ( a ) V ∂ i V I U i ( λ ) , ℓ ( λ ) ≥ 3. \begin{array}{c|l}
\mathrm{I} & U_i^{(d)}\\
\mathrm{II} & U_i^{(\lambda,j,k)},\ \ell(\lambda)=2\\
\mathrm{III} & \theta d_{i,j}\\
\mathrm{IV} & \theta\tilde{d}_{i,j}^{(a)}\\
\mathrm{V} & \partial_i\\
\mathrm{VI} & U_i^{(\lambda)},\ \ell(\lambda)\ge3.
\end{array} I II III IV V VI U i ( d ) U i ( λ , j , k ) , ℓ ( λ ) = 2 θ d i , j θ d ~ i , j ( a ) ∂ i U i ( λ ) , ℓ ( λ ) ≥ 3. We denote the corresponding classes by J I , … , J V I J_{\mathrm{I}},\ldots,J_{\mathrm{VI}} J I , … , J VI U i ( λ ) U_i^{(\lambda)} U i ( λ ) ℓ ( λ ) = 2 \ell(\lambda)=2 ℓ ( λ ) = 2
In this way, we may view each term in the sum as a discrete-time process of polynomials in x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N P ( 0 ) = 1 P(0)=1 P ( 0 ) = 1 t > 0 t>0 t > 0 P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) P(t)=T(t)\cdots T(1)(1) P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) t = 1 , 2 , … , ∣ μ ∣ t=1,2,\ldots,|\mu| t = 1 , 2 , … , ∣ μ ∣ T ( t ) T(t) T ( t ) P ( t ) P(t) P ( t )
Next, we introduce a way to visualize each monomial sequence P ( t ) P(t) P ( t ) t = 1 , 2 , … , ∣ μ ∣ t=1,2,\ldots,|\mu| t = 1 , 2 , … , ∣ μ ∣ T ( t ) T(t) T ( t )
Definition 4.1. For N ∈ Z ≥ 1 N \in\mathbb{Z}_{\ge1} N ∈ Z ≥ 1 let P ( x ⃗ ) = c ⋅ x 1 n 1 ⋯ x N n N P(\vec{x})=c\cdot x_1^{n_1}\cdots x_N^{n_N} P ( x ) = c ⋅ x 1 n 1 ⋯ x N n N be a monomial in x 1 , … , x N x_1,\ldots,x_N x 1 , … , x N We denote
Deg ( P ) : = ∑ i = 1 N n i , Deg i ( P ) : = n i for i = 1 , 2 , … , N . \operatorname{Deg}(P):=\sum_{i=1}^{N}n_i,\qquad\operatorname{Deg}_i(P):=n_i\quad\text{for }i=1,2,\ldots,N. Deg ( P ) := i = 1 ∑ N n i , Deg i ( P ) := n i for i = 1 , 2 , … , N . Observe that after the action of T ( t ) T(t) T ( t ) t ∈ ⟦ 1 , ∣ μ ∣ ⟧ t\in\llbracket1,\lvert\mu\rvert\rrbracket t ∈ [ [ 1 , ∣ μ ∣ ] ] if P ( t ) ≠ 0 P(t)\ne0 P ( t ) = 0 the degree has either
increased by d − 1 d-1 d − 1 T ( t ) = U i ( d ) T(t)=U_i^{(d)} T ( t ) = U i ( d ) U i ( λ ) U_i^{(\lambda)} U i ( λ ) i i i d ≥ 1 d\ge1 d ≥ 1 λ \lambda λ ∣ λ ∣ = d \lvert\lambda\rvert=d ∣ λ ∣ = d
decreased by 1 1 1
Definition 4.2. Fix N ∈ Z ≥ 1 N\in\mathbb{Z}_{\ge1} N ∈ Z ≥ 1 and a partition μ \mu μ of length 1 1 1 or 2 2 2 For each sequence of monomials P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) P(t)=T(t)\cdots T(1)(1) P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) t = 1 , 2 , … , ∣ μ ∣ t=1,2,\ldots,\lvert\mu\rvert t = 1 , 2 , … , ∣ μ ∣ such that P ( ∣ μ ∣ ) ∣ x ⃗ = 0 ≠ 0 \left.P(\lvert\mu\rvert)\right|_{\vec{x}=0}\ne0 P (∣ μ ∣) ∣ x = 0 = 0 let E ( t ) : [ 0 , ∣ μ ∣ ] → Z ≥ 0 E(t):[0,\lvert\mu\rvert]\to\mathbb{Z}_{\ge0} E ( t ) : [ 0 , ∣ μ ∣] → Z ≥ 0 be a càdlàg function defined by
E ( t ) = Deg ( P ( ⌊ t ⌋ ) ) . (24) E(t)=\operatorname{Deg}(P(\lfloor t\rfloor)). \tag*{(24)} E ( t ) = Deg ( P (⌊ t ⌋)) . (24) We refer to E ( t ) E(t) E ( t ) as a walk.
If μ = ( k ) \mu=(k) μ = ( k ) we regard the walk as an element of L ( k ) L(k) L ( k ) If μ = ( m , n ) \mu=(m,n) μ = ( m , n ) we regard it as an element of L ~ ( m , n ) \widetilde{L}(m,n) L ( m , n ) with a distinguished splitting point at time m m m If μ = ( m + n ) \mu=(m+n) μ = ( m + n ) we regard it as an element of L ( m + n ) L(m+n) L ( m + n ) where no splitting point is part of the data.
While the walk records the total degree, we also need to keep track of the degree of each individual variable. Due to the actions of the operators d ~ i , j ( a ) \widetilde{d}_{i,j}^{(a)} d i , j ( a ) each Deg i ( P ( t ) ) \operatorname{Deg}_i(P(t)) Deg i ( P ( t )) changes in certain nontrivial ways, which we record by a collection of blocks q i ( t ) q_i(t) q i ( t ) More precisely, we have the following.
Definition 4.3. Fix N ∈ Z ≥ 1 N\in\mathbb{Z}_{\ge1} N ∈ Z ≥ 1 and a partition μ \mu μ of length 1 1 1 or 2 2 2 For ℓ ( μ ) = 1 \ell(\mu)=1 ℓ ( μ ) = 1 we define μ 2 = 0 \mu_2=0 μ 2 = 0 For each sequence of monomials P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) P(t)=T(t)\cdots T(1)(1) P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) t = 1 , 2 , … , ∣ μ ∣ t=1,2,\ldots,\lvert\mu\rvert t = 1 , 2 , … , ∣ μ ∣ let q j ( t ) : [ 0 , ∣ μ ∣ ] → Z ≥ 0 q_j(t):[0,\lvert\mu\rvert]\to\mathbb{Z}_{\ge0} q j ( t ) : [ 0 , ∣ μ ∣] → Z ≥ 0 be a piecewise function, which is càdlàg on [ 0 , ∣ μ ∣ ] ∖ { 0 , μ 1 , … , ∣ μ ∣ } [0,\lvert\mu\rvert]\setminus\{0,\mu_1,\ldots,\lvert\mu\rvert\} [ 0 , ∣ μ ∣] ∖ { 0 , μ 1 , … , ∣ μ ∣} defined by
q j ( t ) = { Deg j ( P ( ⌊ t ⌋ ) ) , j ≥ 3 , t ∈ [ 0 , ∣ μ ∣ ] , Deg j ( P ( ⌊ t ⌋ ) ) , j = 1 , t ∈ [ 0 , ∣ μ ∣ ] ∖ [ 0 , μ 1 ] , Deg j ( P ( ⌊ t ⌋ ) ) , j = 2 , t ∈ [ 0 , ∣ μ ∣ ] ∖ ( μ 1 , μ 1 + μ 2 ] , 0 , otherwise . (25) q_j(t)=
\begin{cases}
\operatorname{Deg}_j(P(\lfloor t\rfloor)), & j\ge3,\ t\in[0,\lvert\mu\rvert],\\
\operatorname{Deg}_j(P(\lfloor t\rfloor)), & j=1,\ t\in[0,\lvert\mu\rvert]\setminus[0,\mu_1],\\
\operatorname{Deg}_j(P(\lfloor t\rfloor)), & j=2,\ t\in[0,\lvert\mu\rvert]\setminus(\mu_1,\mu_1+\mu_2],\\
0, & \text{otherwise}.
\end{cases} \tag*{(25)} q j ( t ) = ⎩ ⎨ ⎧ Deg j ( P (⌊ t ⌋)) , Deg j ( P (⌊ t ⌋)) , Deg j ( P (⌊ t ⌋)) , 0 , j ≥ 3 , t ∈ [ 0 , ∣ μ ∣] , j = 1 , t ∈ [ 0 , ∣ μ ∣] ∖ [ 0 , μ 1 ] , j = 2 , t ∈ [ 0 , ∣ μ ∣] ∖ ( μ 1 , μ 1 + μ 2 ] , otherwise . (25) Let
q ( t ) = ∑ j = 1 N q j ( t ) , t ∈ [ 0 , ∣ μ ∣ ] . q(t)=\sum_{j=1}^{N}q_j(t),\qquad t\in[0,\lvert\mu\rvert]. q ( t ) = j = 1 ∑ N q j ( t ) , t ∈ [ 0 , ∣ μ ∣] . We call q ⃗ ( t ) = ( q 1 ( t ) , … , q N ( t ) ) \vec{q}(t)=(q_1(t),\ldots,q_N(t)) q ( t ) = ( q 1 ( t ) , … , q N ( t )) the blocks of { P ( t ) } t = 0 ∣ μ ∣ \{P(t)\}_{t=0}^{\lvert\mu\rvert} { P ( t ) } t = 0 ∣ μ ∣
We call ( E , q ⃗ ) (E,\vec{q}) ( E , q ) the configuration of the monomial sequence. During the i i i th interval ∑ j < i μ j < t ≤ ∑ j ≤ i μ j \sum_{j<i}\mu_j<t\le\sum_{j\le i}\mu_j ∑ j < i μ j < t ≤ ∑ j ≤ i μ j the main variable is x i x_i x i The path records total degree, while the blocks record how this degree is distributed among variables that are not currently main. Recall that P ( E , q ⃗ ) = D ( E ) ∖ J ( E , q ⃗ ) P(E,\vec{q})=D(E)\setminus J(E,\vec{q}) P ( E , q ) = D ( E ) ∖ J ( E , q ) is the set of down-step times that are not block jumps. At each such time, the same decrease of total degree may be produced either by Type III or by Type V. Thus, after auxiliary labels are fixed, ( E , q ⃗ ) (E,\vec{q}) ( E , q ) together with the subset t ( ∂ ) ⊂ P ( E , q ⃗ ) t(\partial)\subset P(E,\vec{q}) t ( ∂ ) ⊂ P ( E , q ) of Type V times determines the monomial branch; the complementary times in P ( E , q ⃗ ) P(E,\vec{q}) P ( E , q ) are Type III times.
The right discontinuities at the splitting point merely record the change of main variable. The left discontinuities caused by operator actions are of the following two types. If the discontinuity
is given by an operator d ~ i , j ( a ) \widetilde{d}_{i,j}^{(a)} d i , j ( a ) q j ( t ) q_j(t) q j ( t ) a − 1 a-1 a − 1 a ≥ 2 a \ge2 a ≥ 2 a a a a ≥ 1 a \ge1 a ≥ 1 E ( t ) E(t) E ( t ) jump of x j x_j x j U i ( λ , 1 , k ) U_i^{(\lambda,1,k)} U i ( λ , 1 , k ) U i ( λ , 2 , k ) U_i^{(\lambda,2,k)} U i ( λ , 2 , k ) k ≠ i k \ne i k = i q k ( t ) − q k ( t − ) = λ 2 q_k(t)-q_k(t-) = \lambda_2 q k ( t ) − q k ( t − ) = λ 2 λ 1 \lambda_1 λ 1 nontrivial up step .
Remark 4.4. For i ≠ 2 i \ne2 i = 2 jump of each x i x_i x i q i ( t ) q_i(t) q i ( t ) (21) , E ( t − ) = q ( t − ) E(t-) = q(t-) E ( t − ) = q ( t − )
For a partition μ = ( μ 1 , … , μ ℓ ( μ ) ) \mu= (\mu_1,\ldots,\mu_{\ell(\mu)}) μ = ( μ 1 , … , μ ℓ ( μ ) ) i = ( i 1 , … , i ℓ ( μ ) ) \mathbf{i} = (i_1,\ldots,i_{\ell(\mu)}) i = ( i 1 , … , i ℓ ( μ ) ) P ( μ ; i ) \mathcal{P}(\mu;\mathbf{i}) P ( μ ; i )
P ⃗ = { P ( t ) } t = 0 ∣ μ ∣ , P ( 0 ) = 1 , P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) , \vec{P} = \{P(t)\}_{t=0}^{|\mu|}, \qquad P(0)=1,\qquad P(t)=T(t)\cdots T(1)(1), P = { P ( t ) } t = 0 ∣ μ ∣ , P ( 0 ) = 1 , P ( t ) = T ( t ) ⋯ T ( 1 ) ( 1 ) , obtained from the operator expansion above, where the main variable is x i r x_{i_r} x i r
μ 1 + ⋯ + μ r − 1 < t ≤ μ 1 + ⋯ + μ r . \mu_1+\cdots+\mu_{r-1}<t\le\mu_1+\cdots+\mu_r. μ 1 + ⋯ + μ r − 1 < t ≤ μ 1 + ⋯ + μ r . In particular, introduce the three collections
P m , n 12 : = P ( ( m , n ) ; ( 1 , 2 ) ) , P m + n 1 : = P ( ( m + n ) ; ( 1 ) ) , \mathcal{P}_{m,n}^{12}:=\mathcal{P}((m,n);(1,2)),\qquad\mathcal{P}_{m+n}^{1}:=\mathcal{P}((m+n);(1)), P m , n 12 := P (( m , n ) ; ( 1 , 2 )) , P m + n 1 := P (( m + n ) ; ( 1 )) , and
P m , n × : = P ( ( m ) ; ( 1 ) ) × P ( ( n ) ; ( 1 ) ) . \mathcal{P}_{m,n}^{\times}:=\mathcal{P}((m);(1))\times\mathcal{P}((n);(1)). P m , n × := P (( m ) ; ( 1 )) × P (( n ) ; ( 1 )) . We regard these as disjoint collections. For an element in one of these collections, define its signed covariance weight by
wt N ( P ⃗ ) : = { N ( N − 1 ) N m + n P ( m + n ) ∣ x ⃗ = 0 , P ⃗ ∈ P m , n 12 , N N m + n P ( m + n ) ∣ x ⃗ = 0 , P ⃗ ∈ P m + n 1 , − N 2 N m + n P ( 1 ) ( m ) ∣ x ⃗ = 0 P ( 2 ) ( n ) ∣ x ⃗ = 0 , P ⃗ = ( P ⃗ ( 1 ) , P ⃗ ( 2 ) ) ∈ P m , n × . \operatorname{wt}_{N}(\vec{P}):=
\begin{cases}
\left.\dfrac{N(N-1)}{N^{m+n}}P(m+n)\right|_{\vec{x}=0}, & \vec{P}\in\mathcal{P}_{m,n}^{12},\\[6pt]
\left.\dfrac{N}{N^{m+n}}P(m+n)\right|_{\vec{x}=0}, & \vec{P}\in\mathcal{P}_{m+n}^{1},\\[6pt]
-\left.\dfrac{N^2}{N^{m+n}}P^{(1)}(m)\right|_{\vec{x}=0}\left.P^{(2)}(n)\right|_{\vec{x}=0}, & \vec{P}=(\vec{P}^{(1)},\vec{P}^{(2)})\in\mathcal{P}_{m,n}^{\times}.
\end{cases} wt N ( P ) := ⎩ ⎨ ⎧ N m + n N ( N − 1 ) P ( m + n ) x = 0 , N m + n N P ( m + n ) x = 0 , − N m + n N 2 P ( 1 ) ( m ) x = 0 P ( 2 ) ( n ) x = 0 , P ∈ P m , n 12 , P ∈ P m + n 1 , P = ( P ( 1 ) , P ( 2 ) ) ∈ P m , n × . Proposition 4.5. Let P m , n > 0 \mathcal{P}_{m,n}^{>0} P m , n > 0 P ⃗ ∈ P m , n 12 ⊔ P m + n 1 \vec{P}\in\mathcal{P}_{m,n}^{12}\sqcup\mathcal{P}_{m+n}^{1} P ∈ P m , n 12 ⊔ P m + n 1
none of the operators occurring in P ⃗ \vec{P} P
the associated walk-block configuration ( E , q ⃗ ) (E,\vec{q}) ( E , q )
Let P m , n V I \mathcal{P}_{m,n}^{\mathrm{VI}} P m , n VI
P m , n 12 ⊔ P m + n 1 ⊔ P m , n × \mathcal{P}_{m,n}^{12}\sqcup\mathcal{P}_{m+n}^{1}\sqcup\mathcal{P}_{m,n}^{\times} P m , n 12 ⊔ P m + n 1 ⊔ P m , n × which contain at least one Type VI operator; in the product collection, this means that at least one of the two operator sequences contains a Type VI operator. Then
Cov ( p ~ m ( a ⃗ ) , p ~ n ( a ⃗ ) ) = ∑ P ⃗ ∈ P m , n > 0 wt N ( P ⃗ ) + ∑ P ⃗ ∈ P m , n V I wt N ( P ⃗ ) . (26) \operatorname{Cov}\bigl(\widetilde{p}_m(\vec{a}),\widetilde{p}_n(\vec{a})\bigr)
=
\sum_{\vec{P}\in\mathcal{P}_{m,n}^{>0}}\operatorname{wt}_{N}(\vec{P})
+
\sum_{\vec{P}\in\mathcal{P}_{m,n}^{\mathrm{VI}}}\operatorname{wt}_{N}(\vec{P}). \tag*{(26)} Cov ( p m ( a ) , p n ( a ) ) = P ∈ P m , n > 0 ∑ wt N ( P ) + P ∈ P m , n VI ∑ wt N ( P ) . (26) Proof. Expanding each Dunkl action in Eqn. (18) in terms of the operators of Types I–VI gives
Cov ( p ~ m ( a ⃗ ) , p ~ n ( a ⃗ ) ) = N ( N − 1 ) N m + n ∑ P ⃗ ∈ P m , n 12 P ( m + n ) ∣ x ⃗ = 0 + N N m + n ∑ P ⃗ ∈ P m + n 1 P ( m + n ) ∣ x ⃗ = 0 − N 2 N m + n ∑ P ⃗ ( 1 ) ∈ P ( m ) ; ( 1 ) P ⃗ ( 2 ) ∈ P ( n ) ; ( 1 ) P ( 1 ) ( m ) ∣ x ⃗ = 0 P ( 2 ) ( n ) ∣ x ⃗ = 0 . \operatorname{Cov}\left(\widetilde{p}_m(\vec{a}),\widetilde{p}_n(\vec{a})\right)
= \frac{N(N-1)}{N^{m+n}} \sum_{\vec{P}\in\mathcal{P}^{12}_{m,n}} P(m+n)\bigg|_{\vec{x}=0}
+ \frac{N}{N^{m+n}} \sum_{\vec{P}\in\mathcal{P}^{1}_{m+n}} P(m+n)\bigg|_{\vec{x}=0}
- \frac{N^2}{N^{m+n}} \sum_{\substack{\vec{P}^{(1)}\in\mathcal{P}_{(m);(1)}\\\vec{P}^{(2)}\in\mathcal{P}_{(n);(1)}}} P^{(1)}(m)\bigg|_{\vec{x}=0}P^{(2)}(n)\bigg|_{\vec{x}=0}. Cov ( p m ( a ) , p n ( a ) ) = N m + n N ( N − 1 ) P ∈ P m , n 12 ∑ P ( m + n ) x = 0 + N m + n N P ∈ P m + n 1 ∑ P ( m + n ) x = 0 − N m + n N 2 P ( 1 ) ∈ P ( m ) ; ( 1 ) P ( 2 ) ∈ P ( n ) ; ( 1 ) ∑ P ( 1 ) ( m ) x = 0 P ( 2 ) ( n ) x = 0 . It remains to identify the cancellation of the terms without Type VI operators whose associated walk satisfies E ( m ) = 0 E(m)=0 E ( m ) = 0 P ( m ) P(m) P ( m ) m m m m m m n n n
Conversely, every ordered pair of operator sequences of lengths m m m n n n P m , n 12 \mathcal{P}^{12}_{m,n} P m , n 12 E ( m ) = 0 E(m)=0 E ( m ) = 0 P m + n 1 \mathcal{P}^{1}_{m+n} P m + n 1 E ( m ) = 0 E(m)=0 E ( m ) = 0 x 1 x_1 x 1 x 2 x_2 x 2
N ( N − 1 ) + N − N 2 = 0. N(N-1)+N-N^2=0. N ( N − 1 ) + N − N 2 = 0. Hence all terms without Type VI operators and with E ( m ) = 0 E(m)=0 E ( m ) = 0
The first sum in Eqn. (25) is the configuration contribution that will be analyzed below. The second sum is retained for the moment as an error term. We will prove separately that, under the assumptions of the corresponding fixed-temperature analysis,
∑ P ⃗ ∈ P m , n V I wt N ( P ⃗ ) ∼ o ( 1 ) , N → ∞ . \sum_{\vec{P}\in\mathcal{P}^{\mathrm{VI}}_{m,n}}\operatorname{wt}_{N}(\vec{P})\sim o(1),\qquad N\to\infty. P ∈ P m , n VI ∑ wt N ( P ) ∼ o ( 1 ) , N → ∞. Thus, operator sequences containing a Type VI action do not contribute to the limiting covariance.
Fixed-temperature covariance in Theorem 3.12
We now analyze the three signed sums in Proposition 4.5 under the fixed-temperature assumptions. Lemma 4.6 removes Type VI and all but at most one grouped Type II action. The contribution with no Type II action yields the polynomial in first-order free cumulants; the contribution with one Type II action yields the term involving one second-order free cumulant.
We assume that for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\geq1} d ∈ Z ≥ 1 k 1 , k 2 ∈ Z ≥ 1 k_1,k_2\in\mathbb{Z}_{\geq1} k 1 , k 2 ∈ Z ≥ 1 N → ∞ N\to\infty N → ∞
d N c ( d ) → θ 1 − d κ d ; \frac{d}{N}c_{(d)}\to\theta^{1-d}\kappa_d; N d c ( d ) → θ 1 − d κ d ; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; k_1k_2(1+\mathbf{1}_{k_1=k_2})c_{(k_1,k_2)}\to\theta^{1-k_1-k_2}\kappa_{k_1,k_2}; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; c λ → 0 for ℓ ( λ ) ≥ 3. (27) c_{\lambda}\to0\qquad\text{for }\ell(\lambda)\geq3. \tag*{(27)} c λ → 0 for ℓ ( λ ) ≥ 3. (27) The first observation concerns scaling. In the fixed-temperature regime θ \theta θ N N N D i N \frac{D_i}{N} N D i N − 1 N^{-1} N − 1
Lemma 4.6. In the fixed-temperature regime, the following statements hold for the operator expansion of the covariance in Eqn. (18).
(a) The total weight of all operator strings containing at least one Type VI operator is o ( 1 ) o(1) o ( 1 ) N → ∞ N \to\infty N → ∞
(b) After excluding operator strings containing Type VI operators, the total weight of all strings containing at least two Type II operators is o ( 1 ) o(1) o ( 1 ) N → ∞ N \to\infty N → ∞
Proof. The argument is carried out after grouping together all the monomial terms arising from the same operator U i ( λ ) U_i^{(\lambda)} U i ( λ ) ℓ ( λ ) = 2 \ell(\lambda) = 2 ℓ ( λ ) = 2
U i ( λ ) = ∑ k = 1 N ( U i ( λ , 1 , k ) + U i ( λ , 2 , k ) ) . U_i^{(\lambda)} = \sum_{k=1}^{N} \left(U_i^{(\lambda,1,k)} + U_i^{(\lambda,2,k)}\right). U i ( λ ) = k = 1 ∑ N ( U i ( λ , 1 , k ) + U i ( λ , 2 , k ) ) . Thus, the assertions below concern the total weight of the corresponding grouped terms, rather than each individual monomial sequence.
Let λ \lambda λ
r : = ℓ ( λ ) ≥ 2. r := \ell(\lambda) \ge2. r := ℓ ( λ ) ≥ 2. Each summand in U i ( λ ) U_i^{(\lambda)} U i ( λ )
c λ λ j x i λ j − 1 ∏ 1 ≤ k < r k ≠ j p λ k ( x ⃗ ) , c_\lambda\lambda_j x_i^{\lambda_j-1} \prod_{\substack{1 \leq k < r \\ k \neq j}} p_{\lambda_k}(\vec{x}), c λ λ j x i λ j − 1 1 ≤ k < r k = j ∏ p λ k ( x ) , and hence introduces r − 1 r - 1 r − 1
After the action of U i ( λ ) U_i^{(\lambda)} U i ( λ ) S ( x ⃗ ) S(\vec{x}) S ( x ) S S S
1 − σ a , b x a − x b ( h ( x ⃗ ) S ( x ⃗ ) ) = S ( x ⃗ ) 1 − σ a , b x a − x b h ( x ⃗ ) . \frac{1-\sigma_{a,b}}{x_a-x_b}\left(h(\vec{x})S(\vec{x})\right)
=
S(\vec{x})\frac{1-\sigma_{a,b}}{x_a-x_b}h(\vec{x}). x a − x b 1 − σ a , b ( h ( x ) S ( x ) ) = S ( x ) x a − x b 1 − σ a , b h ( x ) . Consequently, the difference part of a subsequent Dunkl operator passes all the power-sum factors through unchanged. On the other hand,
∂ a ( h ( x ⃗ ) S ( x ⃗ ) ) = ∂ a h ( x ⃗ ) S ( x ⃗ ) + h ( x ⃗ ) ∂ a S ( x ⃗ ) , \partial_a\left(h(\vec{x})S(\vec{x})\right)
=
\partial_a h(\vec{x}) S(\vec{x}) + h(\vec{x})\partial_a S(\vec{x}), ∂ a ( h ( x ) S ( x ) ) = ∂ a h ( x ) S ( x ) + h ( x ) ∂ a S ( x ) , and, in each term of the Leibniz expansion of ∂ a S ( x ⃗ ) \partial_a S(\vec{x}) ∂ a S ( x )
Any power-sum factor that is never acted on by a partial derivative remains present until the end and makes the resulting expression vanish after setting x ⃗ = 0 \vec{x}=0 x = 0 x ⃗ = 0 \vec{x}=0 x = 0 r − 1 r - 1 r − 1 r − 1 r - 1 r − 1 U i ( λ ) U_i^{(\lambda)} U i ( λ )
More generally, if an operator string contains
U i 1 ( λ ( 1 ) ) , … , U i s ( λ ( s ) ) , U_{i_1}^{(\lambda^{(1)})}, \ldots, U_{i_s}^{(\lambda^{(s)})}, U i 1 ( λ ( 1 ) ) , … , U i s ( λ ( s ) ) , then every nonzero contribution contains at least
∑ a = 1 s ( ℓ ( λ ( a ) ) − 1 ) \sum_{a=1}^{s} \left(\ell(\lambda^{(a)}) - 1\right) a = 1 ∑ s ( ℓ ( λ ( a ) ) − 1 ) Type V actions.
We now combine this observation with the fixed-temperature scaling. Each occurrence of U i ( λ ) U_i^{(\lambda)} U i ( λ ) D i / N D_i/N D i / N N − 1 N^{-1} N − 1 ℓ ( λ ) = 2 \ell(\lambda)=2 ℓ ( λ ) = 2 O ( 1 ) O(1) O ( 1 ) ℓ ( λ ) ≥ 3 \ell(\lambda)\geq3 ℓ ( λ ) ≥ 3 o ( 1 ) o(1) o ( 1 ) (13) . Each required Type V action contributes one further factor O ( N − 1 ) O(N^{-1}) O ( N − 1 ) O ( 1 ) O(1) O ( 1 ) (18) are of order at most N 2 N^2 N 2
Suppose first that an operator string contains a Type VI operator U i ( λ ) U_i^{(\lambda)} U i ( λ ) r = ℓ ( λ ) ≥ 3 r=\ell(\lambda)\geq3 r = ℓ ( λ ) ≥ 3 r − 1 r-1 r − 1
o ( N − 1 ) O ( N − ( r − 1 ) ) ∼ o ( N − r ) . o(N^{-1})O(N^{-(r-1)})\sim o(N^{-r}). o ( N − 1 ) O ( N − ( r − 1 ) ) ∼ o ( N − r ) . After multiplication by the exterior O ( N 2 ) O(N^2) O ( N 2 )
o ( N 2 − r ) ∼ o ( 1 ) . o(N^{2-r})\sim o(1). o ( N 2 − r ) ∼ o ( 1 ) . Additional Type II, Type V, or Type VI actions can only decrease the order. This proves part (a).
Next, suppose that an operator string contains s s s s s s
O ( N − s ) O ( N − s ) ∼ O ( N − 2 s ) . O(N^{-s})O(N^{-s})\sim O(N^{-2s}). O ( N − s ) O ( N − s ) ∼ O ( N − 2 s ) . After multiplication by the exterior O ( N 2 ) O(N^2) O ( N 2 )
O ( N 2 − 2 s ) , O(N^{2-2s}), O ( N 2 − 2 s ) , which tends to zero whenever s ≥ 2 s\geq2 s ≥ 2
If there is exactly one Type II operator, the preceding argument forces at least one Type V action. One Type II action together with one Type V action is of order O ( N − 2 ) O(N^{-2}) O ( N − 2 ) N 2 N^2 N 2 N − 1 N^{-1} N − 1
We now record the scaling of the remaining operator types more explicitly.
By Eqn. (13) , the Type I action, which corresponds to a nonnegative increment of E E E O ( 1 ) O(1) O ( 1 )
The Type III action corresponds to a “canonical” down step of E E E N − O ( 1 ) N-O(1) N − O ( 1 ) x j x_j x j 1 − σ i j x i − x j \frac{1-\sigma_{ij}}{x_i-x_j} x i − x j 1 − σ ij O ( 1 ) O(1) O ( 1 ) N − 1 N^{-1} N − 1 x j x_j x j N − O ( 1 ) N-O(1) N − O ( 1 ) j j j N − 1 N^{-1} N − 1 first jump . Every later jump of the same block has only O ( 1 ) O(1) O ( 1 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 ) auxiliary jump . The same O ( N − 1 ) O(N^{-1}) O ( N − 1 ) x 2 x_2 x 2 m m m
By Lemma 4.6, in the remainder of Section 4.1, we will only consider P ⃗ ∈ P m , n > 0 \vec{P}\in\mathcal{P}_{m,n}^{>0} P ∈ P m , n > 0
First-order asymptotics
We first treat strings with no Type II action. Every auxiliary jump and every Type V action costs N − 1 N^{-1} N − 1 O ( 1 ) O(1) O ( 1 ) O ( N 2 ) O(N^2) O ( N 2 ) O ( N ) O(N) O ( N ) E ( m ) > 0 E(m)>0 E ( m ) > 0 x 1 x_1 x 1
From the discussion above, we classify the configurations that might contribute to the limiting covariance into the following groups. Group A consists of two-index configurations with one auxiliary jump and no Type V action, together with single-index configurations with no costly action and no Type V action.
Group B consists of two-index configurations with two auxiliary jumps and no Type V action, together with single-index configurations with one auxiliary jump and no Type V action.
Group C consists of two-index configurations with one auxiliary jump and one Type V action, together with single-index configurations with one Type V action and no auxiliary jump.
Groups A and B cancel completely. In Group C, some terms cancel with each other, while the remaining configurations give I 1 ( 1 ) ( E , q ⃗ ) I_1^{(1)}(E,\vec{q}) I 1 ( 1 ) ( E , q )
Enumeration of first-order configurations
We now enumerate the configurations allowed by this cost bound. In the list, jump means every discontinuity of a block, including its first jump; only the jumps classified above as auxiliary cost N − 1 N^{-1} N − 1
Up to relabeling variables x i x_i x i i ≥ 3 i\geq3 i ≥ 3
E ∈ L ( m + n ) E\in L(m+n) E ∈ L ( m + n )
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 1 x_1 x 1
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 2 x_2 x 2
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x i x_i x i i ≥ 3 i\geq3 i ≥ 3
It will be shown that the above are all the cases that contribute to the limiting covariance. The contributions of the following cases vanish in the limit after certain weight cancellations.
E ∈ L ( m + n ) E\in L(m+n) E ∈ L ( m + n ) x i x_i x i i ≥ 2 i\geq2 i ≥ 2
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 1 x_1 x 1 x 2 x_2 x 2
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 1 x_1 x 1 x i x_i x i i ≥ 3 i\geq3 i ≥ 3
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 2 x_2 x 2 x i x_i x i i ≥ 3 i\geq3 i ≥ 3
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 1 x_1 x 1
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x 2 x_2 x 2
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x i x_i x i i ≥ 3 i\geq3 i ≥ 3
E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) x i x_i x i x j x_j x j 3 ≤ i < j 3\leq i<j 3 ≤ i < j
Definition 4.7. For E 1 , E 2 ∈ L ( m , n ) ∪ L ( m + n ) E_1,E_2\in L(m,n)\cup L(m+n) E 1 , E 2 ∈ L ( m , n ) ∪ L ( m + n ) E 1 ∼ E 2 E_1\sim E_2 E 1 ∼ E 2 [ 0 , m + n ] [0,m+n] [ 0 , m + n ] m m m
Next, we further classify the configurations in the above 12 groups as a collection of disjoint pairs and three triples, such that the Lukasi ewicz paths in the same pair/triple are equivalent.
For brevity, we say a jump is before time m m m t = 1 , 2 , … , m t=1,2,\ldots,m t = 1 , 2 , … , m m m m m + 1 , m + 2 , … , m + n m+1,m+2,\ldots,m+n m + 1 , m + 2 , … , m + n ( E , q ⃗ ) (E,\vec{q}) ( E , q ) E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) q ⃗ \vec{q} q x 1 x_1 x 1 x 2 x_2 x 2
We start with this configuration and transform it into the other elements in the pair/triple, under the following rules:
I. If ( E , q ⃗ ) (E,\vec{q}) ( E , q ) x 1 x_1 x 1 m m m E E E L ( m + n ) L(m+n) L ( m + n )
II. If ( E , q ⃗ ) (E,\vec{q}) ( E , q ) x 2 x_2 x 2 m m m x 2 x_2 x 2 x j x_j x j j ≥ 3 j \ge3 j ≥ 3 q j ( t ) ≡ 0 q_j(t) \equiv0 q j ( t ) ≡ 0 ( E , q ⃗ ) (E,\vec{q}) ( E , q ) x j x_j x j deg ( x j ) \operatorname{deg}(x_j) deg ( x j )
III. If ( E , q ⃗ ) (E,\vec{q}) ( E , q ) x 1 x_1 x 1 m m m E E E L ( m + n ) L(m+n) L ( m + n ) x i x_i x i i ≥ 3 i \ge3 i ≥ 3
Each rule is reversible after the common path and the jump times and heights are fixed. Rule I changes whether the block crossing the splitting point is carried by x 1 x_1 x 1 x 2 x_2 x 2 x 1 x_1 x 1
More precisely, the following pairs use Rule I.
(1.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) q 1 ( t ) = C q_1(t)=C q 1 ( t ) = C ( m , τ ) (m,\tau) ( m , τ )
(1.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n )
Figure 1 (PDF p. 29).
(2.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i t 1 > τ t_1>\tau t 1 > τ i ≥ 3 i \ge3 i ≥ 3 x 1 x_1 x 1 t 2 ∈ [ m + 1 , t 1 − 1 ] t_2 \in[m+1,t_1-1] t 2 ∈ [ m + 1 , t 1 − 1 ]
(2.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n ) x i x_i x i m m m x i x_i x i t 1 > τ t_1>\tau t 1 > τ i ≥ 2 i \ge2 i ≥ 2
Moreover5 q i ( 1 ) ( t ) = q i ( 2 ) ( t ) q_i^{(1)}(t)=q_i^{(2)}(t) q i ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ 0 , m + n ] t \in[0,m+n] t ∈ [ 0 , m + n ]
Figure 2.
(3.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x i x_i x i m m m i ≥ 2 i \ge2 i ≥ 2 x 1 x_1 x 1 τ \tau τ
(3.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n ) x i x_i x i m m m i ≥ 2 i \ge2 i ≥ 2
Moreover, q i ( 1 ) ( t ) = q i ( 2 ) ( t ) q_i^{(1)}(t)=q_i^{(2)}(t) q i ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ 0 , m + n ] t \in[0,m+n] t ∈ [ 0 , m + n ]
Figure 3.1.
(4.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x i x_i x i t 1 , t 2 t_1,t_2 t 1 , t 2 m < t 1 < t 2 < τ m < t_1 < t_2 < \tau m < t 1 < t 2 < τ τ < t 1 < t 2 ≤ m + n \tau< t_1 < t_2 \le m+n τ < t 1 < t 2 ≤ m + n i ≥ 3 i \ge3 i ≥ 3 x 1 x_1 x 1 τ \tau τ q ( 1 ) ( t ) = q ( 2 ) ( t ) q^{(1)}(t)=q^{(2)}(t) q ( 1 ) ( t ) = q ( 2 ) ( t ) [ t 1 , t 2 ) [t_1,t_2) [ t 1 , t 2 )
(4.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n ) x i x_i x i t 1 , t 2 t_1,t_2 t 1 , t 2 m < t 1 < t 2 < τ m < t_1 < t_2 < \tau m < t 1 < t 2 < τ τ < t 1 < t 2 ≤ m + n \tau< t_1 < t_2 \le m+n τ < t 1 < t 2 ≤ m + n i ≥ 2 i \ge2 i ≥ 2
Figure 4.1.
The following configurations use Rule II.
(5.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m
(5.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i m m m i ≥ 3 i \ge3 i ≥ 3
(5.3) E 3 ∈ L ( m + n ) E_3 \in L(m+n) E 3 ∈ L ( m + n ) i ≥ 2 i \ge2 i ≥ 2 x i x_i x i m m m τ \tau τ C = E ( m ) C=E(m) C = E ( m )
More precisely, the block functions in all three configurations agree before the change of main variable:
q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) = q i ( 3 ) ( t ) , 0 ≤ t ≤ m . q_2^{(1)}(t)=q_i^{(2)}(t)=q_i^{(3)}(t), \qquad0\le t\le m. q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) = q i ( 3 ) ( t ) , 0 ≤ t ≤ m . They vanish before the common first-jump time and equal C C C m m m C C C τ \tau τ
Figure 5.1.
(6.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m x 1 x_1 x 1 m m m
(6.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i t 1 > m t_1>m t 1 > m i ≥ 3 i \ge3 i ≥ 3 x 1 x_1 x 1 t 2 > t 1 t_2>t_1 t 2 > t 1
Moreover, q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) q_2^{(1)}(t)=q_i^{(2)}(t) q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ 0 , m ) t\in[0,m) t ∈ [ 0 , m )
Figure 6.1.
Figure 6.2.
(7.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x i x_i x i t 1 , t 2 t_1,t_2 t 1 , t 2 1 ≤ t 1 < t 2 ≤ m 1 \leq t_1 < t_2 \leq m 1 ≤ t 1 < t 2 ≤ m τ < t 1 < t 2 ≤ m + n \tau< t_1 < t_2 \leq m+n τ < t 1 < t 2 ≤ m + n i ≥ 3 i \geq3 i ≥ 3 x 2 x_2 x 2 m m m
(7.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i t 1 , t 2 t_1,t_2 t 1 , t 2 1 ≤ t 1 < t 2 ≤ m 1 \leq t_1 < t_2 \leq m 1 ≤ t 1 < t 2 ≤ m τ < t 1 < t 2 ≤ m + n \tau< t_1 < t_2 \leq m+n τ < t 1 < t 2 ≤ m + n i ≥ 3 i \geq3 i ≥ 3 x j x_j x j m m m x j x_j x j m m m j ≠ 1 , 2 , i j \ne1,2,i j = 1 , 2 , i
Moreover, q i ( 1 ) ( t ) = q i ( 2 ) ( t ) q_i^{(1)}(t)=q_i^{(2)}(t) q i ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ 0 , m + n ] t \in[0,m+n] t ∈ [ 0 , m + n ]
Figure 7.1.
Figure 7.2.
(8.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m x i x_i x i m m m i ≥ 3 i \geq3 i ≥ 3 x i x_i x i τ \tau τ
(8.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i m m m i ≥ 3 i \geq3 i ≥ 3 τ \tau τ
Moreover, q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) q_2^{(1)}(t)=q_i^{(2)}(t) q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ 0 , m ) t \in[0,m) t ∈ [ 0 , m ) q i ( 1 ) ( t ) = q i ( 2 ) ( t ) q_i^{(1)}(t)=q_i^{(2)}(t) q i ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ τ , m + n ] t \in[\tau,m+n] t ∈ [ τ , m + n ]
Figure 8.1.
Figure 8.2.
(9.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m
(9.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i m m m i ≥ 3 i \geq3 i ≥ 3
Moreover, q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) q_2^{(1)}(t)=q_i^{(2)}(t) q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) [ 0 , m ) [0,m) [ 0 , m )
Figure 9.1.
Figure 9.2.
(10.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m x i x_i x i m m m x i x_i x i m m m i ≥ 3 i \ge3 i ≥ 3
(10.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x j x_j x j m m m x j x_j x j m m m j ≥ 3 j \ge3 j ≥ 3 x i x_i x i m m m x i x_i x i m m m i ≠ 1 , 2 , j i \ne1,2,j i = 1 , 2 , j
Moreover, q ( 1 ) ( t ) = q ( 2 ) ( t ) q^{(1)}(t)=q^{(2)}(t) q ( 1 ) ( t ) = q ( 2 ) ( t ) t ∈ [ 0 , m ) t \in[0,m) t ∈ [ 0 , m )
Figure 10.1.
Figure 10.2.
(11.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m x i x_i x i t 1 , t 2 ∈ [ m + 1 , τ − 1 ] t_1,t_2 \in[m+1,\tau-1] t 1 , t 2 ∈ [ m + 1 , τ − 1 ] i ≥ 3 i \ge3 i ≥ 3
(11.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x j x_j x j m m m x i x_i x i t 1 , t 2 ∈ [ m + 1 , τ − 1 ] t_1,t_2 \in[m+1,\tau-1] t 1 , t 2 ∈ [ m + 1 , τ − 1 ] i ≥ 3 i \ge3 i ≥ 3 x j x_j x j τ \tau τ j ≠ 1 , 2 , i j \ne1,2,i j = 1 , 2 , i
Moreover, q ( 1 ) ( t ) = q ( 2 ) ( t ) q^{(1)}(t)=q^{(2)}(t) q ( 1 ) ( t ) = q ( 2 ) ( t ) [ 0 , m ) ∪ [ t 1 , t 2 ) [0,m) \cup[t_1,t_2) [ 0 , m ) ∪ [ t 1 , t 2 )
Figure 11.1.
Figure 11.2.
(12.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 2 x_2 x 2 m m m x i x_i x i t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] x i x_i x i t 2 ∈ [ τ , m + n ] t_2 \in[\tau,m+n] t 2 ∈ [ τ , m + n ] i ≥ 3 i \ge3 i ≥ 3
(12.2) E 2 ∈ L ( m , n ) E_2 \in L(m,n) E 2 ∈ L ( m , n ) x i x_i x i m m m x i x_i x i t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] t 2 t_2 t 2 t 1 < t 2 ≤ m + n t_1 < t_2 \le m+n t 1 < t 2 ≤ m + n i ≥ 3 i \ge3 i ≥ 3
(12.3) E 3 ∈ L ( m , n ) E_3 \in L(m,n) E 3 ∈ L ( m , n ) x i x_i x i m m m s 1 s_1 s 1 x j x_j x j t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] t 2 t_2 t 2 m < t 1 < s 1 < t 2 ≤ m + n m < t_1 < s_1 < t_2 \le m+n m < t 1 < s 1 < t 2 ≤ m + n i ≥ 3 i \ge3 i ≥ 3 j ≠ 1 , 2 , i j \ne1,2,i j = 1 , 2 , i
In (12.2) and (12.3), the last closing time may lie on either side of τ \tau τ t 2 ≥ τ t_2 \ge\tau t 2 ≥ τ
The index in the third configuration is i i i j j j
q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) = q i ( 3 ) ( t ) , 0 ≤ t < m . q^{(1)}_2(t)=q^{(2)}_i(t)=q^{(3)}_i(t), \qquad0 \le t < m. q 2 ( 1 ) ( t ) = q i ( 2 ) ( t ) = q i ( 3 ) ( t ) , 0 ≤ t < m . The x j x_j x j t 1 t_1 t 1
Figure 12.1.
Figure 12.2.
Figure 12.3.
The following pair uses Rule III.
(13.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 1 x_1 x 1 m m m τ \tau τ
(13.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n ) x i x_i x i m m m i ≥ 2 i \ge2 i ≥ 2 τ \tau τ
Moreover, q 1 ( 1 ) ( t ) = q i ( 2 ) ( t ) q_1^{(1)}(t)=q_i^{(2)}(t) q 1 ( 1 ) ( t ) = q i ( 2 ) ( t ) t ∈ [ τ , m + n ] t \in[\tau,m+n] t ∈ [ τ , m + n ]
Figure 13.1.
Figure 13.2.
Finally, the following triple combines Rules I and III.
(14.1) E 1 ∈ L ( m , n ) E_1 \in L(m,n) E 1 ∈ L ( m , n ) x 1 x_1 x 1 s 1 s_1 s 1 x i x_i x i t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] s 2 s_2 s 2 m < t 1 < s 1 < s 2 ≤ m + n m < t_1 < s_1 < s_2 \le m+n m < t 1 < s 1 < s 2 ≤ m + n i ≥ 3 i \ge3 i ≥ 3
(14.2) E 2 ∈ L ( m + n ) E_2 \in L(m+n) E 2 ∈ L ( m + n ) x i x_i x i t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] t 2 ∈ [ τ , m + n ] t_2 \in[\tau,m+n] t 2 ∈ [ τ , m + n ] i ≥ 2 i \ge2 i ≥ 2
(14.3) E 3 ∈ L ( m , n ) E_3 \in L(m,n) E 3 ∈ L ( m , n ) x 1 x_1 x 1 t 1 ∈ [ m + 1 , τ − 1 ] t_1 \in[m+1,\tau-1] t 1 ∈ [ m + 1 , τ − 1 ] t 2 t_2 t 2 t 1 < t 2 ≤ m + n t_1 < t_2 \le m+n t 1 < t 2 ≤ m + n
The last closing time is unrestricted relative to τ \tau τ t 2 ≥ τ t_2 \ge\tau t 2 ≥ τ
No block is present before the splitting point in any of the three configurations:
q ( 1 ) ( t ) = q ( 2 ) ( t ) = q ( 3 ) ( t ) = 0 , 0 ≤ t < m . q^{(1)}(t)=q^{(2)}(t)=q^{(3)}(t)=0,\qquad0\le t<m. q ( 1 ) ( t ) = q ( 2 ) ( t ) = q ( 3 ) ( t ) = 0 , 0 ≤ t < m . Figure 14.1.
Figure 14.2.
Figure 14.3.
These eleven pairs and three triples exhaust the cost-admissible configurations. First record whether a prescribed variable x 1 x_1 x 1 x 2 x_2 x 2 m m m τ \tau τ τ \tau τ
Cancellation of first-order weights
In this subsection, we prove the following lemma.
Lemma 4.8. (a) The total weight of the configurations in pairs 2–4, 6–11, and 13 is of order o ( 1 ) o(1) o ( 1 )
(b) The total weight of the zero-Type V configurations in triples 5 and 12 is of order o ( 1 ) o(1) o ( 1 )
(c) The total weight of the zero-Type V configurations in triple 14 and pair 1 is of order o ( 1 ) o(1) o ( 1 )
Remark 4.9. The Group A patterns are (1.1)–(1.2) and (5.1)–(5.2); with one Type V action they form Group C. Configuration (5.3), the entries in part (a), and triples 12 and 14 use the full Group B cost, so an additional Type V action makes their normalized contribution o ( 1 ) o(1) o ( 1 ) § 7.2.4 .
Proof. For part (a), fix a common path and a matched pair of block patterns, with the correspondence specified in § 7.2.2 . The potentially nonvanishing terms belong to Group B, so each side has normalized weight O ( 1 ) O(1) O ( 1 ) c ( d ) c(d) c ( d )
The auxiliary-label and the canonical down steps give multiplicities of the form N − c N-c N − c c = O ( 1 ) c=O(1) c = O ( 1 ) m , n m,n m , n N N N 1 + O ( N − 1 ) 1+O(N^{-1}) 1 + O ( N − 1 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 ) o ( 1 ) o(1) o ( 1 )
For part (b), fix E E E m m m C = E ( m ) C=E(m) C = E ( m ) m < t < τ m<t<\tau m < t < τ r = r ( t ) : = E ( t − 1 ) − C ≥ 1 r=r(t):=E(t-1)-C\ge1 r = r ( t ) := E ( t − 1 ) − C ≥ 1 N N N θ \theta θ ∏ B i ∈ E κ ∣ B i ∣ \prod_{B_i\in E}\kappa^{|B_i|} ∏ B i ∈ E κ ∣ B i ∣
First combine (5.2) and (5.3). Apart from the choices of auxiliary variable x i x_i x i N − 2 N-2 N − 2 N − 1 N-1 N − 1
N ( N − 1 ) ( N − 2 ) + N ( N − 1 ) N m + n = N ( N − 1 ) 2 N m + n . \frac{N(N-1)(N-2)+N(N-1)}{N^{m+n}}=\frac{N(N-1)^2}{N^{m+n}}. N m + n N ( N − 1 ) ( N − 2 ) + N ( N − 1 ) = N m + n N ( N − 1 ) 2 . This matches the exterior factor of (5.1) and its N − 1 N-1 N − 1 τ \tau τ m m m τ \tau τ m < t < τ m<t<\tau m < t < τ N − 1 N-1 N − 1 N − 1 − 1 { r ( t ) ≤ C } N-1-\mathbf{1}_{\{r(t)\le C\}} N − 1 − 1 { r ( t ) ≤ C }
∑ t ∈ D ( E ) m < t < τ 1 { r ( t ) ≤ C } . \sum_{\substack{t\in D(E)\\m<t<\tau}}\mathbf{1}_{\{r(t)\le C\}}. t ∈ D ( E ) m < t < τ ∑ 1 { r ( t ) ≤ C } . We now compute the total weight of triple 12. Fix its first jump time t 1 = t ∈ D ( E ) t_1=t\in D(E) t 1 = t ∈ D ( E ) m m m m < t < τ m<t<\tau m < t < τ r ′ r' r ′ 1 , … , C 1,\ldots,C 1 , … , C τ \tau τ r ′ = C r'=C r ′ = C − C -C − C C C C { r , … , C − 1 } \{r,\ldots,C-1\} { r , … , C − 1 } r ≤ C r\le C r ≤ C { C + 1 , … , r − 1 } \{C+1,\ldots,r-1\} { C + 1 , … , r − 1 } r > C r>C r > C C − r C-r C − r − ( r − C − 1 ) -(r-C-1) − ( r − C − 1 ) 1 , … , r − 1 1,\ldots,r-1 1 , … , r − 1 r − 1 r-1 r − 1
All three configurations have normalized weight O ( 1 ) O(1) O ( 1 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 )
{ − C + ( C − r ) + ( r − 1 ) = − 1 , r ≤ C , − C − ( r − C − 1 ) + ( r − 1 ) = 0 , r > C . \begin{cases}
-C+(C-r)+(r-1)=-1, & r\le C,\\
-C-(r-C-1)+(r-1)=0, & r>C.
\end{cases} { − C + ( C − r ) + ( r − 1 ) = − 1 , − C − ( r − C − 1 ) + ( r − 1 ) = 0 , r ≤ C , r > C . Summing over the choices of t 1 t_1 t 1
− ∑ t ∈ D ( E ) m < t < τ 1 { r ( t ) ≤ C } . -\sum_{\substack{t\in D(E)\\m<t<\tau}}\mathbf{1}_{\{r(t)\le C\}}. − t ∈ D ( E ) m < t < τ ∑ 1 { r ( t ) ≤ C } . This proves (b).
For part (c), fix E E E C C C r ( t ) r(t) r ( t ) N − 1 N-1 N − 1 τ \tau τ N − 1 N-1 N − 1 τ \tau τ m < t < τ m<t<\tau m < t < τ N − 1 N-1 N − 1 N − 1 − 1 { r ( t ) ≤ C } N-1-\mathbf{1}_{\{r(t)\le C\}} N − 1 − 1 { r ( t ) ≤ C }
∑ t ∈ D ( E ) m < t < τ 1 { r ( t ) ≤ C } . \sum_{\substack{t\in D(E)\\m<t<\tau}}\mathbf{1}_{\{r(t)\le C\}}. t ∈ D ( E ) m < t < τ ∑ 1 { r ( t ) ≤ C } . At a fixed first jump time t 1 = t t_1=t t 1 = t − C -C − C C C C x i x_i x i x 1 x_1 x 1 C C C C − r C-r C − r r ≤ C r\le C r ≤ C − ( r − C − 1 ) -(r-C-1) − ( r − C − 1 ) r > C r>C r > C r − 1 r-1 r − 1 − 1 { r ( t ) ≤ C } -\mathbf{1}_{\{r(t)\le C\}} − 1 { r ( t ) ≤ C } t t t
First-order free cumulants in covariance
It remains to evaluate the one-Type V parts of the pairs (1.1)–(1.2) and (5.1)–(5.2), which belong to Group C. The two-index terms have one costly auxiliary jump and one Type V action, compensated by the exterior factor N ( N − 1 ) ∼ N 2 N(N-1)\sim N^2 N ( N − 1 ) ∼ N 2 N N N
( 1.2 ) ↔ ( a ) , ( 1.1 ) ↔ ( b ) , ( 5.1 ) ↔ ( c ) , ( 5.2 ) ↔ ( d ) . (1.2)\leftrightarrow(a),\qquad(1.1)\leftrightarrow(b),\qquad(5.1)\leftrightarrow(c),\qquad(5.2)\leftrightarrow(d). ( 1.2 ) ↔ ( a ) , ( 1.1 ) ↔ ( b ) , ( 5.1 ) ↔ ( c ) , ( 5.2 ) ↔ ( d ) . Match the two entries of each pair by their common path E E E ∏ B i ∈ E K ∣ B i ∣ \prod_{B_i\in E}K_{|B_i|} ∏ B i ∈ E K ∣ B i ∣
A Type V action at t ∈ P ( E , q ⃗ ) t\in P(E,\vec q) t ∈ P ( E , q ) E ( t − ) − q ( t − ) E(t^-)-q(t^- ) E ( t − ) − q ( t − ) C ( E ) C(E) C ( E ) ( m , τ ) (m,\tau) ( m , τ ) C ( E ) C(E) C ( E )
( { m + 1 , … , τ − 1 } ∩ D ( E ) ) ∪ { τ } . \bigl(\{m+1,\ldots,\tau-1\}\cap D(E)\bigr)\cup\{\tau\}. ( { m + 1 , … , τ − 1 } ∩ D ( E ) ) ∪ { τ } . There are B ( E ) B(E) B ( E ) { τ ˉ + 1 , … , m } ∩ D ( E ) \{\bar{\tau}+1,\ldots,m\}\cap D(E) { τ ˉ + 1 , … , m } ∩ D ( E ) A ( E ) A(E) A ( E )
1 θ A ( E ) B ( E ) C ( E ) ∏ B i ∈ E κ ∣ B i ∣ . \frac{1}{\theta}A(E)B(E)C(E)\prod_{B_i\in E}\kappa_{\lvert B_i\rvert}. θ 1 A ( E ) B ( E ) C ( E ) B i ∈ E ∏ κ ∣ B i ∣ . Summing over E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n )
1 θ ∑ ( E , q ⃗ ) ∈ Conf 1 ( m , n ) I 1 ( 1 ) ( E , q ⃗ ) = 1 θ ∑ E ∈ L ( m , n ) A ( E ) B ( E ) C ( E ) ∏ B i ∈ E κ ∣ B i ∣ . (28) \frac{1}{\theta}\sum_{(E,\vec{q})\in\operatorname{Conf}_1(m,n)}I_1^{(1)}(E,\vec{q})=\frac{1}{\theta}\sum_{E\in L(m,n)}A(E)B(E)C(E)\prod_{B_i\in E}\kappa_{\lvert B_i\rvert}. \tag*{(28)} θ 1 ( E , q ) ∈ Conf 1 ( m , n ) ∑ I 1 ( 1 ) ( E , q ) = θ 1 E ∈ L ( m , n ) ∑ A ( E ) B ( E ) C ( E ) B i ∈ E ∏ κ ∣ B i ∣ . (28) This is the contribution of first-order free cumulants to α m , n \alpha_{m,n} α m , n (14) .
Reduction of the second-order contribution
We now consider the contribution involving the coefficients c λ c_\lambda c λ ℓ ( λ ) = 2 \ell(\lambda)=2 ℓ ( λ ) = 2 Lemma 4.6 , after discarding an o ( 1 ) o(1) o ( 1 )
At the level where the Type II operator is kept in the grouped form U i ( λ ) U_i^{(\lambda)} U i ( λ ) U i ( λ ) U_i^{(\lambda)} U i ( λ ) U i ( λ ) U_i^{(\lambda)} U i ( λ ) U i ( λ , r , k ) U_i^{(\lambda,r,k)} U i ( λ , r , k ) x j x_j x j j ≥ 3 j\ge3 j ≥ 3
We first observe that the single-index term
N ( D 1 N ) m + n exp ( F N , θ ) ∣ x ⃗ = 0 N\left(\frac{\mathcal{D}_1}{N}\right)^{m+n}\left.\exp(F_{N,\theta})\right|_{\vec{x}=0} N ( N D 1 ) m + n exp ( F N , θ ) ∣ x = 0 has vanishing second-order contribution. Indeed, a Type II action and a Type V action together contribute a factor O ( N − 2 ) O(N^{-2}) O ( N − 2 ) x 1 x_1 x 1 N N N O ( N − 1 ) O(N^{-1}) O ( N − 1 ) x j x_j x j j ≠ 1 j\ne1 j = 1 O ( N ) O(N) O ( N ) j j j O ( N − 1 ) O(N^{-1}) O ( N − 1 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 )
N ( N − 1 ) ( D 2 N ) n ( D 1 N ) m exp ( F N , θ ) ∣ x ⃗ = 0 N(N-1)\left(\frac{\mathcal{D}_2}{N}\right)^n\left(\frac{\mathcal{D}_1}{N}\right)^m\left.\exp(F_{N,\theta})\right|_{\vec{x}=0} N ( N − 1 ) ( N D 2 ) n ( N D 1 ) m exp ( F N , θ ) ∣ x = 0 needs to be considered.
Let s s s
c λ λ r x 1 λ r − 1 x k λ 3 − r , r ∈ { 1 , 2 } . (29) c_\lambda\lambda_r x_1^{\lambda_r-1}x_k^{\lambda_3-r},\qquad r\in\{1,2\}. \tag*{(29)} c λ λ r x 1 λ r − 1 x k λ 3 − r , r ∈ { 1 , 2 } . (29) We claim that, at leading order,
s ≤ m and k ≠ 1. s\le m\qquad\text{and}\qquad k\ne1. s ≤ m and k = 1. Indeed, if s > m s>m s > m E ( m ) > 0 E(m)>0 E ( m ) > 0 x 1 x_1 x 1 m m m m m m x 2 x_2 x 2 x 1 x_1 x 1 O ( N − 1 ) O(N^{-1}) O ( N − 1 )
Similarly, if k = 1 k=1 k = 1 E ( m ) = 0 E(m)=0 E ( m ) = 0 E ( m ) > 0 E(m)>0 E ( m ) > 0 o ( 1 ) o(1) o ( 1 )
It follows that the unique Type II action occurs at a time s ≤ m s\le m s ≤ m k = 2 k=2 k = 2 k = j ≥ 3 k=j\ge3 k = j ≥ 3
Fix the associated path E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n )
C = C ( E ) : = E ( m ) > 0. C=C(E):=E(m)>0. C = C ( E ) := E ( m ) > 0. At time m m m O ( 1 ) O(1) O ( 1 ) o ( 1 ) o(1) o ( 1 ) x 1 x_1 x 1 m m m x 2 x_2 x 2 x 1 x_1 x 1 C ( E ) = E ( m ) C(E)=E(m) C ( E ) = E ( m ) (29) . Therefore, the power assigned to the auxiliary variable in Eqn. (29) must equal C C C
λ 3 − r = C . \lambda_{3-r}=C. λ 3 − r = C . Since the Type II action gives an increment λ r + C − 1 \lambda_r+C-1 λ r + C − 1
k 1 ( E , s ) : = E ( s ) − E ( s − ) − C ( E ) + 1 , k 2 ( E ) : = C ( E ) . (30) k_1(E,s):=E(s)-E(s-)-C(E)+1,\qquad k_2(E):=C(E). \tag*{(30)} k 1 ( E , s ) := E ( s ) − E ( s − ) − C ( E ) + 1 , k 2 ( E ) := C ( E ) . (30) Thus λ r = k 1 ( E , s ) \lambda_r=k_1(E,s) λ r = k 1 ( E , s ) λ 3 − r = k 2 ( E ) \lambda_{3-r}=k_2(E) λ 3 − r = k 2 ( E )
The Type II action is an up or flat step, so s ∈ U ( E ) s\in U(E) s ∈ U ( E ) C C C q ( t ) ≤ E ( t ) q(t)\le E(t) q ( t ) ≤ E ( t ) E ( t ) ≥ C E(t)\ge C E ( t ) ≥ C m m m
s ∈ { τ ˉ , … , m } ∩ U ( E ) . s\in\{\bar{\tau},\ldots,m\}\cap U(E). s ∈ { τ ˉ , … , m } ∩ U ( E ) . As before, we use the convention that κ k 1 ( E , s ) , k 2 ( E ) = 0 \kappa_{k_1(E,s),k_2(E)}=0 κ k 1 ( E , s ) , k 2 ( E ) = 0 k 1 ( E , s ) ≤ 0 k_1(E,s)\le0 k 1 ( E , s ) ≤ 0
The two monomial realizations
Fix E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n )
s ∈ { τ ˉ , … , m } ∩ U ( E ) . s\in\{\bar{\tau},\ldots,m\}\cap U(E). s ∈ { τ ˉ , … , m } ∩ U ( E ) . There are precisely two leading-order monomial realizations of the power-sum factor introduced by the Type II action.
Case I: the monomial indexed by k = 2 k=2 k = 2
In this case the Type II action introduces the factor x 2 C x_2^C x 2 C m m m
q 2 ( t ) = { C , t ∈ [ s , m ] , 0 , otherwise , q i ( t ) ≡ 0 , i ≠ 2. q_2(t)=
\begin{cases}
C, & t\in[s,m],\\
0, & \text{otherwise},
\end{cases}
\qquad q_i(t)\equiv0,\quad i\ne2. q 2 ( t ) = { C , 0 , t ∈ [ s , m ] , otherwise , q i ( t ) ≡ 0 , i = 2. This is exactly the limiting second-order configuration in case (e) of Definition 3.9 .
**Case II: the monomial indexed by k = j ≥ 3 k=j\ge3 k = j ≥ 3 x j C x_j^C x j C j ≥ 3 j \ge3 j ≥ 3
q j j ( t ) = C , t ∈ [ s , m ] . q_j^j(t)=C,\qquad t\in[s,m]. q j j ( t ) = C , t ∈ [ s , m ] . Since x j x_j x j
q j j ( t ) = { C , t ∈ [ s , τ ) , 0 , otherwise , q i j ( t ) = 0 , i ≠ j . q_j^j(t)=
\begin{cases}
C, & t\in[s,\tau),\\
0, & \text{otherwise},
\end{cases}
\qquad q_i^j(t)=0,\quad i\ne j. q j j ( t ) = { C , 0 , t ∈ [ s , τ ) , otherwise , q i j ( t ) = 0 , i = j . A minus sign enters the weight at the closing Type IV action at time τ \tau τ q j q_j q j C C C
There are N − 2 N-2 N − 2 j j j j = 3 j=3 j = 3
Any further Type IV action creates an additional auxiliary-jump factor O ( N − 1 ) O(N^{-1}) O ( N − 1 ) o ( 1 ) o(1) o ( 1 )
Second-order asymptotics
Fix E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n )
s ∈ { τ ˉ , … , m } ∩ U ( E ) , s\in\{\bar{\tau},\ldots,m\}\cap U(E), s ∈ { τ ˉ , … , m } ∩ U ( E ) , and write
k 1 = k 1 ( E , s ) : = E ( s ) − E ( s − ) − C ( E ) + 1 , k 2 = k 2 ( E ) : = C ( E ) . k_1=k_1(E,s):=E(s)-E(s-)-C(E)+1,\qquad k_2=k_2(E):=C(E). k 1 = k 1 ( E , s ) := E ( s ) − E ( s − ) − C ( E ) + 1 , k 2 = k 2 ( E ) := C ( E ) . Let q ⃗ ( 2 ) \vec{q}^{(2)} q ( 2 ) q ⃗ ( 3 ) \vec{q}^{(3)} q ( 3 ) k = 2 k=2 k = 2 k = j ≥ 3 k=j\ge3 k = j ≥ 3
Both classes have finite limits. Indeed, by Eqn. (13) ,
k 1 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) ⟶ θ 1 − k 1 − k 2 κ k 1 , k 2 k 2 , k_1(1+\mathbf{1}_{k_1=k_2})c(k_1,k_2)\longrightarrow\theta^{1-k_1-k_2}\frac{\kappa_{k_1,k_2}}{k_2}, k 1 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) ⟶ θ 1 − k 1 − k 2 k 2 κ k 1 , k 2 , while every other up or flat step contributes the corresponding first-order free cumulant. In the k = 2 k=2 k = 2
N ( N − 1 ) N 2 ⟶ 1. \frac{N(N-1)}{N^2}\longrightarrow1. N 2 N ( N − 1 ) ⟶ 1. In the k = j ≥ 3 k=j\ge3 k = j ≥ 3 N − 2 N-2 N − 2 N − 1 N^{-1} N − 1 τ \tau τ
N ( N − 1 ) ( N − 2 ) N 3 ⟶ 1. \frac{N(N-1)(N-2)}{N^3}\longrightarrow1. N 3 N ( N − 1 ) ( N − 2 ) ⟶ 1. The latter class has the opposite sign because its closing jump has negative jump height. Consequently, after summing over all fresh indices j ≥ 3 j\ge3 j ≥ 3
1 θ ( I 1 ( 2 ) ( E , q ⃗ ( 2 ) ) + I 1 ( 2 ) ( E , q ⃗ ( 3 ) ) ) = 1 θ κ k 1 , k 2 k 2 ∏ B i ∈ E t ( B i ) ≠ s κ ∣ B i ∣ [ ∑ t ∈ D ( E ) ( E ( t − ) − q ( 2 ) ( t − ) ) − ∑ t ∈ D ( E ) ∖ { τ } ( E ( t − ) − q ( 3 ) ( t − ) ) ] . (31) \frac{1}{\theta}\left(I_1^{(2)}(E,\vec{q}^{(2)})+I_1^{(2)}(E,\vec{q}^{(3)})\right)
=
\frac{1}{\theta}\frac{\kappa_{k_1,k_2}}{k_2}
\prod_{\substack{B_i\in E\\t(B_i)\ne s}}\kappa_{\lvert B_i\rvert}
\left[
\sum_{t\in D(E)}\left(E(t-)-q^{(2)}(t-)\right)
-
\sum_{t\in D(E)\setminus\{\tau\}}\left(E(t-)-q^{(3)}(t-)\right)
\right]. \tag*{(31)} θ 1 ( I 1 ( 2 ) ( E , q ( 2 ) ) + I 1 ( 2 ) ( E , q ( 3 ) ) ) = θ 1 k 2 κ k 1 , k 2 B i ∈ E t ( B i ) = s ∏ κ ∣ B i ∣ t ∈ D ( E ) ∑ ( E ( t − ) − q ( 2 ) ( t − ) ) − t ∈ D ( E ) ∖ { τ } ∑ ( E ( t − ) − q ( 3 ) ( t − ) ) . (31) It remains to evaluate the difference in brackets. The two block functions agree before time m m m
q ( 2 ) ( t ) = q ( 3 ) ( t ) , t < m , q^{(2)}(t)=q^{(3)}(t), \qquad t<m, q ( 2 ) ( t ) = q ( 3 ) ( t ) , t < m , and both vanish after time τ \tau τ ( m , τ ) (m,\tau) ( m , τ )
q ( 2 ) ( t ) = 0 , q ( 3 ) ( t ) = C ( E ) . q^{(2)}(t)=0, \qquad q^{(3)}(t)=C(E). q ( 2 ) ( t ) = 0 , q ( 3 ) ( t ) = C ( E ) . Moreover, the time τ \tau τ q ⃗ ( 2 ) \vec{q}^{(2)} q ( 2 ) q ⃗ ( 3 ) \vec{q}^{(3)} q ( 3 )
E ( τ − ) = C ( E ) . E(\tau-)=C(E). E ( τ − ) = C ( E ) . Therefore,
∑ t ∈ D ( E ) ( E ( t − ) − q ( 2 ) ( t − ) ) − ∑ t ∈ D ( E ) ∖ { τ } ( E ( t − ) − q ( 3 ) ( t − ) ) = C ( E ) ( ∣ { m + 1 , … , τ − 1 } ∩ D ( E ) ∣ + 1 ) = C ( E ) B ( E ) . \begin{aligned}
\sum_{t\in D(E)}\left(E(t-)-q^{(2)}(t-)\right)-\sum_{t\in D(E)\setminus\{\tau\}}\left(E(t-)-q^{(3)}(t-)\right)
&=C(E)\left(\left|\{m+1,\ldots,\tau-1\}\cap D(E)\right|+1\right)\\
&=C(E)B(E).
\end{aligned} t ∈ D ( E ) ∑ ( E ( t − ) − q ( 2 ) ( t − ) ) − t ∈ D ( E ) ∖ { τ } ∑ ( E ( t − ) − q ( 3 ) ( t − ) ) = C ( E ) ( ∣ { m + 1 , … , τ − 1 } ∩ D ( E ) ∣ + 1 ) = C ( E ) B ( E ) . Since k 2 = C ( E ) k_2=C(E) k 2 = C ( E ) C ( E ) C(E) C ( E ) k 2 k_2 k 2 (31) . Thus, for every fixed pair ( E , s ) (E,s) ( E , s )
1 θ B ( E ) κ k 1 ( E , s ) , k 2 ( E ) ∏ B i ∈ E t ( B i ) ≠ s κ ∣ B i ∣ . \frac{1}{\theta}B(E)\kappa_{k_1(E,s),k_2(E)}\prod_{\substack{B_i\in E\\t(B_i)\ne s}}\kappa_{\lvert B_i\rvert}. θ 1 B ( E ) κ k 1 ( E , s ) , k 2 ( E ) B i ∈ E t ( B i ) = s ∏ κ ∣ B i ∣ . Finally, summing over all admissible paths and all possible positions of the Type II action gives
1 θ ∑ ( E , q ⃗ ) ∈ Conf 2 ( m , n ) I 1 ( 2 ) ( E , q ⃗ ) = 1 θ ∑ E ∈ L ( m , n ) ∑ s ∈ { τ , … , m ‾ } ∩ U ( E ) B ( E ) κ k 1 ( E , s ) , k 2 ( E ) ∏ B i ∈ E t ( B i ) ≠ s κ ∣ B i ∣ . (32) \frac{1}{\theta}\sum_{(E,\vec{q})\in\operatorname{Conf}_2(m,n)}I_1^{(2)}(E,\vec{q})
=\frac{1}{\theta}\sum_{E\in\mathcal{L}(m,n)}\sum_{s\in\{\tau,\ldots,\overline{m}\}\cap U(E)}B(E)\kappa_{k_1(E,s),k_2(E)}\prod_{\substack{B_i\in E\\t(B_i)\ne s}}\kappa_{\lvert B_i\rvert}. \tag*{(32)} θ 1 ( E , q ) ∈ Conf 2 ( m , n ) ∑ I 1 ( 2 ) ( E , q ) = θ 1 E ∈ L ( m , n ) ∑ s ∈ { τ , … , m } ∩ U ( E ) ∑ B ( E ) κ k 1 ( E , s ) , k 2 ( E ) B i ∈ E t ( B i ) = s ∏ κ ∣ B i ∣ . (32) This is precisely the second-order contribution in Eqn. (14) .
High-temperature covariance in Theorem 3.13
Recall that
p k N = 1 N p k ( a ⃗ ) = 1 N ∑ i = 1 N a i k . p_k^N=\frac{1}{N}p_k(\vec{a})=\frac{1}{N}\sum_{i=1}^{N}a_i^k. p k N = N 1 p k ( a ) = N 1 i = 1 ∑ N a i k . Therefore, by Eqn. (19) , the covariance appearing in the high-temperature CLT is
N Cov ( p m N , p n N ) = ( N − 1 ) D 2 n D 1 m exp ( F N , θ ) ∣ x ⃗ = 0 + D 1 m + n exp ( F N , θ ) ∣ x ⃗ = 0 − N D 1 m exp ( F N , θ ) ∣ x ⃗ = 0 D 1 n exp ( F N , θ ) ∣ x ⃗ = 0 . (33) \begin{aligned}
N\operatorname{Cov}(p_m^N,p_n^N)
={}&(N-1)\left.\mathcal{D}_2^n\mathcal{D}_1^m\exp(F_{N,\theta})\right|_{\vec{x}=0}
+\left.\mathcal{D}_1^{m+n}\exp(F_{N,\theta})\right|_{\vec{x}=0}\\
&-N\left.\mathcal{D}_1^m\exp(F_{N,\theta})\right|_{\vec{x}=0}
\left.\mathcal{D}_1^n\exp(F_{N,\theta})\right|_{\vec{x}=0}. \tag*{(33)}
\end{aligned} N Cov ( p m N , p n N ) = ( N − 1 ) D 2 n D 1 m exp ( F N , θ ) ∣ x = 0 + D 1 m + n exp ( F N , θ ) x = 0 − N D 1 m exp ( F N , θ ) ∣ x = 0 D 1 n exp ( F N , θ ) ∣ x = 0 . (33) Thus, the exterior factors in the three terms are respectively of orders N N N 1 1 1 N N N
We assume throughout this subsection that
d c ( d ) ⟶ κ d γ , N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) ⟶ κ k 1 , k 2 γ , N ℓ ( λ ) / 2 c λ ⟶ 0 for ℓ ( λ ) ≥ 3. d\,c_{(d)}\longrightarrow\kappa_d^\gamma,\qquad Nk_1k_2(1+1_{k_1=k_2})c_{(k_1,k_2)}\longrightarrow\kappa_{k_1,k_2}^\gamma,\qquad N^{\ell(\lambda)/2}c_\lambda\longrightarrow0\quad\text{for }\ell(\lambda)\ge3. d c ( d ) ⟶ κ d γ , N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) ⟶ κ k 1 , k 2 γ , N ℓ ( λ ) /2 c λ ⟶ 0 for ℓ ( λ ) ≥ 3. The terms with E ( m ) = 0 E(m)=0 E ( m ) = 0 O ( 1 ) O(1) O ( 1 ) θ ( N − \theta(N- θ ( N − O ( 1 ) ) → γ O(1)) \to\gamma O ( 1 )) → γ ± θ ∼ O ( N − 1 ) \pm\theta\sim O(N^{-1}) ± θ ∼ O ( N − 1 ) N − O ( 1 ) N-O(1) N − O ( 1 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 )
To control Types II and VI, retain their power-sum factors as in the proof of Lemma 4.6. The difference part passes every such symmetric factor through unchanged, so each factor must be hit by a Type V action before evaluation at the origin. These derivatives introduce no free-index multiplicity, and the remaining grouped actions are O ( 1 ) O(1) O ( 1 ) r r r O ( N − r ) O(N^{-r}) O ( N − r ) λ \lambda λ o ( N − ℓ ( λ ) / 2 ) o(N^{-\ell(\lambda)/2}) o ( N − ℓ ( λ ) /2 ) O ( N ) O(N) O ( N )
Without Type II, the two-index term allows at most one costly auxiliary jump, and the one-index term allows none. Hence Group B, including (5.3), is o ( 1 ) o(1) o ( 1 ) C o n f 1 ( m , n ) Conf_1(m,n) C o n f 1 ( m , n )
Now suppose there is exactly one Type II action. Its single-index contribution is O ( N − 1 ) O(N^{-1}) O ( N − 1 ) s ≤ m s \le m s ≤ m m m m
k 1 c ( k 1 , k 2 ) x 1 k 1 − 1 p k 2 = k 1 c ( k 1 , k 2 ) ∑ i = 1 N x 1 k 1 − 1 x i k 2 . k_1 c_{(k_1,k_2)} x_1^{k_1-1}p_{k_2}
=
k_1 c_{(k_1,k_2)}\sum_{i=1}^{N}x_1^{k_1-1}x_i^{k_2}. k 1 c ( k 1 , k 2 ) x 1 k 1 − 1 p k 2 = k 1 c ( k 1 , k 2 ) i = 1 ∑ N x 1 k 1 − 1 x i k 2 . If i = 1 i=1 i = 1 x 1 x_1 x 1 m m m O ( N − 1 ) O(N^{-1}) O ( N − 1 ) i = 2 i=2 i = 2 O ( 1 ) O(1) O ( 1 ) i ≥ 3 i\ge3 i ≥ 3 N − 1 N^{-1} N − 1 N − 2 N-2 N − 2 i i i O ( 1 ) O(1) O ( 1 ) O ( N ) O(N) O ( N ) O ( N − 1 ) O(N^{-1}) O ( N − 1 ) o ( 1 ) o(1) o ( 1 )
In the last two cases the x 1 x_1 x 1 m m m
k 2 = C = E ( m ) , k 1 = E ( s ) − E ( s − ) − C + 1. k_2=C=E(m),\qquad k_1=E(s)-E(s-)-C+1. k 2 = C = E ( m ) , k 1 = E ( s ) − E ( s − ) − C + 1. The i = 2 i=2 i = 2 [ s , m ] [s,m] [ s , m ] i ≥ 3 i\ge3 i ≥ 3 [ s , τ ) [s,\tau) [ s , τ ) τ \tau τ Definition 3.9 , with the same admissible Type II times as in § 7.2.6 . Pair them by their common path and Type II time, retaining both Type II summands with their multiplicities. With no Type V action, the closing factor θ ( N − 2 ) \theta(N-2) θ ( N − 2 ) i ≥ 3 i\ge3 i ≥ 3 θ ( N − O ( 1 ) ) \theta(N-O(1)) θ ( N − O ( 1 )) i = 2 i=2 i = 2 τ \tau τ o ( 1 ) o(1) o ( 1 ) C o n f 2 ( m , n ) Conf_2(m,n) C o n f 2 ( m , n )
Fix one such configuration ( E , q ⃗ ) (E,\vec q) ( E , q )
t 1 < ⋯ < t r , t 1 , … , t r ∈ P ( E , q ⃗ ) . t_1<\cdots<t_r,\qquad t_1,\ldots,t_r\in P(E,\vec q). t 1 < ⋯ < t r , t 1 , … , t r ∈ P ( E , q ) . The r r r
∏ j = 1 r ( E ( t j − ) − q ( t j − ) ) . \prod_{j=1}^{r}\bigl(E(t_j-)-q(t_j-)\bigr). j = 1 ∏ r ( E ( t j − ) − q ( t j − ) ) . The remaining ∣ D ( E ) ∣ − r |D(E)|-r ∣ D ( E ) ∣ − r θ \theta θ
γ ∣ D ( E ) ∣ − r . \gamma^{|D(E)|-r}. γ ∣ D ( E ) ∣ − r . The up and flat steps contribute the first-order free γ \gamma γ γ \gamma γ r r r
{ t 1 , … , t r } ⊂ P ( E , q ⃗ ) , \{t_1,\ldots,t_r\}\subset P(E,\vec{q}), { t 1 , … , t r } ⊂ P ( E , q ) , (the subset sum is exactly the one built into Definitions 3.10 and 3.11), the limiting contribution is
γ ∣ D ( E ) ∣ − r I r ( 1 ) ( E , q ⃗ ; γ ) \gamma^{|D(E)|-r}I_r^{(1)}(E,\vec{q};\gamma) γ ∣ D ( E ) ∣ − r I r ( 1 ) ( E , q ; γ ) for ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) (E,\vec{q})\in Conf_1(m,n) ( E , q ) ∈ C o n f 1 ( m , n )
γ ∣ D ( E ) ∣ − r I r ( 2 ) ( E , q ⃗ ; γ ) \gamma^{|D(E)|-r}I_r^{(2)}(E,\vec{q};\gamma) γ ∣ D ( E ) ∣ − r I r ( 2 ) ( E , q ; γ ) for ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) (E,\vec{q})\in Conf_2(m,n) ( E , q ) ∈ C o n f 2 ( m , n )
The r = 0 r=0 r = 0 P ( E , q ⃗ ) P(E,\vec{q}) P ( E , q )
Consequently,
α m , n = ∑ ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) ∑ r = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − r I r ( 1 ) ( E , q ⃗ ; γ ) + ∑ ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) ∑ r = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − r I r ( 2 ) ( E , q ⃗ ; γ ) . (34) \begin{aligned}
\alpha_{m,n}={}&\sum_{(E,\vec{q})\in Conf_1(m,n)}
\sum_{r=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-r}I_r^{(1)}(E,\vec{q};\gamma)\\
&+\sum_{(E,\vec{q})\in Conf_2(m,n)}
\sum_{r=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-r}I_r^{(2)}(E,\vec{q};\gamma). \tag*{(34)}
\end{aligned} α m , n = ( E , q ) ∈ C o n f 1 ( m , n ) ∑ r = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − r I r ( 1 ) ( E , q ; γ ) + ( E , q ) ∈ C o n f 2 ( m , n ) ∑ r = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − r I r ( 2 ) ( E , q ; γ ) . (34) This is exactly the covariance formula in Eqn. (16) . The configuration classes are the same as in the fixed-temperature regime. The difference is that, at fixed temperature, every Type V action contributes an additional factor N − 1 N^{-1} N − 1 O ( 1 ) O(1) O ( 1 ) P ( E , q ⃗ ) P(E,\vec{q}) P ( E , q )
The ``only if'' direction of Theorems 1.3 and 1.5
We start with the following combinatorial result on the coefficients of ln B a ( x ⃗ ; θ ) \ln B_a(\vec{x};\theta) ln B a ( x ; θ ) ln B a ( x ⃗ ; θ ) \ln B_a(\vec{x};\theta) ln B a ( x ; θ ) a a a x ⃗ \vec{x} x p μ 1 ( a / N ) N ℓ ( μ 1 ) p μ 2 ( x ⃗ ) \frac{p_{\mu_1}(a/N)}{N^{\ell(\mu_1)}}p_{\mu_2}(\vec{x}) N ℓ ( μ 1 ) p μ 1 ( a / N ) p μ 2 ( x ) ∣ μ 1 ∣ = ∣ μ 2 ∣ = d |\mu_1|=|\mu_2|=d ∣ μ 1 ∣ = ∣ μ 2 ∣ = d d ∈ Z ≥ 1 d\in\mathbb{Z}_{\geq1} d ∈ Z ≥ 1 C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2
Lemma 5.1. In the fixed-temperature regime, for μ 1 , μ 2 ⊢ d \mu_1,\mu_2\vdash d μ 1 , μ 2 ⊢ d N N N C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2 N ℓ ( μ 2 ) − 2 C μ 1 , μ 2 N^{\ell(\mu_2)-2}C_{\mu_1,\mu_2} N ℓ ( μ 2 ) − 2 C μ 1 , μ 2
Proof. This follows directly from Theorem 2.8 .
We shall use the following product formula for cumulants, usually called the Leonov–Shiryaev formula; see [42 , 62 ] .
Lemma 5.2. Let I I I ( ξ i ) i ∈ I (\xi_i)_{i\in I} ( ξ i ) i ∈ I
τ = { I 1 , … , I r } \tau=\{I_1,\ldots,I_r\} τ = { I 1 , … , I r } be a partition of I I I
Y n = ∏ i ∈ I n ξ i , n = 1 , … , r . Y_n=\prod_{i\in I_n}\xi_i,\qquad n=1,\ldots,r. Y n = i ∈ I n ∏ ξ i , n = 1 , … , r . Then
c r ( Y 1 , … , Y r ) = ∑ π ∈ Π ( I ) π ∨ τ = 1 ^ ∏ B ∈ π c ∣ B ∣ ( ξ i : i ∈ B ) . c_r(Y_1,\ldots,Y_r)=\sum_{\substack{\pi\in\Pi(I)\\\pi\vee\tau=\hat{1}}}\prod_{B\in\pi}c_{|B|}(\xi_i:i\in B). c r ( Y 1 , … , Y r ) = π ∈ Π ( I ) π ∨ τ = 1 ^ ∑ B ∈ π ∏ c ∣ B ∣ ( ξ i : i ∈ B ) . Here Π ( I ) \Pi(I) Π ( I ) I I I 1 ^ \hat{1} 1 ^ I I I π ∨ τ \pi\vee\tau π ∨ τ π \pi π τ \tau τ
Theorem 5.4. In the fixed-temperature regime, if the CLT of Definition 1.2 holds, then for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\ge1} d ∈ Z ≥ 1 k 1 , k 2 ∈ Z ≥ 1 k_1,k_2\in\mathbb{Z}_{\ge1} k 1 , k 2 ∈ Z ≥ 1
d N c ( d ) → θ 1 − d κ d ; \frac{d}{N}c^{(d)}\to\theta^{1-d}\kappa_d; N d c ( d ) → θ 1 − d κ d ; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; k_1k_2(1+\mathbf{1}_{k_1=k_2})c_{(k_1,k_2)}\to\theta^{1-k_1-k_2}\kappa_{k_1,k_2}; k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → θ 1 − k 1 − k 2 κ k 1 , k 2 ; c λ → 0 for ℓ ( λ ) ≥ 3 , (35) c_\lambda\to0\qquad\text{for }\ell(\lambda)\ge3, \tag*{(35)} c λ → 0 for ℓ ( λ ) ≥ 3 , (35) for certain constants κ d , κ k 1 , k 2 \kappa_d,\kappa_{k_1,k_2} κ d , κ k 1 , k 2 N → ∞ N\to\infty N → ∞
Proof. The first line of Eqn. (35) follows directly from [22 Theorem 3.4] , since CLT implies LLN. For the third line of Eqn. (35) , recall that
ln G θ , N ( x ⃗ ) = ln E [ exp ( ln ( B a ( x ⃗ ; θ ) ) ) ] = ln E [ exp ( ∑ μ 2 ( ∑ μ 1 C μ 1 , μ 2 p μ 1 ( a / N ) N ℓ ( μ 1 ) ) p μ 2 ( x ⃗ ) ) ] . (36) \ln G_{\theta,N}(\vec{x})=\ln\mathbb{E}[\exp(\ln(B_a(\vec{x};\theta)))]
=\ln\mathbb{E}\left[\exp\left(\sum_{\mu_2}\left(\sum_{\mu_1}C_{\mu_1,\mu_2}\frac{p_{\mu_1}(a/N)}{N^{\ell(\mu_1)}}\right)p_{\mu_2}(\vec{x})\right)\right]. \tag*{(36)} ln G θ , N ( x ) = ln E [ exp ( ln ( B a ( x ; θ )))] = ln E [ exp ( μ 2 ∑ ( μ 1 ∑ C μ 1 , μ 2 N ℓ ( μ 1 ) p μ 1 ( a / N ) ) p μ 2 ( x ) ) ] . (36) Note that for d ∈ Z ≥ 1 d\in\mathbb{Z}_{\ge1} d ∈ Z ≥ 1 μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 d d d c λ c_\lambda c λ
c l ( C μ 1 ( 1 ) , μ 2 ( 1 ) p μ 1 ( 1 ) ( a / N ) N ℓ ( μ 1 ( 1 ) ) , … , C μ 1 ( l ) , μ 2 ( l ) p μ 1 ( l ) ( a / N ) N ℓ ( μ 1 ( l ) ) ) = N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ⋅ N l c l ( p μ 1 ( 1 ) ( a / N ) N ℓ ( μ 1 ( 1 ) ) , … , p μ 1 ( l ) ( a / N ) N ℓ ( μ 1 ( l ) ) ) , (37) c_l\left(C_{\mu_1(1),\mu_2(1)}\frac{p_{\mu_1(1)}(a/N)}{N^{\ell(\mu_1(1))}},\ldots,C_{\mu_1(l),\mu_2(l)}\frac{p_{\mu_1(l)}(a/N)}{N^{\ell(\mu_1(l))}}\right)
=N^{-l}\prod_{j=1}^{l}C_{\mu_1(j),\mu_2(j)}\cdot N^l c_l\left(\frac{p_{\mu_1(1)}(a/N)}{N^{\ell(\mu_1(1))}},\ldots,\frac{p_{\mu_1(l)}(a/N)}{N^{\ell(\mu_1(l))}}\right), \tag*{(37)} c l ( C μ 1 ( 1 ) , μ 2 ( 1 ) N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a / N ) , … , C μ 1 ( l ) , μ 2 ( l ) N ℓ ( μ 1 ( l )) p μ 1 ( l ) ( a / N ) ) = N − l j = 1 ∏ l C μ 1 ( j ) , μ 2 ( j ) ⋅ N l c l ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a / N ) , … , N ℓ ( μ 1 ( l )) p μ 1 ( l ) ( a / N ) ) , (37) where λ = μ 2 ( 1 ) ⊔ ⋯ ⊔ μ 2 ( l ) \lambda=\mu_2(1)\sqcup\cdots\sqcup\mu_2(l) λ = μ 2 ( 1 ) ⊔ ⋯ ⊔ μ 2 ( l )
The remaining task is to estimate the order of the joint cumulants of products using Lemma 5.2 . More precisely, if l ≥ 3 l\geq3 l ≥ 3 Lemma 5.1 , N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ≲ O ( 1 ) N^{-l}\prod_{j=1}^{l}C_{\mu_1(j),\mu_2(j)}\lesssim O(1) N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ≲ O ( 1 ) Lemma 5.2 and the CLT assumption, N l c l ( p μ 1 ( 1 ) ( a / N ) N ℓ ( μ 1 ( 1 ) ) , … , p μ 1 ( l ) ( a / N ) N ℓ ( μ 1 ( l ) ) ) ∼ o ( 1 ) N^l c_l\left(\frac{p_{\mu_1(1)}(a/N)}{N^{\ell(\mu_1(1))}},\ldots,\frac{p_{\mu_1(l)}(a/N)}{N^{\ell(\mu_1(l))}}\right)\sim o(1) N l c l ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a / N ) , … , N ℓ ( μ 1 ( l )) p μ 1 ( l ) ( a / N ) ) ∼ o ( 1 )
If l = 2 l=2 l = 2 Lemma 5.1 N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ∼ o ( 1 ) N^{-l}\prod_{j=1}^{l}C_{\mu_1(j),\mu_2(j)}\sim o(1) N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ∼ o ( 1 ) μ 2 ( j ) \mu_2(j) μ 2 ( j ) Lemma 5.2 and the CLT assumption, N 2 c 2 ( p μ 1 ( 1 ) ( a / N ) N ℓ ( μ 1 ( 1 ) ) , p μ 1 ( 2 ) ( a / N ) N ℓ ( μ 1 ( 2 ) ) ) ≲ O ( 1 ) N^2c_2\left(\frac{p_{\mu_1(1)}(a/N)}{N^{\ell(\mu_1(1))}},\frac{p_{\mu_1(2)}(a/N)}{N^{\ell(\mu_1(2))}}\right)\lesssim O(1) N 2 c 2 ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a / N ) , N ℓ ( μ 1 ( 2 )) p μ 1 ( 2 ) ( a / N ) ) ≲ O ( 1 )
If l = 1 l=1 l = 1 ℓ ( μ 2 ) ≥ 3 \ell(\mu_2)\geq3 ℓ ( μ 2 ) ≥ 3 Lemma 5.1 C μ 1 , μ 2 ∼ o ( 1 ) C_{\mu_1,\mu_2}\sim o(1) C μ 1 , μ 2 ∼ o ( 1 ) c 1 ( p μ 1 ( a / N ) / N ℓ ( μ 1 ) ) ≲ O ( 1 ) c_1(p_{\mu_1}(a/N)/N^{\ell(\mu_1)})\lesssim O(1) c 1 ( p μ 1 ( a / N ) / N ℓ ( μ 1 ) ) ≲ O ( 1 ) (35) .
The second line of Eqn. (35) follows from a similar argument. Note that l l l l = 1 l=1 l = 1 ℓ ( μ 2 ) = 2 \ell(\mu_2)=2 ℓ ( μ 2 ) = 2 Lemma 5.1 C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2 c 1 ( p μ 1 ( a / N ) / N ℓ ( μ 1 ) ) c_1(p_{\mu_1}(a/N)/N^{\ell(\mu_1)}) c 1 ( p μ 1 ( a / N ) / N ℓ ( μ 1 ) ) N → ∞ N\to\infty N → ∞ l = 2 l=2 l = 2 N − 2 ∏ j = 1 2 C μ 1 ( j ) , μ 2 ( j ) N^{-2}\prod_{j=1}^{2}C_{\mu_1(j),\mu_2(j)} N − 2 ∏ j = 1 2 C μ 1 ( j ) , μ 2 ( j ) Lemma 5.1 , and the terms N 2 c 2 ( p μ 1 ( 1 ) ( a / N ) N ℓ ( μ 1 ( 1 ) ) , p μ 1 ( 2 ) ( a / N ) N ℓ ( μ 1 ( 2 ) ) ) N^2c_2\left(\frac{p_{\mu_1(1)}(a/N)}{N^{\ell(\mu_1(1))}},\frac{p_{\mu_1(2)}(a/N)}{N^{\ell(\mu_1(2))}}\right) N 2 c 2 ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a / N ) , N ℓ ( μ 1 ( 2 )) p μ 1 ( 2 ) ( a / N ) ) Lemma 5.2 and the CLT assumptions.
Next, we move on to the high-temperature regime. In this regime, let C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2 ln B a ( x ⃗ ; θ ) \ln B_a(\vec{x};\theta) ln B a ( x ; θ ) p μ 1 ( a ) N ℓ ( μ 1 ) p μ 2 ( x ⃗ ) \frac{p_{\mu_1}(a)}{N^{\ell(\mu_1)}}p_{\mu_2}(\vec{x}) N ℓ ( μ 1 ) p μ 1 ( a ) p μ 2 ( x ) ν ρ ( M , c ) \nu_\rho(M,c) ν ρ ( M , c ) ν ( M ) \nu(M) ν ( M )
Lemma 5.5. In the high-temperature regime, for μ 1 , μ 2 ⊢ d \mu_1,\mu_2\vdash d μ 1 , μ 2 ⊢ d N N N C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2
1 − ℓ ( μ 2 ) , 1-\ell(\mu_2), 1 − ℓ ( μ 2 ) , and N ℓ ( μ 2 ) − 1 C μ 1 , μ 2 N^{\ell(\mu_2)-1}C_{\mu_1,\mu_2} N ℓ ( μ 2 ) − 1 C μ 1 , μ 2
Proof. Set
u = N θ , α = θ − 1 = N u . u=N\theta,\qquad\alpha=\theta^{-1}=\frac{N}{u}. u = N θ , α = θ − 1 = u N . First, consider u > d u>d u > d N > d N>d N > d n = d n=d n = d θ = u / N \theta=u/N θ = u / N p μ 1 ( a ) N ℓ ( μ 1 ) p μ 2 ( x ⃗ ) \frac{p_{\mu_1}(a)}{N^{\ell(\mu_1)}}p_{\mu_2}(\vec{x}) N ℓ ( μ 1 ) p μ 1 ( a ) p μ 2 ( x )
∑ M ∈ C d , ∘ ( ∞ ) : M n o r m a l , μ 1 ( M ) = μ 1 , μ 2 ( M ) = μ 2 1 d ( N u ) d − 1 N 2 − 2 g ( M ) ( N u − 1 ) ν ( M ) N − ℓ ( μ 2 ) N − d f η ( M ) ( { ( − 1 ) k } k ≥ 1 ) . (38) \sum_{\substack{\mathrm{M}\in\mathcal{C}^{(\infty)}_{d,\circ}:\\ \mathrm{M\ normal},\ \mu_1(\mathrm{M})=\mu_1,\ \mu_2(\mathrm{M})=\mu_2}}
\frac{1}{d}\left(\frac{N}{u}\right)^{d-1}N^{2-2g(\mathrm{M})}
\left(\frac{N}{u}-1\right)^{\nu(\mathrm{M})}
N^{-\ell(\mu_2)}N^{-d}f_\eta(\mathrm{M})\left(\{(-1)^k\}_{k\geq1}\right). \tag*{(38)} M ∈ C d , ∘ ( ∞ ) : M normal , μ 1 ( M ) = μ 1 , μ 2 ( M ) = μ 2 ∑ d 1 ( u N ) d − 1 N 2 − 2 g ( M ) ( u N − 1 ) ν ( M ) N − ℓ ( μ 2 ) N − d f η ( M ) ( {( − 1 ) k } k ≥ 1 ) . (38) We claim that this is a formal power series in the variables u − 1 u^{-1} u − 1 N − 1 N^{-1} N − 1 M \mathrm{M} M u u u N N N 1 − 2 g ( M ) + ν ( M ) − ℓ ( μ 2 ) 1-2g(\mathrm{M})+\nu(\mathrm{M})-\ell(\mu_2) 1 − 2 g ( M ) + ν ( M ) − ℓ ( μ 2 ) [18 Definitions 3.6–3.8] that 0 ≤ ν ( M ) ≤ 2 g ( M ) 0\leq\nu(\mathrm{M})\leq2g(\mathrm{M}) 0 ≤ ν ( M ) ≤ 2 g ( M )
Moreover, for real u > 0 u>0 u > 0 N ≥ u N\geq u N ≥ u M \mathrm{M} M ν ( M ) ∈ Z ≥ 0 \nu(\mathrm{M})\in\mathbb{Z}_{\geq0} ν ( M ) ∈ Z ≥ 0 2 g ( M ) − ν ( M ) ∈ Z ≥ 0 2g(\mathrm{M})-\nu(\mathrm{M})\in\mathbb{Z}_{\geq0} 2 g ( M ) − ν ( M ) ∈ Z ≥ 0
(1).
∣ f η ( { ( − 1 ) − k } k ≥ 1 ) ∣ ≤ ( 8 e ) ∣ η ∣ ≤ ( 8 e ) 2 g + 2 d , \left|f_\eta\left(\{(-1)^{-k}\}_{k\geq1}\right)\right|\leq(8e)^{|\eta|}\leq(8e)^{2g+2d}, f η ( {( − 1 ) − k } k ≥ 1 ) ≤ ( 8 e ) ∣ η ∣ ≤ ( 8 e ) 2 g + 2 d , and
(2). The total number of constellations in the above sum with genus g g g d 4 d ( d + g ) d^{4d(d+g)} d 4 d ( d + g ) Proposition 2.7 and [22 Remark 5.7] for the proofs of these two facts. Based on (1) and (2), we can upper bound the absolute value of the inner sum of M \mathrm{M} M
1 d u 1 − d N 1 − ℓ ( μ 2 ) d 4 d 2 ( 8 e ) 2 d ∑ ν ( M ) = 0 ∞ d 2 d ν ( M ) ( u / 8 e ) − ν ( M ) ∑ 2 g ( M ) − ν ( M ) = 0 ∞ d 2 d ( 2 g − ν ( M ) ) ( N / 8 e ) − ( 2 g − ν ( M ) ) , \frac{1}{d}u^{1-d}N^{1-\ell(\mu_2)}d^{4d^2}(8e)^{2d}\sum_{\nu(\mathrm{M})=0}^{\infty}d^{2d\nu(\mathrm{M})}(u/8e)^{-\nu(\mathrm{M})}\sum_{2g(\mathrm{M})-\nu(\mathrm{M})=0}^{\infty}d^{2d(2g-\nu(\mathrm{M}))}(N/8e)^{-(2g-\nu(\mathrm{M}))}, d 1 u 1 − d N 1 − ℓ ( μ 2 ) d 4 d 2 ( 8 e ) 2 d ν ( M ) = 0 ∑ ∞ d 2 d ν ( M ) ( u /8 e ) − ν ( M ) 2 g ( M ) − ν ( M ) = 0 ∑ ∞ d 2 d ( 2 g − ν ( M )) ( N /8 e ) − ( 2 g − ν ( M )) , which converges when N N N u > 8 e d 2 d u > 8ed^{2d} u > 8 e d 2 d
It remains to extend the coefficient bound to arbitrary u u u ℜ u > 0 \Re u > 0 ℜ u > 0 d d d C μ 1 , μ 2 C_{\mu_1,\mu_2} C μ 1 , μ 2 θ = u / N \theta=u/N θ = u / N z = N − 1 z=N^{-1} z = N − 1 u u u z = 0 z=0 z = 0
C μ 1 , μ 2 = ∑ r ≤ R N r h r ( u ) C_{\mu_1,\mu_2}=\sum_{r\le R}N^r h_r(u) C μ 1 , μ 2 = r ≤ R ∑ N r h r ( u ) for some finite integer R R R h r h_r h r { ℜ u > 0 } \{\Re u>0\} { ℜ u > 0 } N N N u > 8 e d 2 d u>8ed^{2d} u > 8 e d 2 d O ( N 1 − ℓ ( μ 2 ) ) O(N^{1-\ell(\mu_2)}) O ( N 1 − ℓ ( μ 2 ) ) h r ( u ) = 0 h_r(u)=0 h r ( u ) = 0 r > 1 − ℓ ( μ 2 ) r>1-\ell(\mu_2) r > 1 − ℓ ( μ 2 )
h r ( u ) = 0 for all u > 0 , r > 1 − ℓ ( μ 2 ) . h_r(u)=0\qquad\text{for all }u>0,\ r>1-\ell(\mu_2). h r ( u ) = 0 for all u > 0 , r > 1 − ℓ ( μ 2 ) . In the high-temperature regime, N → ∞ N\to\infty N → ∞ u = u N → γ > 0 u=u_N\to\gamma>0 u = u N → γ > 0
deg N C μ 1 , μ 2 ≤ 1 − ℓ ( μ 2 ) . \deg_N C_{\mu_1,\mu_2}\le1-\ell(\mu_2). deg N C μ 1 , μ 2 ≤ 1 − ℓ ( μ 2 ) . The convergence as N → ∞ N\to\infty N → ∞ h r ( u ) h_r(u) h r ( u ) u → γ u\to\gamma u → γ
Theorem 5.6. In the high-temperature regime, if the CLT of Definition 1.4 holds, then for all d ∈ Z ≥ 1 d\in\mathbb{Z}_{\ge1} d ∈ Z ≥ 1 k 1 , k 2 ∈ Z ≥ 1 k_1,k_2\in\mathbb{Z}_{\ge1} k 1 , k 2 ∈ Z ≥ 1
d ⋅ c ( d ) → κ d γ ; d\cdot c_{(d)}\to\kappa_d^\gamma; d ⋅ c ( d ) → κ d γ ; N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → κ k 1 , k 2 γ ; Nk_1k_2(1+\mathbf{1}_{k_1=k_2})c_{(k_1,k_2)}\to\kappa_{k_1,k_2}^\gamma; N k 1 k 2 ( 1 + 1 k 1 = k 2 ) c ( k 1 , k 2 ) → κ k 1 , k 2 γ ; N ℓ ( λ ) / 2 c λ → 0 for ℓ ( λ ) ≥ 3. (39) N^{\ell(\lambda)/2}c_\lambda\to0\qquad\text{for }\ell(\lambda)\ge3. \tag*{(39)} N ℓ ( λ ) /2 c λ → 0 for ℓ ( λ ) ≥ 3. (39) for certain constants κ d γ , κ k 1 , k 2 γ \kappa_d^\gamma,\kappa_{k_1,k_2}^\gamma κ d γ , κ k 1 , k 2 γ
Proof. The proof is similar to the one in the fixed-temperature regime, but one must keep track of the different scaling. The first line of Eqn. (39) again follows directly. For the third line, by exactly the same argument as in the proof of Theorem 5.4, it remains to estimate the asymptotic order of the terms of the form
N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ⋅ N l c l ( p μ 1 ( 1 ) ( a ) N ℓ ( μ 1 ( 1 ) ) , … , p μ 1 ( l ) ( a ) N ℓ ( μ 1 ( l ) ) ) . N^{-l}\prod_{j=1}^{l}C_{\mu_1(j),\mu_2(j)}\cdot N^l c_l\left(\frac{p_{\mu_1(1)}(a)}{N^{\ell(\mu_1(1))}},\ldots,\frac{p_{\mu_1(l)}(a)}{N^{\ell(\mu_1(l))}}\right). N − l j = 1 ∏ l C μ 1 ( j ) , μ 2 ( j ) ⋅ N l c l ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a ) , … , N ℓ ( μ 1 ( l )) p μ 1 ( l ) ( a ) ) . If l ≥ 3 l\ge3 l ≥ 3 N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ≲ N − l + ∑ j = 1 l ( 1 − ℓ ( μ 2 ( j ) ) ) N^{-l}\prod_{j=1}^{l}C_{\mu_1(j),\mu_2(j)}\lesssim N^{-l+\sum_{j=1}^{l}(1-\ell(\mu_2(j)))} N − l ∏ j = 1 l C μ 1 ( j ) , μ 2 ( j ) ≲ N − l + ∑ j = 1 l ( 1 − ℓ ( μ 2 ( j )))
N l c l ( p μ 1 ( 1 ) ( a ) N ℓ ( μ 1 ( 1 ) ) , … , p μ 1 ( l ) ( a ) N ℓ ( μ 1 ( l ) ) ) = o ( N l / 2 ) . N^l c_l\left(\frac{p_{\mu_1(1)}(a)}{N^{\ell(\mu_1(1))}},\ldots,\frac{p_{\mu_1(l)}(a)}{N^{\ell(\mu_1(l))}}\right)=o(N^{l/2}). N l c l ( N ℓ ( μ 1 ( 1 )) p μ 1 ( 1 ) ( a ) , … , N ℓ ( μ 1 ( l )) p μ 1 ( l ) ( a ) ) = o ( N l /2 ) . Then the product of these two parts is ≲ N − ∑ j = 1 l ℓ ( μ 2 ( j ) ) + l / 2 ≤ N − ℓ ( λ ) / 2 \lesssim N^{-\sum_{j=1}^{l}\ell(\mu_2(j))+l/2}\le N^{-\ell(\lambda)/2} ≲ N − ∑ j = 1 l ℓ ( μ 2 ( j )) + l /2 ≤ N − ℓ ( λ ) /2 l = 2 l=2 l = 2 N − ∑ j = 1 l ℓ ( μ 2 ( j ) ) + l / 2 N^{-\sum_{j=1}^{l}\ell(\mu_2(j))+l/2} N − ∑ j = 1 l ℓ ( μ 2 ( j )) + l /2 μ 2 ( j ) \mu_2(j) μ 2 ( j ) − ∑ j = 1 l ℓ ( μ 2 ( j ) ) + l / 2 < − ∑ j = 1 l ℓ ( μ 2 ( j ) ) 2 = − ℓ ( λ ) 2 -\sum_{j=1}^{l}\ell(\mu_2(j))+l/2<-\frac{\sum_{j=1}^{l}\ell(\mu_2(j))}{2}=-\frac{\ell(\lambda)}{2} − ∑ j = 1 l ℓ ( μ 2 ( j )) + l /2 < − 2 ∑ j = 1 l ℓ ( μ 2 ( j )) = − 2 ℓ ( λ )
If l = 1 l=1 l = 1 λ = μ 2 \lambda=\mu_2 λ = μ 2 ℓ ( μ 2 ) ≥ 3 \ell(\mu_2)\ge3 ℓ ( μ 2 ) ≥ 3 C μ 1 , μ 2 ∼ O ( N 1 − ℓ ( μ 2 ) ) C_{\mu_1,\mu_2}\sim O(N^{1-\ell(\mu_2)}) C μ 1 , μ 2 ∼ O ( N 1 − ℓ ( μ 2 ) ) c 1 ( p μ 1 ( a ) / N ℓ ( μ 1 ) ) ≲ O ( 1 ) c_1(p_{\mu_1}(a)/N^{\ell(\mu_1)})\lesssim O(1) c 1 ( p μ 1 ( a ) / N ℓ ( μ 1 ) ) ≲ O ( 1 ) N ℓ ( μ 2 ) / 2 N^{\ell(\mu_2)/2} N ℓ ( μ 2 ) /2 O ( N 1 − ℓ ( μ 2 ) / 2 ) = o ( 1 ) O(N^{1-\ell(\mu_2)/2})=o(1) O ( N 1 − ℓ ( μ 2 ) /2 ) = o ( 1 ) (39) .
The second line of Eqn. (39) also follows similarly from Lemma 5.5, the CLT assumption and the above argument.
Vanishing of the higher joint cumulants
In this section we complete the proofs of Theorems 3.12 and 3.13 by proving that, under the corresponding fixed- and high-temperature scalings, every classical joint cumulant of order at least three formed from the power sums converges to zero as N → ∞ N\to\infty N → ∞
Theorem 6.1. Assume the fixed-temperature coefficient conditions in Eqn. (13) . Then, for every s ≥ 3 s\ge3 s ≥ 3 k 1 , … , k s ≥ 1 k_1,\ldots,k_s\ge1 k 1 , … , k s ≥ 1
N s c s ( p ~ k 1 N , … , p ~ k s N ) → 0. N^s c_s\left(\widetilde{p}_{k_1}^N,\ldots,\widetilde{p}_{k_s}^N\right)\to0. N s c s ( p k 1 N , … , p k s N ) → 0. Assume instead the high-temperature coefficient conditions in Eqn. (15) . Then
N s / 2 c s ( p k 1 N , … , p k s N ) → 0. N^{s/2}c_s\left(p_{k_1}^N,\ldots,p_{k_s}^N\right)\to0. N s /2 c s ( p k 1 N , … , p k s N ) → 0. Consequently, the moment–cumulant relations, together with the first- and second-cumulant limits proved earlier, give the moment convergence in Definitions 1.2 and 1.4.
Proof. The proof has four steps. We first expand each c λ c_\lambda c λ d d d d d d o ( 1 ) o(1) o ( 1 )
Step 1. Cumulant expansion and coefficient estimates. We use the same expansion that appeared in the proofs of Theorems 5.4 and 5.6 , but invert its highest-total-degree linear part. Write
Z k F , N : = p ~ k N , Z k H , N : = p k N , Z μ ⋄ , N : = ∏ i = 1 ℓ ( μ ) Z μ i ⋄ , N , ⋄ ∈ { F , H } . Z_k^{\mathrm{F},N}:=\widetilde{p}_k^N,\qquad Z_k^{\mathrm{H},N}:=p_k^N,\qquad Z_\mu^{\diamond,N}:=\prod_{i=1}^{\ell(\mu)}Z_{\mu_i}^{\diamond,N},\qquad\diamond\in\{\mathrm{F},\mathrm{H}\}. Z k F , N := p k N , Z k H , N := p k N , Z μ ⋄ , N := i = 1 ∏ ℓ ( μ ) Z μ i ⋄ , N , ⋄ ∈ { F , H } . Thus
Z μ F , N = p μ ( a / N ) N ℓ ( μ ) , Z μ H , N = p μ ( a ) N ℓ ( μ ) . Z_\mu^{\mathrm{F},N}=\frac{p_\mu(a/N)}{N^{\ell(\mu)}},\qquad Z_\mu^{\mathrm{H},N}=\frac{p_\mu(a)}{N^{\ell(\mu)}}. Z μ F , N = N ℓ ( μ ) p μ ( a / N ) , Z μ H , N = N ℓ ( μ ) p μ ( a ) . For the two regimes, respectively, write directly
log B a ( x ⃗ ; θ ) = ∑ ν ∑ μ ⊢ ∣ ν ∣ C μ , ν ⋄ , N Z μ ⋄ , N p ν ( x ⃗ ) . \log B_a(\vec{x};\theta)=\sum_{\nu}\sum_{\mu\vdash|\nu|}C_{\mu,\nu}^{\diamond,N}Z_\mu^{\diamond,N}p_\nu(\vec{x}). log B a ( x ; θ ) = ν ∑ μ ⊢ ∣ ν ∣ ∑ C μ , ν ⋄ , N Z μ ⋄ , N p ν ( x ) . For a fixed total degree only finitely many terms occur. Since G N , θ = E [ exp ( log B a ) ] G_{N,\theta}=\mathbb{E}[\exp(\log B_a)] G N , θ = E [ exp ( log B a )] λ ⊢ d \lambda\vdash d λ ⊢ d
c λ = ∑ r = 1 ℓ ( λ ) 1 r ! ∑ ν ( 1 ) , … , ν ( r ) ≠ ∅ ν ( 1 ) ⊔ ⋯ ⊔ ν ( r ) = λ ∑ μ ( j ) ⊢ ∣ ν ( j ) ∣ 1 ≤ j ≤ r ( ∏ j = 1 r C μ ( j ) , ν ( j ) ⋄ , N ) c r ( Z μ ( 1 ) ⋄ , N , … , Z μ ( r ) ⋄ , N ) . (40) c_\lambda=\sum_{r=1}^{\ell(\lambda)}\frac{1}{r!}\sum_{\substack{\nu^{(1)},\ldots,\nu^{(r)}\ne\varnothing\\ \nu^{(1)}\sqcup\cdots\sqcup\nu^{(r)}=\lambda}}\sum_{\substack{\mu^{(j)}\vdash|\nu^{(j)}|\\1\le j\le r}}\left(\prod_{j=1}^{r}C_{\mu^{(j)},\nu^{(j)}}^{\diamond,N}\right)c_r\left(Z_{\mu^{(1)}}^{\diamond,N},\ldots,Z_{\mu^{(r)}}^{\diamond,N}\right). \tag*{(40)} c λ = r = 1 ∑ ℓ ( λ ) r ! 1 ν ( 1 ) , … , ν ( r ) = ∅ ν ( 1 ) ⊔ ⋯ ⊔ ν ( r ) = λ ∑ μ ( j ) ⊢ ∣ ν ( j ) ∣ 1 ≤ j ≤ r ∑ ( j = 1 ∏ r C μ ( j ) , ν ( j ) ⋄ , N ) c r ( Z μ ( 1 ) ⋄ , N , … , Z μ ( r ) ⋄ , N ) . (40) Here ⊔ \sqcup ⊔ ν ( 1 ) , … , ν ( r ) \nu^{(1)},\ldots,\nu^{(r)} ν ( 1 ) , … , ν ( r ) 1 / r ! 1/r! 1/ r ! λ \lambda λ
We now separate the two kinds of terms that arise from Eqn. (40) . Apply the Leonov–Shiryaev formula, Lemma 5.2 , to each classical cumulant in that equation. For a fixed summand, write
ρ = μ ( 1 ) ⊔ ⋯ ⊔ μ ( r ) , \rho=\mu^{(1)}\sqcup\cdots\sqcup\mu^{(r)}, ρ = μ ( 1 ) ⊔ ⋯ ⊔ μ ( r ) , and let τ \tau τ ℓ ( ρ ) \ell(\rho) ℓ ( ρ ) r r r μ ( 1 ) , … , μ ( r ) \mu^{(1)},\ldots,\mu^{(r)} μ ( 1 ) , … , μ ( r ) π \pi π π ∨ τ = 1 ^ \pi\vee\tau=\hat{1} π ∨ τ = 1 ^
We call the contribution with π = 1 ^ \pi=\hat{1} π = 1 ^ linear part , because it contains one joint cumulant of all ℓ ( ρ ) \ell(\rho) ℓ ( ρ ) ℓ ( ρ ) ≥ 3 \ell(\rho)\ge3 ℓ ( ρ ) ≥ 3 ℓ ( ρ ) ≤ 2 \ell(\rho)\le2 ℓ ( ρ ) ≤ 2 o ( 1 ) o(1) o ( 1 )
We call the contributions with more than one block in π \pi π nonlinear part . By the Leonov–Shiryaev formula, each such contribution is a product of joint cumulants, and every factor has total degree strictly smaller than d d d d d d o ( 1 ) o(1) o ( 1 )
We need two facts about the coefficients in Eqn. (40) . Lemmas 5.1 and 5.5 give
C μ , ν F , N ∼ O ( N 2 r − ℓ ( ν ) ) , C μ , ν H , N ∼ O ( N 1 − ℓ ( ν ) ) . (41) C_{\mu,\nu}^{\mathrm{F},N}\sim O\left(N^{2r-\ell(\nu)}\right),\qquad C_{\mu,\nu}^{\mathrm{H},N}\sim O\left(N^{1-\ell(\nu)}\right). \tag*{(41)} C μ , ν F , N ∼ O ( N 2 r − ℓ ( ν ) ) , C μ , ν H , N ∼ O ( N 1 − ℓ ( ν ) ) . (41) The triangular moment–cumulant relations in Theorem 3.2 and [7 Theorem 3.10] , together with the first lines of Eqns. (5) and (7) , imply that the normalized one-row diagonal coefficients converge to nonzero limits:
C ( k ) , ( k ) F , N N ⟶ a nonzero limit , C ( k ) , ( k ) H , N ⟶ a nonzero limit . (42) \frac{C_{(k),(k)}^{\mathrm{F},N}}{N}\longrightarrow\text{a nonzero limit},\qquad C_{(k),(k)}^{\mathrm{H},N}\longrightarrow\text{a nonzero limit}. \tag*{(42)} N C ( k ) , ( k ) F , N ⟶ a nonzero limit , C ( k ) , ( k ) H , N ⟶ a nonzero limit . (42) Step 2. The linear system in total degree d d d Fix d ≥ 3 d\ge3 d ≥ 3 d d d
P ≥ 3 ( d ) : = { ρ ⊢ d : ℓ ( ρ ) ≥ 3 } . \mathcal{P}_{\ge3}(d):=\{\rho\vdash d:\ell(\rho)\ge3\}. P ≥ 3 ( d ) := { ρ ⊢ d : ℓ ( ρ ) ≥ 3 } . For ρ ∈ P ≥ 3 ( d ) \rho\in\mathcal{P}_{\ge3}(d) ρ ∈ P ≥ 3 ( d )
K ρ F , N : = N ℓ ( ρ ) c ℓ ( ρ ) ( Z ρ 1 F , N , … , Z ρ ℓ ( ρ ) F , N ) , K_{\rho}^{\mathrm{F},N}:=N^{\ell(\rho)}c_{\ell(\rho)}\left(Z_{\rho_1}^{\mathrm{F},N},\ldots,Z_{\rho_{\ell(\rho)}}^{\mathrm{F},N}\right), K ρ F , N := N ℓ ( ρ ) c ℓ ( ρ ) ( Z ρ 1 F , N , … , Z ρ ℓ ( ρ ) F , N ) , K ρ H , N : = N ℓ ( ρ ) / 2 c ℓ ( ρ ) ( Z ρ 1 H , N , … , Z ρ ℓ ( ρ ) H , N ) . K_{\rho}^{\mathrm{H},N}:=N^{\ell(\rho)/2}c_{\ell(\rho)}\left(Z_{\rho_1}^{\mathrm{H},N},\ldots,Z_{\rho_{\ell(\rho)}}^{\mathrm{H},N}\right). K ρ H , N := N ℓ ( ρ ) /2 c ℓ ( ρ ) ( Z ρ 1 H , N , … , Z ρ ℓ ( ρ ) H , N ) . We shall obtain a square linear system for the vectors K d ⋄ , N : = ( K ρ ⋄ , N ) ρ ∈ P ≥ 3 ( d ) K_d^{\diamond,N}:=(K_\rho^{\diamond,N})_{\rho\in\mathcal{P}_{\ge3}(d)} K d ⋄ , N := ( K ρ ⋄ , N ) ρ ∈ P ≥ 3 ( d )
We first treat the linear terms with ℓ ( ρ ) ≥ 3 \ell(\rho)\ge3 ℓ ( ρ ) ≥ 3 K ρ F , N K_\rho^{\mathrm{F},N} K ρ F , N K ρ F , N K_\rho^{\mathrm{F},N} K ρ F , N (41) ,
O ( N 2 r − ℓ ( λ ) − ℓ ( ρ ) ) . O\left(N^{2r-\ell(\lambda)-\ell(\rho)}\right). O ( N 2 r − ℓ ( λ ) − ℓ ( ρ ) ) . Indeed, the product of the C C C O ( N 2 r − ℓ ( λ ) ) O(N^{2r-\ell(\lambda)}) O ( N 2 r − ℓ ( λ ) ) N − ℓ ( ρ ) K ρ F , N N^{-\ell(\rho)}K_\rho^{\mathrm{F},N} N − ℓ ( ρ ) K ρ F , N r ≤ ℓ ( λ ) r\le\ell(\lambda) r ≤ ℓ ( λ ) r ≤ ℓ ( ρ ) r\le\ell(\rho) r ≤ ℓ ( ρ ) r = ℓ ( λ ) = ℓ ( ρ ) r=\ell(\lambda)=\ell(\rho) r = ℓ ( λ ) = ℓ ( ρ ) ν ( j ) \nu^{(j)} ν ( j ) μ ( j ) \mu^{(j)} μ ( j ) ρ = λ \rho=\lambda ρ = λ o ( 1 ) o(1) o ( 1 )
The same argument in the high-temperature regime is applied after multiplying both sides of Eqn. (40) by N ℓ ( λ ) / 2 N^{\ell(\lambda)/2} N ℓ ( λ ) /2 N ℓ ( λ ) / 2 c λ → 0 N^{\ell(\lambda)/2}c_\lambda\to0 N ℓ ( λ ) /2 c λ → 0 K ρ H , N K_\rho^{\mathrm{H},N} K ρ H , N
O ( N r − ℓ ( λ ) / 2 − ℓ ( ρ ) / 2 ) ∼ O ( N ( 2 r − ℓ ( λ ) − ℓ ( ρ ) ) / 2 ) , O\left(N^{r-\ell(\lambda)/2-\ell(\rho)/2}\right)\sim O\left(N^{(2r-\ell(\lambda)-\ell(\rho))/2}\right), O ( N r − ℓ ( λ ) /2 − ℓ ( ρ ) /2 ) ∼ O ( N ( 2 r − ℓ ( λ ) − ℓ ( ρ )) /2 ) , and equality in the exponent again occurs only for the diagonal term r = ℓ ( λ ) = ℓ ( ρ ) r=\ell(\lambda)=\ell(\rho) r = ℓ ( λ ) = ℓ ( ρ ) ρ = λ \rho=\lambda ρ = λ
It remains only to note that the diagonal limit is nonzero. The ordered single-row decompositions of λ \lambda λ 1 / ℓ ( λ ) ! 1/\ell(\lambda)! 1/ ℓ ( λ )! (40) , contribute a fixed nonzero combinatorial factor depending on the multiplicities of the parts of λ \lambda λ
∏ i = 1 ℓ ( λ ) C ( λ i ) , ( λ i ) F , N N . \prod_{i=1}^{\ell(\lambda)} \frac{C^{\mathrm{F},N}_{(\lambda_i),(\lambda_i)}}{N}. i = 1 ∏ ℓ ( λ ) N C ( λ i ) , ( λ i ) F , N . and the high-temperature diagonal entry is
∏ i = 1 ℓ ( λ ) C ( λ i ) , ( λ i ) H , N . \prod_{i=1}^{\ell(\lambda)} C^{\mathrm{H},N}_{(\lambda_i),(\lambda_i)}. i = 1 ∏ ℓ ( λ ) C ( λ i ) , ( λ i ) H , N . Both converge to nonzero limits by Eqn. (42) .
We finally treat the linear terms with ℓ ( ρ ) ≤ 2 \ell(\rho)\leq2 ℓ ( ρ ) ≤ 2 K d ∘ , N K_d^{\circ,N} K d ∘ , N ρ ∈ P ≥ 3 ( d ) \rho\in\mathcal{P}_{\geq3}(d) ρ ∈ P ≥ 3 ( d ) ℓ ( ρ ) = 1 \ell(\rho)=1 ℓ ( ρ ) = 1 ℓ ( ρ ) = 2 \ell(\rho)=2 ℓ ( ρ ) = 2 ℓ ( ρ ) = 2 \ell(\rho)=2 ℓ ( ρ ) = 2 K ρ ∘ , N ∼ O ( 1 ) K_\rho^{\circ,N}\sim O(1) K ρ ∘ , N ∼ O ( 1 ) ℓ ( ρ ) ≥ 3 \ell(\rho)\geq3 ℓ ( ρ ) ≥ 3 K ρ ∘ , N K_\rho^{\circ,N} K ρ ∘ , N O ( N 2 r − ℓ ( λ ) − ℓ ( ρ ) ) O(N^{2r-\ell(\lambda)-\ell(\rho)}) O ( N 2 r − ℓ ( λ ) − ℓ ( ρ ) ) O ( N r − ℓ ( λ ) / 2 − ℓ ( ρ ) / 2 ) O(N^{r-\ell(\lambda)/2-\ell(\rho)/2}) O ( N r − ℓ ( λ ) /2 − ℓ ( ρ ) /2 ) o ( 1 ) o(1) o ( 1 ) r ≤ ℓ ( ρ ) r\leq\ell(\rho) r ≤ ℓ ( ρ ) ℓ ( λ ) ≥ 3 \ell(\lambda)\geq3 ℓ ( λ ) ≥ 3 ℓ ( ρ ) = 2 \ell(\rho)=2 ℓ ( ρ ) = 2 o ( 1 ) o(1) o ( 1 ) ℓ ( ρ ) = 1 \ell(\rho)=1 ℓ ( ρ ) = 1 r = 1 r=1 r = 1 (40) , and the corresponding terms in c λ c_\lambda c λ N ℓ ( λ ) / 2 c λ N^{\ell(\lambda)/2}c_\lambda N ℓ ( λ ) /2 c λ C ( d ) , λ F , N ⋅ E [ Z d ∘ , N ] C^{\mathrm{F},N}_{(d),\lambda}\cdot\mathbb{E}[Z_d^{\circ,N}] C ( d ) , λ F , N ⋅ E [ Z d ∘ , N ] N ℓ ( λ ) / 2 C ( d ) , λ H , N ⋅ E [ Z d ∘ , N ] N^{\ell(\lambda)/2}C^{\mathrm{H},N}_{(d),\lambda}\cdot\mathbb{E}[Z_d^{\circ,N}] N ℓ ( λ ) /2 C ( d ) , λ H , N ⋅ E [ Z d ∘ , N ] E [ Z d ∘ , N ] ∼ O ( 1 ) \mathbb{E}[Z_d^{\circ,N}]\sim O(1) E [ Z d ∘ , N ] ∼ O ( 1 ) C ( d ) , λ F , N C^{\mathrm{F},N}_{(d),\lambda} C ( d ) , λ F , N N ℓ ( λ ) / 2 C ( d ) , λ H , N N^{\ell(\lambda)/2}C^{\mathrm{H},N}_{(d),\lambda} N ℓ ( λ ) /2 C ( d ) , λ H , N o ( 1 ) o(1) o ( 1 ) (41) . Thus the contribution of linear terms with ℓ ( ρ ) = 1 \ell(\rho)=1 ℓ ( ρ ) = 1 o ( 1 ) o(1) o ( 1 )
Step 3. The nonlinear remainder and the induction step. The goal of this step is to show that the nonlinear part defined after Eqn. (40) is o ( 1 ) o(1) o ( 1 ) π \pi π r = 1 r=1 r = 1 r ≥ 2 r\geq2 r ≥ 2
We first treat the nonlinear terms with r = 1 r=1 r = 1 π \pi π π \pi π O ( 1 ) O(1) O ( 1 ) ν ( 1 ) = λ \nu^{(1)}=\lambda ν ( 1 ) = λ (40) , including its coefficient C μ ( 1 ) , λ F , N C^{\mathrm{F},N}_{\mu^{(1)},\lambda} C μ ( 1 ) , λ F , N O ( N 2 − ℓ ( λ ) ) ∼ o ( 1 ) O(N^{2-\ell(\lambda)})\sim o(1) O ( N 2 − ℓ ( λ ) ) ∼ o ( 1 ) C μ ( 1 ) , λ H , N C^{\mathrm{H},N}_{\mu^{(1)},\lambda} C μ ( 1 ) , λ H , N N ℓ ( λ ) / 2 N^{\ell(\lambda)/2} N ℓ ( λ ) /2 O ( N 1 − ℓ ( λ ) / 2 ) ∼ o ( 1 ) O(N^{1-\ell(\lambda)/2})\sim o(1) O ( N 1 − ℓ ( λ ) /2 ) ∼ o ( 1 )
We next set up the estimate for r ≥ 2 r\geq2 r ≥ 2
S ( π ) : = ∑ B ∈ π ∣ B ∣ ≥ 2 ∣ B ∣ . S(\pi):=\sum_{\substack{B\in\pi\\ |B|\geq2}} |B|. S ( π ) := B ∈ π ∣ B ∣ ≥ 2 ∑ ∣ B ∣. Recall from Remark 5.3 that S ( π ) ≥ r S(\pi)\geq r S ( π ) ≥ r π \pi π ρ \rho ρ B ∈ π B\in\pi B ∈ π
∑ i ∈ B ρ i ≤ d − 1. (43) \sum_{i\in B}\rho_i\leq d-1. \tag*{(43)} i ∈ B ∑ ρ i ≤ d − 1. (43) This is the point that makes induction on the total degree available: every cumulant factor in the nonlinear term has strictly smaller total degree than the cumulant currently under consideration.
We now carry out the induction on d d d d = 3 d=3 d = 3 ∣ B ∣ ≥ 3 |B| \ge3 ∣ B ∣ ≥ 3 (43) is at least 3, contradicting that it is at most 2. Hence every block of π \pi π O ( 1 ) O(1) O ( 1 ) O ( N − 2 ) O(N^{-2}) O ( N − 2 ) O ( N − 1 ) O(N^{-1}) O ( N − 1 ) O ( N − S ( π ) ) O(N^{-S(\pi)}) O ( N − S ( π ) ) O ( N − S ( π ) / 2 ) O(N^{-S(\pi)/2}) O ( N − S ( π ) /2 )
We next combine these base-case bounds with the coefficients of Eqn. (40) . The whole fixed-temperature term is
O ( N 2 r − ℓ ( λ ) − S ( π ) ) , O(N^{2r-\ell(\lambda)-S(\pi)}), O ( N 2 r − ℓ ( λ ) − S ( π ) ) , and, after the high-temperature normalization, the whole high-temperature term is
O ( N r − ℓ ( λ ) / 2 − S ( π ) / 2 ) . O(N^{r-\ell(\lambda)/2-S(\pi)/2}). O ( N r − ℓ ( λ ) /2 − S ( π ) /2 ) . Both exponents are nonpositive because ℓ ( λ ) ≥ r \ell(\lambda) \ge r ℓ ( λ ) ≥ r S ( π ) ≥ r S(\pi) \ge r S ( π ) ≥ r ℓ ( λ ) = S ( π ) = r \ell(\lambda)=S(\pi)=r ℓ ( λ ) = S ( π ) = r Remark 5.3 , this would require a non-singleton block of size r = ℓ ( λ ) ≥ 3 r=\ell(\lambda)\ge3 r = ℓ ( λ ) ≥ 3 o ( 1 ) o(1) o ( 1 ) d = 3 d=3 d = 3
For the induction step, let d ≥ 4 d \ge4 d ≥ 4 d − 1 d-1 d − 1 (43) , together with ρ i ≥ 1 \rho_i \ge1 ρ i ≥ 1
3 ≤ ∣ B ∣ ≤ ∑ i ∈ B ρ i ≤ d − 1. 3 \le|B| \le\sum_{i\in B}\rho_i \le d-1. 3 ≤ ∣ B ∣ ≤ i ∈ B ∑ ρ i ≤ d − 1. We may therefore apply the induction hypothesis separately to each block B B B 3 ≤ ∣ B ∣ ≤ d − 1 3 \le|B| \le d-1 3 ≤ ∣ B ∣ ≤ d − 1
c ∣ B ∣ ( Z ρ i F , N : i ∈ B ) = { O ( 1 ) , ∣ B ∣ = 1 , O ( N − 2 ) , ∣ B ∣ = 2 , o ( N − ∣ B ∣ ) , 3 ≤ ∣ B ∣ ≤ d − 1. \mathfrak{c}_{|B|}\left(Z_{\rho_i}^{\mathrm{F},N}:i\in B\right)=
\begin{cases}
O(1), & |B|=1,\\
O(N^{-2}), & |B|=2,\\
o(N^{-|B|}), & 3\le|B|\le d-1.
\end{cases} c ∣ B ∣ ( Z ρ i F , N : i ∈ B ) = ⎩ ⎨ ⎧ O ( 1 ) , O ( N − 2 ) , o ( N − ∣ B ∣ ) , ∣ B ∣ = 1 , ∣ B ∣ = 2 , 3 ≤ ∣ B ∣ ≤ d − 1. and
c ∣ B ∣ ( Z ρ i H , N : i ∈ B ) = { O ( 1 ) , ∣ B ∣ = 1 , O ( N − 1 ) , ∣ B ∣ = 2 , o ( N − ∣ B ∣ / 2 ) , 3 ≤ ∣ B ∣ ≤ d − 1. \mathfrak{c}_{|B|}\left(Z_{\rho_i}^{\mathrm{H},N}:i\in B\right)=
\begin{cases}
O(1), & |B|=1,\\
O(N^{-1}), & |B|=2,\\
o(N^{-|B|/2}), & 3\le|B|\le d-1.
\end{cases} c ∣ B ∣ ( Z ρ i H , N : i ∈ B ) = ⎩ ⎨ ⎧ O ( 1 ) , O ( N − 1 ) , o ( N − ∣ B ∣/2 ) , ∣ B ∣ = 1 , ∣ B ∣ = 2 , 3 ≤ ∣ B ∣ ≤ d − 1. These are precisely the induction estimates needed for the individual factors of the nonlinear term.
We then multiply the estimates over all blocks of π \pi π O ( N − S ( π ) ) O(N^{-S(\pi)}) O ( N − S ( π ) ) O ( N − S ( π ) / 2 ) O(N^{-S(\pi)/2}) O ( N − S ( π ) /2 ) (41) , the complete terms have the same two bounds as in the base case:
O ( N 2 r − ℓ ( λ ) − S ( π ) ) and O ( N r − ℓ ( λ ) / 2 − S ( π ) / 2 ) , O(N^{2r-\ell(\lambda)-S(\pi)}) \quad\text{and} \quad O(N^{r-\ell(\lambda)/2-S(\pi)/2}), O ( N 2 r − ℓ ( λ ) − S ( π ) ) and O ( N r − ℓ ( λ ) /2 − S ( π ) /2 ) , respectively. Again, both exponents are nonpositive.
It remains to handle the critical case where one of these exponents is zero. This forces ℓ ( λ ) = S ( π ) = r \ell(\lambda)=S(\pi)=r ℓ ( λ ) = S ( π ) = r Remark 5.3 , π \pi π π \pi π d − 1 d-1 d − 1 3 ≤ r = ℓ ( λ ) ≤ d − 1 3 \le r = \ell(\lambda) \le d-1 3 ≤ r = ℓ ( λ ) ≤ d − 1 o o o o ( 1 ) o(1) o ( 1 ) O O O
We have now treated every nonlinear term. For fixed d d d o ( 1 ) o(1) o ( 1 ) d − 1 d-1 d − 1 d d d
Step 4. Inversion of the coefficient matrix. We have proved, for the two regimes respectively,
( c λ ) λ ∈ P ≥ 3 ( d ) = A d F , N K d F , N + o ( 1 ) , (c_\lambda)_{\lambda\in\mathcal{P}_{\geq3}(d)}=A_d^{\mathrm{F},N}K_d^{\mathrm{F},N}+o(1), ( c λ ) λ ∈ P ≥ 3 ( d ) = A d F , N K d F , N + o ( 1 ) , ( N ℓ ( λ ) / 2 c λ ) λ ∈ P ≥ 3 ( d ) = A d H , N K d H , N + o ( 1 ) , (N^{\ell(\lambda)/2}c_\lambda)_{\lambda\in\mathcal{P}_{\geq3}(d)}=A_d^{\mathrm{H},N}K_d^{\mathrm{H},N}+o(1), ( N ℓ ( λ ) /2 c λ ) λ ∈ P ≥ 3 ( d ) = A d H , N K d H , N + o ( 1 ) , where Step 2 shows that the two coefficient matrices converge entrywise to diagonal matrices with nonzero diagonal entries. For fixed d d d A d α , N A_d^{\alpha,N} A d α , N N N N K d F , N → 0 K_d^{\mathrm{F},N}\to0 K d F , N → 0 K d H , N → 0 K_d^{\mathrm{H},N}\to0 K d H , N → 0
Combinatorial identification of classical second-order free cumulants
In this section, we show that in the fixed-temperature regime, up to the constant factor 1 θ \frac{1}{\theta} θ 1 [20 ] ; see also [49 Chapter 5] . In particular, our proof gives a direct correspondence between these combinatorial objects and the walk-block configurations of Lukasiewicz paths.
For simplicity, for the rest of this section we set θ = 1 \theta=1 θ = 1
We first recall the following classical notions. For n ∈ Z ≥ 1 n \in\mathbb{Z}_{\geq1} n ∈ Z ≥ 1 S n S_n S n n n n m , n ∈ Z ≥ 1 m,n \in\mathbb{Z}_{\geq1} m , n ∈ Z ≥ 1
Definition 7.1. For n ∈ Z ≥ 1 n \in\mathbb{Z}_{\geq1} n ∈ Z ≥ 1 π = ⨆ i B i \pi=\bigsqcup_i B_i π = ⨆ i B i { 1 , 2 , … , n } \{1,2,\ldots,n\} { 1 , 2 , … , n } i 1 , i 2 ∈ B i i_1,i_2 \in B_i i 1 , i 2 ∈ B i j 1 , j 2 ∈ B j j_1,j_2 \in B_j j 1 , j 2 ∈ B j i 1 < j 1 < i 2 < j 2 i_1<j_1<i_2<j_2 i 1 < j 1 < i 2 < j 2 B i B_i B i π \pi π
There is a canonical way to draw each partition as a graph on a horizontal line or circle, where there are n n n 1 , 2 , … , n 1,2,\ldots,n 1 , 2 , … , n min ( B i ) \min(B_i) min ( B i ) B i B_i B i π \pi π { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } N C ( n ) \mathrm{NC}(n) NC ( n )
Definition 7.2. [49 Section 5.1] The ( m , n ) (m,n) ( m , n ) 1 1 1 m m m m + 1 m+1 m + 1 m + n m+n m + n π \pi π S m + n S_{m+n} S m + n ( m , n ) (m,n) ( m , n ) π \pi π
(i). the cycles do not cross,
(ii). each cycle encloses a region between the circles homeomorphic to a disk with boundary oriented clockwise, and
(iii). at least one cycle connects the two circles.
We denote the set of non-crossing permutations on the ( m , n ) (m,n) ( m , n ) S N C ( m , n ) \mathrm{SNC}(m,n) SNC ( m , n ) 2 2 2 ( m , n ) (m,n) ( m , n ) N C 2 ( m , n ) \mathrm{NC}_{2}(m,n) NC 2 ( m , n )
Proposition 7.3. [49 Section 5.1] Consider π ∈ S N C ( m , n ) \pi\in\mathrm{SNC}(m,n) π ∈ SNC ( m , n ) π \pi π π \pi π { 1 , 2 , … , m + n } \{1,2,\ldots,m+n\} { 1 , 2 , … , m + n } S N C ( m , n ) \mathrm{SNC}(m,n) SNC ( m , n ) { 1 , 2 , … , m } \{1,2,\ldots,m\} { 1 , 2 , … , m } { m + 1 , … , m + n } \{m+1,\ldots,m+n\} { m + 1 , … , m + n }
Example 7.4. For m = n = 2 m=n=2 m = n = 2 S N C ( 2 , 2 ) \mathrm{SNC}(2,2) SNC ( 2 , 2 ) B 1 = { 1 , 2 , 3 , 4 } B_1=\{1,2,3,4\} B 1 = { 1 , 2 , 3 , 4 }
( 1 , 2 , 3 , 4 ) (1,2,3,4) ( 1 , 2 , 3 , 4 ) ( 1 , 2 , 4 , 3 ) (1,2,4,3) ( 1 , 2 , 4 , 3 ) ( 1 , 3 , 4 , 2 ) (1,3,4,2) ( 1 , 3 , 4 , 2 ) ( 1 , 4 , 3 , 2 ) (1,4,3,2) ( 1 , 4 , 3 , 2 )
The four annular drawings are shown in Figure 2.
Figure 2 (PDF p. 50). The four elements of S N C ( 2 , 2 ) \mathrm{SNC}(2,2) SNC ( 2 , 2 )
Definition 7.5. [49 Section 5.4] The second-order non-crossing annular partitions, [ N C ( m ) × N C ( n ) ] [\mathrm{NC}(m)\times\mathrm{NC}(n)] [ NC ( m ) × NC ( n )] σ = ( π 1 , M 1 ) × ( π 2 , M 2 ) \sigma=(\pi_1,M_1)\times(\pi_2,M_2) σ = ( π 1 , M 1 ) × ( π 2 , M 2 ) π 1 × π 2 ∈ N C ( m ) × N C ( n ) \pi_1\times\pi_2\in\mathrm{NC}(m)\times\mathrm{NC}(n) π 1 × π 2 ∈ NC ( m ) × NC ( n ) M 1 ∈ π 1 M_1\in\pi_1 M 1 ∈ π 1 M 2 ∈ π 2 M_2\in\pi_2 M 2 ∈ π 2 M 1 M_1 M 1 M 2 M_2 M 2 π 1 \pi_1 π 1 π 2 \pi_2 π 2
Theorem 7.6. The classical covariance–cumulant formula is due to Collins, Mingo, Śniady, and Speicher [20 ] ; see also [49 Section 5.4] . In the fixed-temperature regime, for θ = 1 \theta=1 θ = 1 { μ N } N ≥ 1 \{\mu_N\}_{N\geq1} { μ N } N ≥ 1 α m , n \alpha_{m,n} α m , n { κ l } l ≥ 1 \{\kappa_l\}_{l\geq1} { κ l } l ≥ 1 { κ k 1 , k 2 } k 1 , k 2 ≥ 1 \{\kappa_{k_1,k_2}\}_{k_1,k_2\geq1} { κ k 1 , k 2 } k 1 , k 2 ≥ 1
α m , n = ∑ π ∈ S N C ( m , n ) ∏ B i ∈ π κ ∣ B i ∣ + ∑ σ ∈ [ N C ( m ) × N C ( n ) ] κ σ . (44) \alpha_{m,n} =
\sum_{\pi\in\mathrm{SNC}(m,n)}
\prod_{B_i\in\pi}\kappa_{\lvert B_i\rvert}
+
\sum_{\sigma\in[\mathrm{NC}(m)\times\mathrm{NC}(n)]}\kappa_\sigma. \tag*{(44)} α m , n = π ∈ SNC ( m , n ) ∑ B i ∈ π ∏ κ ∣ B i ∣ + σ ∈ [ NC ( m ) × NC ( n )] ∑ κ σ . (44) where for each σ ∈ [ N C ( m ) × N C ( n ) ] \sigma\in[\mathrm{NC}(m)\times\mathrm{NC}(n)] σ ∈ [ NC ( m ) × NC ( n )] B 1 , i B_{1,i} B 1 , i N C ( m ) \mathrm{NC}(m) NC ( m ) M 1 M_1 M 1 k 1 k_1 k 1 B 2 , j B_{2,j} B 2 , j N C ( n ) \mathrm{NC}(n) NC ( n ) M 2 M_2 M 2 k 2 k_2 k 2
κ σ = ∏ i κ ∣ B 1 , i ∣ ∏ j κ ∣ B 2 , j ∣ ⋅ κ k 1 , k 2 . (45) \kappa_\sigma=
\prod_i\kappa_{\lvert B_{1,i}\rvert}
\prod_j\kappa_{\lvert B_{2,j}\rvert}\cdot\kappa_{k_1,k_2}. \tag*{(45)} κ σ = i ∏ κ ∣ B 1 , i ∣ j ∏ κ ∣ B 2 , j ∣ ⋅ κ k 1 , k 2 . (45) The main result of this section is the following.
Theorem 7.7. For any m , n ∈ Z ≥ 1 m,n \in\mathbb{Z}_{\ge1} m , n ∈ Z ≥ 1 (14) and (44) are equivalent, when θ = 1 \theta=1 θ = 1
∑ E ∈ L ( m , n ) A ( E ) B ( E ) C ( E ) ∏ B i ∈ E κ ∣ B i ∣ = ∑ π ∈ S N C ( m , n ) ∏ B i ∈ π κ ∣ B i ∣ , (46) \sum_{E\in L(m,n)} A(E)B(E)C(E)\prod_{B_i\in E}\kappa_{\lvert B_i\rvert}
=
\sum_{\pi\in SNC(m,n)}\prod_{B_i\in\pi}\kappa_{\lvert B_i\rvert}, \tag*{(46)} E ∈ L ( m , n ) ∑ A ( E ) B ( E ) C ( E ) B i ∈ E ∏ κ ∣ B i ∣ = π ∈ S N C ( m , n ) ∑ B i ∈ π ∏ κ ∣ B i ∣ , (46) and
∑ E ∈ L ( m , n ) ∑ s ∈ { τ , … , m } ∩ U ( E ) B ( E ) κ k 1 ( E , s ) , k 2 ( E ) ∏ B i ∈ E t ( B i ) ≠ s κ ∣ B i ∣ = ∑ σ ∈ [ N C ( m ) × N C ( n ) ] κ σ . (47) \sum_{E\in L(m,n)}\sum_{s\in\{\tau,\ldots,m\}\cap U(E)}B(E)\kappa_{k_1(E,s),k_2(E)}
\prod_{\substack{B_i\in E\\t(B_i)\ne s}}\kappa_{\lvert B_i\rvert}
=
\sum_{\sigma\in[NC(m)\times NC(n)]}\kappa_\sigma. \tag*{(47)} E ∈ L ( m , n ) ∑ s ∈ { τ , … , m } ∩ U ( E ) ∑ B ( E ) κ k 1 ( E , s ) , k 2 ( E ) B i ∈ E t ( B i ) = s ∏ κ ∣ B i ∣ = σ ∈ [ N C ( m ) × N C ( n )] ∑ κ σ . (47) Proof. The equivalence between Eqn. (14) and (44) follows directly from Eqn. (46) and (47) , and we start by proving Eqn. (46) . First, it is classical (see, e.g., [50 Proposition 9.8] ) that one can associate any π ∈ N C ( m + n ) \pi\in NC(m+n) π ∈ N C ( m + n ) E ∈ L ( m + n ) E\in L(m+n) E ∈ L ( m + n )
For each block B i ∈ π B_i\in\pi B i ∈ π t i ∈ { 1 , 2 , … , m + n } t_i\in\{1,2,\ldots,m+n\} t i ∈ { 1 , 2 , … , m + n } B i B_i B i E E E π \pi π t i t_i t i ∣ B i ∣ − 1 \lvert B_i\rvert-1 ∣ B i ∣ − 1 t i t_i t i E E E − 1 -1 − 1
One can check directly that the above map is bijective.
Therefore we identify E E E π ∈ N C ( m + n ) \pi\in NC(m+n) π ∈ N C ( m + n ) π ~ ∈ S N C ( m , n ) \widetilde{\pi}\in SNC(m,n) π ∈ S N C ( m , n ) π ∈ N C ( m + n ) \pi\in NC(m+n) π ∈ N C ( m + n ) π ~ \widetilde{\pi} π π \pi π { 1 , 2 , … , m + n } \{1,2,\ldots,m+n\} { 1 , 2 , … , m + n } π ~ \widetilde{\pi} π π \pi π
After all these preparations, it suffices to check that for a fixed π ∈ N C ( m + n ) \pi\in NC(m+n) π ∈ N C ( m + n ) π ~ \widetilde{\pi} π S N C ( m , n ) SNC(m,n) S N C ( m , n ) π \pi π E E E A ( E ) B ( E ) C ( E ) A(E)B(E)C(E) A ( E ) B ( E ) C ( E )
The trivial case is when C ( E ) = 0 C(E)=0 C ( E ) = 0 ( m , n ) (m,n) ( m , n ) S N C ( m , n ) SNC(m,n) S N C ( m , n )
For π \pi π C ( E ) ≥ 1 C(E)\ge1 C ( E ) ≥ 1 π ~ 0 \widetilde{\pi}_0 π 0 S N C ( m , n ) SNC(m,n) S N C ( m , n ) π \pi π { 1 , 2 , … , m + n } \{1,2,\ldots,m+n\} { 1 , 2 , … , m + n } π ~ 0 \widetilde{\pi}_0 π 0 π \pi π 6 { m + 1 , … , τ − 1 } \{m+1,\ldots,\tau-1\} { m + 1 , … , τ − 1 } τ \tau τ
This default drawing can be obtained directly from the disk drawing of π \pi π m m m m + 1 m+1 m + 1 m + 1 , … , m + n m+1,\ldots,m+n m + 1 , … , m + n E ( m ) = C ( E ) E(m)=C(E) E ( m ) = C ( E ) C ( E ) > 0 C(E)>0 C ( E ) > 0 π ~ 0 \widetilde{\pi}_0 π 0 π \pi π
Inner reconnection. Let V 0 V_0 V 0 π ~ 0 \widetilde{\pi}_0 π 0 ∣ V 0 ∣ = C ( E ) \lvert V_0\rvert=C(E) ∣ V 0 ∣ = C ( E ) min V 0 = τ \min V_0=\tau min V 0 = τ τ \tau τ V 0 V_0 V 0 τ \tau τ e i ∈ { m + 1 , … , τ − 1 } ∩ D ( E ) e_i \in\{m+1,\ldots,\tau-1\}\cap D(E) e i ∈ { m + 1 , … , τ − 1 } ∩ D ( E ) e i e_i e i τ \tau τ r r r V 0 V_0 V 0
e i = u 1 < ⋯ < u r + C ( E ) . e_i=u_1<\cdots<u_{r+C(E)}. e i = u 1 < ⋯ < u r + C ( E ) . Reassign the selected attachments at u j u_j u j u C ( E ) + j u_{C(E)+j} u C ( E ) + j 1 ≤ j ≤ r 1\leq j\leq r 1 ≤ j ≤ r
V = { u 1 , … , u C ( E ) } . V=\{u_1,\ldots,u_{C(E)}\}. V = { u 1 , … , u C ( E ) } . All other attachments stay fixed, and overlap between the source and receiving sets is allowed. For the default choice, set V = V 0 V=V_0 V = V 0 ∣ V ∣ = C ( E ) |V|=C(E) ∣ V ∣ = C ( E )
The selected blocks form a nested chain. Their attachments retain their order, while every untouched block, together with its interior blocks, remains in a gap between the reconnected legs. Thus the reconnection is non-crossing and preserves all first legs and block sizes. Its choice is recovered from the least inner position belonging to a through-cycle: this is τ \tau τ e i e_i e i
1 + ∣ { m + 1 , … , τ − 1 } ∩ D ( E ) ∣ = B ( E ) 1+\lvert\{m+1,\ldots,\tau-1\}\cap D(E)\rvert=B(E) 1 + ∣{ m + 1 , … , τ − 1 } ∩ D ( E )∣ = B ( E ) inner choices. They determine all purely inner cycles and the set V V V
Outer reconnection. Recall that τ ˉ \bar{\tau} τ ˉ
e ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) . e\in\{\bar{\tau}+1,\ldots,m\}\cap D(E). e ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) . Select e e e V V V
For precision, the cyclic reassignment can be specified as follows. List the r r r C ( E ) C(E) C ( E ) V V V u 1 , … , u r + C ( E ) u_1,\ldots,u_{r+C(E)} u 1 , … , u r + C ( E ) u 1 = e u_1=e u 1 = e u j ↦ u j + r u_j\mapsto u_{j+r} u j ↦ u j + r r + C ( E ) r+C(E) r + C ( E ) V V V π ~ 0 − 1 ( e ) \widetilde{\pi}_0^{-1}(e) π 0 − 1 ( e ) default predecessor of e e e π ~ 0 − 1 ( τ ) \widetilde{\pi}_0^{-1}(\tau) π 0 − 1 ( τ )
1 + ∣ { τ ˉ + 1 , … , m } ∩ D ( E ) ∣ = A ( E ) 1+\lvert\{\bar{\tau}+1,\ldots,m\}\cap D(E)\rvert=A(E) 1 + ∣{ τ ˉ + 1 , … , m } ∩ D ( E )∣ = A ( E ) outer choices, each leaving C ( E ) C(E) C ( E ) Figure 3 illustrates the special case in which the selected attachments belong to a single block.
Figure 3. Single-block examples of the reconnections; every displayed cycle is oriented clockwise. Top: the attachments at e 1 , e 2 e_1,e_2 e 1 , e 2 v 1 , v 2 v_1,v_2 v 1 , v 2 e , f e,f e , f a , e , f > τ ˉ a,e,f>\bar{\tau} a , e , f > τ ˉ a = π ~ 0 − 1 ( e ) a=\tilde{\pi}_0^{-1}(e) a = π ~ 0 − 1 ( e ) e ′ = v 1 e'=v_1 e ′ = v 1 a → e ′ a\to e' a → e ′
Cyclic gluing and inverse recovery. For fixed inner and outer choices, choose e ′ ∈ V e'\in V e ′ ∈ V
e ′ = { π ~ ( π ~ 0 − 1 ( e ) ) , for the outer choice e , π ~ ( π ~ 0 − 1 ( τ ) ) , for the default outer choice. e'=
\begin{cases}
\widetilde{\pi}(\widetilde{\pi}_0^{-1}(e)), & \text{for the outer choice }e,\\
\widetilde{\pi}(\widetilde{\pi}_0^{-1}(\tau)), & \text{for the default outer choice.}
\end{cases} e ′ = { π ( π 0 − 1 ( e )) , π ( π 0 − 1 ( τ )) , for the outer choice e , for the default outer choice. The remaining connections are then made in cyclic order. Thus there are exactly C ( E ) C(E) C ( E )
To recover the choices from π ~ \tilde{\pi} π ~ u 1 u_1 u 1 V V V π ~ \tilde{\pi} π ~ π ~ 0 \tilde{\pi}_0 π ~ 0
e = min { j ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) : π ~ − 1 ( j ) ≠ π ~ 0 − 1 ( j ) } . e=\min\{j\in\{\bar{\tau}+1,\ldots,m\}\cap D(E):\tilde{\pi}^{-1}(j)\ne\tilde{\pi}_0^{-1}(j)\}. e = min { j ∈ { τ ˉ + 1 , … , m } ∩ D ( E ) : π ~ − 1 ( j ) = π ~ 0 − 1 ( j )} . with an empty set indicating the default choice. Indeed, every smaller outer auxiliary position retains its default predecessor, whereas the default predecessor of e e e V V V e ′ e' e ′ ( u 1 , e , e ′ ) (u_1,e,e') ( u 1 , e , e ′ )
To verify that these rules exhaust all annular permutations corresponding to π \pi π [ τ ˉ , m ] [\bar{\tau},m] [ τ ˉ , m ] τ \tau τ [ m + 1 , τ ) [m+1,\tau) [ m + 1 , τ ) τ − m − 1 \tau-m-1 τ − m − 1
First, outer auxiliary positions at or before τ ˉ \bar{\tau} τ ˉ e e e j j j τ ˉ \bar{\tau} τ ˉ b b b a a a a a a a < b a<b a < b b b b k k k j j j a a a j j j b b b k k k [ τ ˉ , m ] [\bar{\tau},m] [ τ ˉ , m ] a a a j j j a a a a a a j j j [ τ ˉ , m ] [\bar{\tau},m] [ τ ˉ , m ] τ ˉ \bar{\tau} τ ˉ
Second, at the first changed outer auxiliary position e e e e e e e e e e e e u > τ u>\tau u > τ b b b τ < b < u \tau<b<u τ < b < u τ \tau τ [ b , u ) [b,u) [ b , u ) [ b , u ) [b,u) [ b , u ) u − b u-b u − b u u u e ′ e' e ′ V 0 ∪ [ m + 1 , … , τ − 1 ∩ D ( E ) ] V_0\cup[\\{m+1,\ldots,\tau-1\\}\cap D(E)] V 0 ∪ [ m + 1 , … , τ − 1 ∩ D ( E )] e ′ ∈ V e'\in V e ′ ∈ V
Once the least inner through-position is fixed, non-crossing forces the inner reconnection along its enclosing chain: a block off that chain stays in an intervening gap, while skipping a participating enclosing block gives alternating endpoints. The same argument, starting at e e e e ′ e' e ′ π ~ 0 \widetilde{\pi}_0 π 0 B ( E ) B(E) B ( E ) A ( E ) A(E) A ( E ) C ( E ) C(E) C ( E ) (46) .
To prove Eqn. (47) , we fix a Lukasiewicz path E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n ) s ∈ { τ ‾ , … , m ‾ } ∩ U ( E ) s\in\{ \overline{\tau},\ldots,\overline{m}\}\cap U(E) s ∈ { τ , … , m } ∩ U ( E ) k 1 ( E , s ) ≥ 1 k_1(E,s)\ge1 k 1 ( E , s ) ≥ 1 σ \sigma σ ( E , s ) (E,s) ( E , s )
The marked block M 1 ∈ π 1 M_1\in\pi_1 M 1 ∈ π 1 k 1 ( E , s ) ≥ 1 k_1(E,s)\ge1 k 1 ( E , s ) ≥ 1 s s s M 2 ∈ π 2 M_2\in\pi_2 M 2 ∈ π 2 k 2 ( E ) = C ( E ) k_2(E)=C(E) k 2 ( E ) = C ( E )
As before, the first-leg positions and sizes of the unmarked blocks B 1 , i B_{1,i} B 1 , i B 2 , i B_{2,i} B 2 , i E E E m m m s s s
Conversely, every σ = ( π 1 , M 1 ) × ( π 2 , M 2 ) \sigma=(\pi_1,M_1)\times(\pi_2,M_2) σ = ( π 1 , M 1 ) × ( π 2 , M 2 ) ( E , s ) (E,s) ( E , s ) s = min M 1 s=\min M_1 s = min M 1 B B B E E E ∣ B ∣ − 1 |B|-1 ∣ B ∣ − 1 s s s ∣ M 1 ∣ + ∣ M 2 ∣ − 1 |M_1|+|M_2|-1 ∣ M 1 ∣ + ∣ M 2 ∣ − 1 m m m π 1 \pi_1 π 1 ∣ M 2 ∣ |M_2| ∣ M 2 ∣ s s s E ( m ) = ∣ M 2 ∣ E(m)=|M_2| E ( m ) = ∣ M 2 ∣ s ∈ { τ ‾ , … , m ‾ } ∩ U ( E ) s\in\{\overline{\tau},\ldots,\overline{m}\}\cap U(E) s ∈ { τ , … , m } ∩ U ( E ) m m m π 2 \pi_2 π 2 ∣ M 2 ∣ |M_2| ∣ M 2 ∣ M 2 M_2 M 2 E ∈ L ( m , n ) E\in L(m,n) E ∈ L ( m , n )
k 1 ( E , s ) = ∣ M 1 ∣ , k 2 ( E ) = ∣ M 2 ∣ . k_1(E,s)=|M_1|,\qquad k_2(E)=|M_2|. k 1 ( E , s ) = ∣ M 1 ∣ , k 2 ( E ) = ∣ M 2 ∣. This construction is inverse to the two rules above. Thus the pairs ( E , s ) (E,s) ( E , s ) (47) , and it remains only to count the partitions corresponding to one fixed pair.
For fixed ( E , s ) (E,s) ( E , s ) N C ( m ) × N C ( n ) NC(m)\times NC(n) N C ( m ) × N C ( n ) C ( E ) C(E) C ( E ) M 2 ( 0 ) M_2^{(0)} M 2 ( 0 ) min M 2 ( 0 ) = τ \min M_2^{(0)}=\tau min M 2 ( 0 ) = τ τ \tau τ
Number these unmarked blocks by increasing first-leg position. Choose an auxiliary position e < τ e<\tau e < τ e e e τ \tau τ e e e r r r M 2 ( 0 ) M_2^{(0)} M 2 ( 0 )
u 1 < ⋯ < u r + C ( E ) . u_1<\cdots<u_{r+C(E)}. u 1 < ⋯ < u r + C ( E ) . Move the selected leg at u i u_i u i u C ( E ) + i u_{C(E)+i} u C ( E ) + i u 1 , … , u C ( E ) u_1,\ldots,u_{C(E)} u 1 , … , u C ( E ) M 2 M_2 M 2 min M 2 = e \min M_2=e min M 2 = e
Example 7.8. Let m ≥ 1 m \ge1 m ≥ 1 M 1 = [ m ] M_1=[m] M 1 = [ m ] n = 9 n=9 n = 9 m + j m+j m + j j j j
B 2 , 1 ( 0 ) = { 1 , 2 , 5 } , B 2 , 2 ( 0 ) = { 3 , 4 } , M 2 ( 0 ) = { 6 , 7 , 8 , 9 } . B^{(0)}_{2,1}=\{1,2,5\},\qquad B^{(0)}_{2,2}=\{3,4\},\qquad M^{(0)}_2=\{6,7,8,9\}. B 2 , 1 ( 0 ) = { 1 , 2 , 5 } , B 2 , 2 ( 0 ) = { 3 , 4 } , M 2 ( 0 ) = { 6 , 7 , 8 , 9 } . Then C ( E ) = 4 C(E)=4 C ( E ) = 4 τ = m + 6 \tau=m+6 τ = m + 6 B ( E ) = 4 B(E)=4 B ( E ) = 4 2 , 4 , 5 2,4,5 2 , 4 , 5 { 2 , 5 } \{2,5\} { 2 , 5 } { 4 , 5 } \{4,5\} { 4 , 5 } { 5 } \{5\} { 5 } 4 4 4 4 4 4 8 8 8 5 5 5 9 9 9
B 2 , 1 = { 1 , 2 , 9 } , B 2 , 2 = { 3 , 8 } , M 2 = { 4 , 5 , 6 , 7 } . B_{2,1}=\{1,2,9\},\qquad B_{2,2}=\{3,8\},\qquad M_2=\{4,5,6,7\}. B 2 , 1 = { 1 , 2 , 9 } , B 2 , 2 = { 3 , 8 } , M 2 = { 4 , 5 , 6 , 7 } . Figure 4 shows all four completions. Their total weight is 4 κ m , 4 κ 3 κ 2 4\kappa_{m,4}\kappa_3\kappa_2 4 κ m , 4 κ 3 κ 2
By viewing such an operation as a map from the default partition to partitions of n n n
It is injective.
The first-leg position of every unmarked block is unchanged.
Any moved unmarked block has its last fixed leg to the left of u 1 u_1 u 1 u C ( E ) u_{C(E)} u C ( E ) M 2 M_2 M 2 M 2 ( 0 ) M^{(0)}_2 M 2 ( 0 ) M 2 M_2 M 2 M 2 M_2 M 2
Conversely, in any partition corresponding to ( E , s ) (E,s) ( E , s ) M 2 M_2 M 2 τ \tau τ τ \tau τ τ − m − 1 \tau-m-1 τ − m − 1 τ \tau τ e e e
By definition, the range of the map above has B ( E ) − 1 B(E)-1 B ( E ) − 1 B ( E ) B(E) B ( E ) (47) .
Figure 4. The four inner partitions in Example 7.8, with local labels increasing from left to right. Arcs join consecutive legs of each block: solid arcs give the unmarked blocks, and dashed arcs give M 2 M_2 M 2 τ \tau τ
Second-order free probability
High-temperature CLTs for Gaussian and Laguerre ensembles
Let θ N → 0 \theta_N \to0 θ N → 0 N θ N → γ > 0 N\theta_N \to\gamma>0 N θ N → γ > 0 N N N β \beta β β = 2 θ N \beta=2\theta_N β = 2 θ N a 1 ≤ ⋯ ≤ a N a_1\le\cdots\le a_N a 1 ≤ ⋯ ≤ a N
1 Z N , θ N G ∏ 1 ≤ i < j ≤ N ( a j − a i ) 2 θ N ∏ i = 1 N e − a i 2 / 2 d a ⃗ . (48) \frac{1}{Z^G_{N,\theta_N}}\prod_{1\le i<j\le N}(a_j-a_i)^{2\theta_N}\prod_{i=1}^{N}e^{-a_i^2/2}\,d\vec a. \tag*{(48)} Z N , θ N G 1 1 ≤ i < j ≤ N ∏ ( a j − a i ) 2 θ N i = 1 ∏ N e − a i 2 /2 d a . (48) Its Bessel generating function is
G N , θ N G ( x ⃗ ) = exp ( 1 2 ∑ i = 1 N x i 2 ) = exp ( 1 2 p 2 ( x ⃗ ) ) . (49) G^G_{N,\theta_N}(\vec{x})=\exp\left(\frac{1}{2}\sum_{i=1}^{N}x_i^2\right)=\exp\left(\frac{1}{2}p_2(\vec{x})\right). \tag*{(49)} G N , θ N G ( x ) = exp ( 2 1 i = 1 ∑ N x i 2 ) = exp ( 2 1 p 2 ( x ) ) . (49) For the Laguerre ensemble, let M N ≥ N M_N \ge N M N ≥ N M N / N → ρ ≥ 1 M_N/N \to\rho\ge1 M N / N → ρ ≥ 1 0 < a 1 ≤ ⋯ ≤ a N 0 < a_1 \le\cdots\le a_N 0 < a 1 ≤ ⋯ ≤ a N
1 Z N , M N , θ N L ∏ 1 ≤ i < j ≤ N ( a j − a i ) 2 θ N ∏ i = 1 N a i θ N ( M N − N + 1 ) − 1 e − a i d a ⃗ . (50) \frac{1}{Z^{\mathrm{L}}_{N,M_N,\theta_N}}\prod_{1\le i<j\le N}(a_j-a_i)^{2\theta_N}\prod_{i=1}^{N}a_i^{\theta_N(M_N-N+1)-1}e^{-a_i}\,d\vec{a}. \tag*{(50)} Z N , M N , θ N L 1 1 ≤ i < j ≤ N ∏ ( a j − a i ) 2 θ N i = 1 ∏ N a i θ N ( M N − N + 1 ) − 1 e − a i d a . (50) The corresponding Bessel generating function, for x ⃗ \vec{x} x
G N , M N , θ N L ( x ⃗ ) = ∏ i = 1 N ( 1 − x i ) − M N θ N = exp ( M N θ N ∑ d ≥ 1 p d ( x ⃗ ) d ) . (51) G^{\mathrm{L}}_{N,M_N,\theta_N}(\vec{x})=\prod_{i=1}^{N}(1-x_i)^{-M_N\theta_N}=\exp\left(M_N\theta_N\sum_{d\ge1}\frac{p_d(\vec{x})}{d}\right). \tag*{(51)} G N , M N , θ N L ( x ) = i = 1 ∏ N ( 1 − x i ) − M N θ N = exp ( M N θ N d ≥ 1 ∑ d p d ( x ) ) . (51) These Bessel generating functions are standard; see [7 Section 4.1 and Remark 4.11] .
Theorem 8.1. Under the scaling above, both the Gaussian ensemble (46) and the Laguerre ensemble (48) satisfy the high-temperature CLT. Their first-order free γ \gamma γ
κ d G , γ = { 1 , d = 2 , 0 , d ≠ 2 , κ d L , γ = λ , λ : = γ ρ , \kappa^{\mathrm{G},\gamma}_d=
\begin{cases}
1, & d=2,\\
0, & d\ne2,
\end{cases}
\qquad
\kappa^{\mathrm{L},\gamma}_d=\lambda,\qquad\lambda:=\gamma\rho, κ d G , γ = { 1 , 0 , d = 2 , d = 2 , κ d L , γ = λ , λ := γ ρ , and all their second-order free γ \gamma γ
κ r , s G , γ = κ r , s L , γ = 0 , r , s ≥ 1. \kappa^{\mathrm{G},\gamma}_{r,s}=\kappa^{\mathrm{L},\gamma}_{r,s}=0,\qquad r,s\ge1. κ r , s G , γ = κ r , s L , γ = 0 , r , s ≥ 1. Consequently, their limiting covariance matrices are obtained from (16) by substituting the respective first-order free cumulants above; in both cases the Conf 2 ( m , n ) \operatorname{Conf}_2(m,n) Conf 2 ( m , n )
Proof. Taking logarithms in (47) and (49) gives
log G N , θ N G = 1 2 p 2 , log G N , M N , θ N L = M N θ N ∑ d ≥ 1 p d d . \log G^{\mathrm{G}}_{N,\theta_N}=\frac{1}{2}p_2,\qquad
\log G^{\mathrm{L}}_{N,M_N,\theta_N}=M_N\theta_N\sum_{d\ge1}\frac{p_d}{d}. log G N , θ N G = 2 1 p 2 , log G N , M N , θ N L = M N θ N d ≥ 1 ∑ d p d . Hence c ( 2 ) G = 1 / 2 c^{\mathrm{G}}_{(2)}=1/2 c ( 2 ) G = 1/2 c ( d ) G = 0 c^{\mathrm{G}}_{(d)}=0 c ( d ) G = 0 d ≠ 2 d\ne2 d = 2
c ( d ) L = M N θ N d , d ≥ 1. c^{\mathrm{L}}_{(d)}=\frac{M_N\theta_N}{d},\qquad d\ge1. c ( d ) L = d M N θ N , d ≥ 1. For both ensembles, c ν = 0 c_\nu=0 c ν = 0 ℓ ( ν ) ≥ 2 \ell(\nu)\ge2 ℓ ( ν ) ≥ 2 M N θ N → γ ρ = λ M_N\theta_N\to\gamma\rho=\lambda M N θ N → γ ρ = λ
d c ( d ) G ⟶ 1 { d = 2 } , d c ( d ) L ⟶ λ , d\,c^{\mathrm{G}}_{(d)}\longrightarrow\mathbf{1}_{\{d=2\}},\qquad
d\,c^{\mathrm{L}}_{(d)}\longrightarrow\lambda, d c ( d ) G ⟶ 1 { d = 2 } , d c ( d ) L ⟶ λ , together with
N r s c ( r , s ) G = N r s c ( r , s ) L = 0 , N ℓ ( ν ) / 2 c ν G = N ℓ ( ν ) / 2 c ν L = 0 for ℓ ( ν ) ≥ 3. Nrs\,c^{\mathrm{G}}_{(r,s)}=Nrs\,c^{\mathrm{L}}_{(r,s)}=0,\qquad
N^{\ell(\nu)/2}c^{\mathrm{G}}_\nu=N^{\ell(\nu)/2}c^{\mathrm{L}}_\nu=0
\quad\text{for }\ell(\nu)\ge3. N r s c ( r , s ) G = N r s c ( r , s ) L = 0 , N ℓ ( ν ) /2 c ν G = N ℓ ( ν ) /2 c ν L = 0 for ℓ ( ν ) ≥ 3. Thus all assumptions of the high-temperature “if” theorem, Theorem 3.13 , hold. The stated CLTs and cumulants follow immediately, and substituting them into (16) gives the covariance assertion. ∎
CLT for fixed- and high-temperature β \beta β
We use the notation a + θ b a+_{\theta}b a + θ b θ \theta θ [22 Section 6.1] , so that β = 2 θ \beta=2\theta β = 2 θ [7 Section 2.4] , we restrict its inputs to compactly supported real symmetric generalized functions, with the normalization of Section 2.3. The addition is characterized by the multiplicativity of Bessel generating functions: if c = a + θ b c=a+_{\theta}b c = a + θ b
G N , θ c ( x ⃗ ) = G N , θ a ( x ⃗ ) G N , θ b ( x ⃗ ) . (52) G^{c}_{N,\theta}(\vec{x})=G^{a}_{N,\theta}(\vec{x})G^{b}_{N,\theta}(\vec{x}). \tag*{(52)} G N , θ c ( x ) = G N , θ a ( x ) G N , θ b ( x ) . (52) Theorem 8.2. Let a ( N ) a(N) a ( N ) b ( N ) b(N) b ( N ) N N N R N \mathbb{R}^{N} R N N ! N! N !
Fixed temperature. Fix θ > 0 \theta>0 θ > 0 a ( N ) a(N) a ( N ) b ( N ) b(N) b ( N )
{ κ d a , κ r , s a } d , r , s ≥ 1 and { κ d b , κ r , s b } d , r , s ≥ 1 , \{\kappa^{a}_{d},\kappa^{a}_{r,s}\}_{d,r,s\geq1}
\qquad\text{and}\qquad
\{\kappa^{b}_{d},\kappa^{b}_{r,s}\}_{d,r,s\geq1}, { κ d a , κ r , s a } d , r , s ≥ 1 and { κ d b , κ r , s b } d , r , s ≥ 1 , respectively, and set
κ d c : = κ d a + κ d b , κ r , s c : = κ r , s a + κ r , s b . \kappa^{c}_{d}:=\kappa^{a}_{d}+\kappa^{b}_{d},\qquad
\kappa^{c}_{r,s}:=\kappa^{a}_{r,s}+\kappa^{b}_{r,s}. κ d c := κ d a + κ d b , κ r , s c := κ r , s a + κ r , s b . Then c ( N ) : = a ( N ) + θ b ( N ) c(N):=a(N)+_{\theta}b(N) c ( N ) := a ( N ) + θ b ( N ) κ d c \kappa^{c}_{d} κ d c κ r , s c \kappa^{c}_{r,s} κ r , s c
α m , n c = 1 θ ∑ E ∈ L ( m , n ) A ( E ) B ( E ) C ( E ) ∏ B i ∈ E κ ∣ B i ∣ c + 1 θ ∑ E ∈ L ( m , n ) ∑ s ∈ { τ , … , m ‾ } ∩ U ( E ) B ( E ) κ k 1 ( E ) , k 2 ( E ) c ∏ B i ∈ E t ( B i ) ≠ s κ ∣ B i ∣ c . (53) \alpha^{c}_{m,n}
=
\frac{1}{\theta}
\sum_{E\in L(m,n)}
A(E)B(E)C(E)
\prod_{B_i\in E}\kappa^{c}_{|B_i|}
+
\frac{1}{\theta}
\sum_{E\in L(m,n)}
\sum_{s\in\{\tau,\ldots,\overline{m}\}\cap U(E)}
B(E)\kappa^{c}_{k_1(E),k_2(E)}
\prod_{\substack{B_i\in E\\t(B_i)\ne s}}\kappa^{c}_{|B_i|}. \tag*{(53)} α m , n c = θ 1 E ∈ L ( m , n ) ∑ A ( E ) B ( E ) C ( E ) B i ∈ E ∏ κ ∣ B i ∣ c + θ 1 E ∈ L ( m , n ) ∑ s ∈ { τ , … , m } ∩ U ( E ) ∑ B ( E ) κ k 1 ( E ) , k 2 ( E ) c B i ∈ E t ( B i ) = s ∏ κ ∣ B i ∣ c . (53) where k 1 ( E ) k_1(E) k 1 ( E ) k 2 ( E ) k_2(E) k 2 ( E )
High temperature. Let θ = θ N → 0 \theta=\theta_N\to0 θ = θ N → 0 N θ N → γ > 0 N\theta_N\to\gamma>0 N θ N → γ > 0 a ( N ) a(N) a ( N ) b ( N ) b(N) b ( N ) γ \gamma γ
{ κ d a , γ , κ r , s a , γ } d , r , s ≥ 1 and { κ d b , γ , κ r , s b , γ } d , r , s ≥ 1 , \{\kappa^{a,\gamma}_{d},\kappa^{a,\gamma}_{r,s}\}_{d,r,s\geq1}
\qquad\text{and}\qquad
\{\kappa^{b,\gamma}_{d},\kappa^{b,\gamma}_{r,s}\}_{d,r,s\geq1}, { κ d a , γ , κ r , s a , γ } d , r , s ≥ 1 and { κ d b , γ , κ r , s b , γ } d , r , s ≥ 1 , and set
κ d c , γ : = κ d a , γ + κ d b , γ , κ r , s c , γ : = κ r , s a , γ + κ r , s b , γ . \kappa^{c,\gamma}_{d}:=\kappa^{a,\gamma}_{d}+\kappa^{b,\gamma}_{d},\qquad
\kappa^{c,\gamma}_{r,s}:=\kappa^{a,\gamma}_{r,s}+\kappa^{b,\gamma}_{r,s}. κ d c , γ := κ d a , γ + κ d b , γ , κ r , s c , γ := κ r , s a , γ + κ r , s b , γ . Then c ( N ) : = a ( N ) + θ N b ( N ) c(N):=a(N)+_{\theta_N}b(N) c ( N ) := a ( N ) + θ N b ( N ) γ \gamma γ κ d c , γ \kappa^{c,\gamma}_{d} κ d c , γ κ r , s c , γ \kappa^{c,\gamma}_{r,s} κ r , s c , γ j = 1 , 2 j=1,2 j = 1 , 2 I k ( j ) , c ( E , q ⃗ ; γ ) I^{(j),c}_{k}(E,\vec{q};\gamma) I k ( j ) , c ( E , q ; γ ) I k ( j ) ( E , q ⃗ ; γ ) I^{(j)}_{k}(E,\vec{q};\gamma) I k ( j ) ( E , q ; γ )
α m , n c , γ = ∑ ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − k I k ( 1 ) , c ( E , q ⃗ ; γ ) + ∑ ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ ∣ D ( E ) ∣ − k I k ( 2 ) , c ( E , q ⃗ ; γ ) . (54) \alpha^{c,\gamma}_{m,n}
=
\sum_{(E,\vec{q})\in Conf_1(m,n)}
\sum_{k=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-k}I^{(1),c}_{k}(E,\vec{q};\gamma)
+
\sum_{(E,\vec{q})\in Conf_2(m,n)}
\sum_{k=1}^{|P(E,\vec{q})|}
\gamma^{|D(E)|-k}I^{(2),c}_{k}(E,\vec{q};\gamma). \tag*{(54)} α m , n c , γ = ( E , q ) ∈ C o n f 1 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − k I k ( 1 ) , c ( E , q ; γ ) + ( E , q ) ∈ C o n f 2 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ ∣ D ( E ) ∣ − k I k ( 2 ) , c ( E , q ; γ ) . (54) Proof. An analogous argument appeared in [7 ] and [22 ] , so we only give a sketch. For u ∈ { a , b , c } u \in\{a,b,c\} u ∈ { a , b , c } c λ u , N c_{\lambda}^{u,N} c λ u , N p λ ( x ⃗ ) p_{\lambda}(\vec{x}) p λ ( x ) ln G N , θ u ( x ⃗ ) \ln G_{N,\theta}^{u}(\vec{x}) ln G N , θ u ( x ) (52) gives
c λ c , N = c λ a , N + c λ b , N c_{\lambda}^{c,N}=c_{\lambda}^{a,N}+c_{\lambda}^{b,N} c λ c , N = c λ a , N + c λ b , N for every partition λ \lambda λ
The only-if statements in Theorems 1.3 and 1.5 give the required coefficient asymptotics for the two inputs. The displayed identity shows that the same asymptotics hold for c ( N ) c(N) c ( N ) Theorems 1.3 and 1.5 respectively. Substituting the summed cumulants into Eqns. (14) and (16) yields Eqns. (53) and (54) , respectively.
gamma to infinity transition
We conclude this section by giving a limit transition from the high-temperature covariance formula to its fixed-temperature counterpart.
Theorem 8.3 (High-to-fixed-temperature transition). Fix θ > 0 \theta> 0 θ > 0 γ > 0 \gamma> 0 γ > 0
{ κ d γ } d ≥ 1 and { κ r , s γ } r , s ≥ 1 \{\kappa_d^\gamma\}_{d\geq1} \qquad\text{and} \qquad\{\kappa_{r,s}^\gamma\}_{r,s\geq1} { κ d γ } d ≥ 1 and { κ r , s γ } r , s ≥ 1 be first- and second-order free γ \gamma γ α m , n γ \alpha_{m,n}^{\gamma} α m , n γ
{ κ d } d ≥ 1 and { κ r , s } r , s ≥ 1 \{\kappa_d\}_{d\geq1} \qquad\text{and} \qquad\{\kappa_{r,s}\}_{r,s\geq1} { κ d } d ≥ 1 and { κ r , s } r , s ≥ 1 be fixed-temperature free cumulants, and denote by α m , n f i x , θ \alpha_{m,n}^{\mathrm{fix},\theta} α m , n fix , θ
Assume that, for every d , r , s ≥ 1 d,r,s\geq1 d , r , s ≥ 1
γ d − 1 κ d γ ⟶ κ d , γ r + s − 1 κ r , s γ ⟶ κ r , s ( γ → ∞ ) . (55) \gamma^{d-1}\kappa_d^\gamma\longrightarrow\kappa_d,\qquad\gamma^{r+s-1}\kappa_{r,s}^\gamma\longrightarrow\kappa_{r,s}\qquad(\gamma\to\infty). \tag*{(55)} γ d − 1 κ d γ ⟶ κ d , γ r + s − 1 κ r , s γ ⟶ κ r , s ( γ → ∞ ) . (55) Then, for every m , n ≥ 1 m,n\geq1 m , n ≥ 1
lim γ → ∞ γ α m , n γ = θ α m , n f i x , θ . (56) \lim_{\gamma\to\infty}\gamma\alpha_{m,n}^\gamma=\theta\alpha_{m,n}^{\mathrm{fix},\theta}. \tag*{(56)} γ → ∞ lim γ α m , n γ = θ α m , n fix , θ . (56) Equivalently,
α m , n f i x , θ = 1 θ lim γ → ∞ γ α m , n γ . \alpha_{m,n}^{\mathrm{fix},\theta}=\frac{1}{\theta}\lim_{\gamma\to\infty}\gamma\alpha_{m,n}^\gamma. α m , n fix , θ = θ 1 γ → ∞ lim γ α m , n γ . Proof. The first relation in (55) is the first-order transition of [7 Section 8.2] ; the second relation is its second-order analogue. We show that these two relations give termwise convergence of the covariance formula.
Fix m , n m,n m , n Definition 3.5 that
D ( E ) = { t : E ( t ) − E ( t − ) = − 1 } D(E)=\{t:E(t)-E(t-)= -1\} D ( E ) = { t : E ( t ) − E ( t − ) = − 1 } is the set of down-step times of E E E U ( E ) U(E) U ( E ) Definitions 3.10 and 3.11 , B i B_i B i t ( B i ) ∈ U ( E ) t(B_i)\in U(E) t ( B i ) ∈ U ( E )
E ( t ( B i ) ) − E ( t ( B i ) − ) = ∣ B i ∣ − 1. E(t(B_i))-E(t(B_i)-)=|B_i|-1. E ( t ( B i )) − E ( t ( B i ) − ) = ∣ B i ∣ − 1. Since a Lukasiewicz path starts and ends at zero, the total height of its nonnegative increments equals the number of its down steps. Hence, for ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) (E,\vec{q}) \in Conf_1(m,n) ( E , q ) ∈ C o n f 1 ( m , n )
∣ D ( E ) ∣ = ∑ B i ∈ E ( ∣ B i ∣ − 1 ) . |D(E)| = \sum_{B_i\in E} (|B_i|-1). ∣ D ( E ) ∣ = B i ∈ E ∑ ( ∣ B i ∣ − 1 ) . By (53), the cumulant attached to B i B_i B i
κ ∣ B i ∣ γ = γ − ( ∣ B i ∣ − 1 ) ( κ ∣ B i ∣ + o ( 1 ) ) . \kappa^{\gamma}_{|B_i|}=\gamma^{-(|B_i|-1)}\left(\kappa_{|B_i|}+o(1)\right). κ ∣ B i ∣ γ = γ − ( ∣ B i ∣ − 1 ) ( κ ∣ B i ∣ + o ( 1 ) ) . Thus a nonnegative increment of height ∣ B i ∣ − 1 |B_i|-1 ∣ B i ∣ − 1 γ − ( ∣ B i ∣ − 1 ) \gamma^{-(|B_i|-1)} γ − ( ∣ B i ∣ − 1 ) γ − ∣ D ( E ) ∣ \gamma^{-|D(E)|} γ − ∣ D ( E ) ∣
For ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) (E,\vec{q}) \in Conf_2(m,n) ( E , q ) ∈ C o n f 2 ( m , n ) s = s ( E , q ⃗ ) s=s(E,\vec{q}) s = s ( E , q )
∣ D ( E ) ∣ = ∑ t ( B i ) ≠ s ( ∣ B i ∣ − 1 ) + k 1 ( E , q ⃗ ) + k 2 ( E ) − 1. |D(E)|=\sum_{t(B_i)\ne s}(|B_i|-1)+k_1(E,\vec{q})+k_2(E)-1. ∣ D ( E ) ∣ = t ( B i ) = s ∑ ( ∣ B i ∣ − 1 ) + k 1 ( E , q ) + k 2 ( E ) − 1. Indeed, k 1 ( E , q ⃗ ) + k 2 ( E ) − 1 = E ( s ) − E ( s − ) k_1(E,\vec{q})+k_2(E)-1=E(s)-E(s-) k 1 ( E , q ) + k 2 ( E ) − 1 = E ( s ) − E ( s − ) s s s s s s γ − ( k 1 + k 2 − 1 ) \gamma^{-(k_1+k_2-1)} γ − ( k 1 + k 2 − 1 ) γ − ( ∣ B i ∣ − 1 ) \gamma^{-(|B_i|-1)} γ − ( ∣ B i ∣ − 1 ) γ − ∣ D ( E ) ∣ \gamma^{-|D(E)|} γ − ∣ D ( E ) ∣ j = 1 , 2 j=1,2 j = 1 , 2
I k ( j ) ( E , q ⃗ ; γ ) = γ − ∣ D ( E ) ∣ ( I k ( j ) ( E , q ⃗ ) + o ( 1 ) ) . I_k^{(j)}(E,\vec{q};\gamma)=\gamma^{-|D(E)|}\left(I_k^{(j)}(E,\vec{q})+o(1)\right). I k ( j ) ( E , q ; γ ) = γ − ∣ D ( E ) ∣ ( I k ( j ) ( E , q ) + o ( 1 ) ) . where the functional on the right is evaluated at the limiting fixed-temperature free cumulants.
The explicit factor γ ∣ D ( E ) ∣ − k \gamma^{|D(E)|-k} γ ∣ D ( E ) ∣ − k k k k P ( E , q ⃗ ) = D ( E ) ∖ J ( E , q ⃗ ) P(E,\vec{q})=D(E)\setminus J(E,\vec{q}) P ( E , q ) = D ( E ) ∖ J ( E , q ) ∣ D ( E ) ∣ − k |D(E)|-k ∣ D ( E ) ∣ − k γ \gamma γ γ ∣ D ( E ) ∣ \gamma^{|D(E)|} γ ∣ D ( E ) ∣ γ \gamma γ γ 1 − k \gamma^{1-k} γ 1 − k
For fixed m , n m,n m , n
γ α m , n γ = ∑ ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ 1 − k ( I k ( 1 ) ( E , q ⃗ ) + o ( 1 ) ) + ∑ ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) ∑ k = 1 ∣ P ( E , q ⃗ ) ∣ γ 1 − k ( I k ( 2 ) ( E , q ⃗ ) + o ( 1 ) ) . \begin{aligned}
\gamma\alpha^\gamma_{m,n}
={}&\sum_{(E,\vec{q})\in Conf_1(m,n)}\sum_{k=1}^{|P(E,\vec{q})|}\gamma^{1-k}\left(I_k^{(1)}(E,\vec{q})+o(1)\right)\\
&+\sum_{(E,\vec{q})\in Conf_2(m,n)}\sum_{k=1}^{|P(E,\vec{q})|}\gamma^{1-k}\left(I_k^{(2)}(E,\vec{q})+o(1)\right).
\end{aligned} γ α m , n γ = ( E , q ) ∈ C o n f 1 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ 1 − k ( I k ( 1 ) ( E , q ) + o ( 1 ) ) + ( E , q ) ∈ C o n f 2 ( m , n ) ∑ k = 1 ∑ ∣ P ( E , q ) ∣ γ 1 − k ( I k ( 2 ) ( E , q ) + o ( 1 ) ) . Every term with k ≥ 2 k\ge2 k ≥ 2 k = 1 k=1 k = 1
lim γ → ∞ γ α m , n γ = ∑ ( E , q ⃗ ) ∈ C o n f 1 ( m , n ) I 1 ( 1 ) ( E , q ⃗ ) + ∑ ( E , q ⃗ ) ∈ C o n f 2 ( m , n ) I 1 ( 2 ) ( E , q ⃗ ) = θ α m , n f i x , θ . \lim_{\gamma\to\infty}\gamma\alpha^\gamma_{m,n}
=\sum_{(E,\vec{q})\in Conf_1(m,n)}I_1^{(1)}(E,\vec{q})+\sum_{(E,\vec{q})\in Conf_2(m,n)}I_1^{(2)}(E,\vec{q})=\theta\alpha^{fix,\theta}_{m,n}. γ → ∞ lim γ α m , n γ = ( E , q ) ∈ C o n f 1 ( m , n ) ∑ I 1 ( 1 ) ( E , q ) + ( E , q ) ∈ C o n f 2 ( m , n ) ∑ I 1 ( 2 ) ( E , q ) = θ α m , n f i x , θ . where the last equality is (14). This proves (54).
Together with the additivity of first- and second-order free cumulants, this gives the covariance-level transition from second-order γ \gamma γ θ = 1 \theta=1 θ = 1 θ \theta θ 1 / θ 1/\theta 1/ θ
Proof of the topological expansion
For completeness, we reproduce the proof of [22 Theorem 5.6] here, following the notation in § 5.4 . Consider the sequences of variables p = ( p 1 , p 2 , … ) \mathbf{p}=(p_1,p_2,\ldots) p = ( p 1 , p 2 , … ) q = ( q 1 , q 2 , … ) \mathbf{q}=(q_1,q_2,\ldots) q = ( q 1 , q 2 , … ) ( g 1 , g 2 , … ) (g_1,g_2,\ldots) ( g 1 , g 2 , … ) J λ ( p ; θ ) J_\lambda(\mathbf{p};\theta) J λ ( p ; θ ) p \mathbf{p} p ∣ λ ∣ |\lambda| ∣ λ ∣ p k p_k p k k k k J λ ( q ; θ ) J_\lambda(\mathbf{q};\theta) J λ ( q ; θ ) G ( z ) = 1 + ∑ k ≥ 1 g k z k G(z)=1+\sum_{k\geq1}g_kz^k G ( z ) = 1 + ∑ k ≥ 1 g k z k [18 ] , define the formal power series
F ^ G ( t , p , q , g 1 , g 2 , … ; θ ) : = ∑ d ≥ 0 t d ∑ ∣ λ ∣ = d J λ ( p ; θ ) J λ ( q ; θ ) ∏ ( i , j ) ∈ λ ( λ i − j ) + θ ( λ j ′ − i ) + θ ( λ i − j ) + θ ( λ j ′ − i ) + 1 ∏ ( i , j ) ∈ λ G ( c θ ( i , j ) ) , (57) \widehat{F}^{G}(t,\mathbf{p},\mathbf{q},g_1,g_2,\ldots;\theta):=\sum_{d\geq0}t^d\sum_{|\lambda|=d}J_\lambda(\mathbf{p};\theta)J_\lambda(\mathbf{q};\theta)\prod_{(i,j)\in\lambda}\frac{(\lambda_i-j)+\theta(\lambda'_j-i)+\theta}{(\lambda_i-j)+\theta(\lambda'_j-i)+1}\prod_{(i,j)\in\lambda}G(c_\theta(i,j)), \tag*{(57)} F G ( t , p , q , g 1 , g 2 , … ; θ ) := d ≥ 0 ∑ t d ∣ λ ∣ = d ∑ J λ ( p ; θ ) J λ ( q ; θ ) ( i , j ) ∈ λ ∏ ( λ i − j ) + θ ( λ j ′ − i ) + 1 ( λ i − j ) + θ ( λ j ′ − i ) + θ ( i , j ) ∈ λ ∏ G ( c θ ( i , j )) , (57) where λ j ′ \lambda'_j λ j ′ j j j λ \lambda λ c θ ( i , j ) : = θ − 1 ( j − 1 ) − ( i − 1 ) c_\theta(i,j):=\theta^{-1}(j-1)-(i-1) c θ ( i , j ) := θ − 1 ( j − 1 ) − ( i − 1 ) [18 Theorem 6.2] , recalled in [22 Theorem 5.5] , states that there exists a non-orientability statistic ν ρ ( M , c ) \nu_\rho(M,c) ν ρ ( M , c )
ln F ^ G = θ ∑ d ≥ 1 t d d ∑ ( M , c ) ∈ C d ( ∞ ) M n o r m a l ( θ − 1 − 1 ) ν ρ ( M , c ) p μ 1 ( M ) q μ 2 ( M ) f η ( M ) ( g 1 , g 2 , … ) . (58) \ln\widehat{F}^{G}=\theta\sum_{d\geq1}\frac{t^d}{d}\sum_{\substack{(M,c)\in\mathcal{C}^{(\infty)}_d\\ M\ \mathrm{normal}}}(\theta^{-1}-1)^{\nu_\rho(M,c)}p_{\mu_1(M)}q_{\mu_2(M)}f_{\eta(M)}(g_1,g_2,\ldots). \tag*{(58)} ln F G = θ d ≥ 1 ∑ d t d ( M , c ) ∈ C d ( ∞ ) M normal ∑ ( θ − 1 − 1 ) ν ρ ( M , c ) p μ 1 ( M ) q μ 2 ( M ) f η ( M ) ( g 1 , g 2 , … ) . (58) Here p μ : = ∏ j p μ j p_\mu:=\prod_jp_{\mu_j} p μ := ∏ j p μ j q μ : = ∏ j q μ j q_\mu:=\prod_jq_{\mu_j} q μ := ∏ j q μ j f η f_\eta f η Theorem 2.8 . (58) is an identity of formal power series; the analytic convergence needed below is a separate part of the proof.
Proof of Theorem 2.8. The multivariate Bessel functions B a ( x ⃗ ; θ ) B_a(\vec{x};\theta) B a ( x ; θ ) ( a , x ⃗ ) ∈ C N × C N (a,\vec{x})\in\mathbb{C}^N\times\mathbb{C}^N ( a , x ) ∈ C N × C N [52 Sec. 4] ):
B a ( x ⃗ ; θ ) = ∑ λ ℓ ( λ ) ≤ N ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + θ ) ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + 1 ) ( N θ + j − 1 − θ ( i − 1 ) ) J λ ( x ⃗ ; θ ) J λ ( a ; θ ) . (59) B_a(\vec{x};\theta)=\sum_{\substack{\lambda\\ \ell(\lambda)\leq N}}\frac{\prod_{(i,j)\in\lambda}\left((\lambda_i-j)+\theta(\lambda'_j-i)+\theta\right)}{\prod_{(i,j)\in\lambda}\left((\lambda_i-j)+\theta(\lambda'_j-i)+1\right)(N\theta+j-1-\theta(i-1))}J_\lambda(\vec{x};\theta)J_\lambda(a;\theta). \tag*{(59)} B a ( x ; θ ) = λ ℓ ( λ ) ≤ N ∑ ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + 1 ) ( N θ + j − 1 − θ ( i − 1 )) ∏ ( i , j ) ∈ λ ( ( λ i − j ) + θ ( λ j ′ − i ) + θ ) J λ ( x ; θ ) J λ ( a ; θ ) . (59) On the other hand, consider the power series F ^ G \widehat{F}^{G} F G g k = ( − N ) − k g_k=(-N)^{-k} g k = ( − N ) − k G ( z ) = ( 1 + z / N ) − 1 G(z)=(1+z/N)^{-1} G ( z ) = ( 1 + z / N ) − 1 t = ( N θ ) − 1 t=(N\theta)^{-1} t = ( N θ ) − 1 x ⃗ = ( x 1 , … , x N ) ∈ C N \vec{x}=(x_1,\ldots,x_N)\in\mathbb{C}^N x = ( x 1 , … , x N ) ∈ C N a = ( a 1 , … , a N ) ∈ C N a=(a_1,\ldots,a_N)\in\mathbb{C}^N a = ( a 1 , … , a N ) ∈ C N p k = p k ( a ) = ∑ i = 1 N a i k p_k=p_k(a)=\sum_{i=1}^N a_i^k p k = p k ( a ) = ∑ i = 1 N a i k q k = p k ( x ⃗ ) = ∑ i = 1 N x i k q_k=p_k(\vec{x})=\sum_{i=1}^N x_i^k q k = p k ( x ) = ∑ i = 1 N x i k
[ F ^ ( 1 + z / N ) − 1 ( 1 N θ , p ( a ) , q ( x ⃗ ) , { ( − N ) − k } k ≥ 1 ; θ ) ] ≤ n = [ B a ( x ⃗ ; θ ) ] ≤ n . (60) \left[\widehat{F}^{(1+z/N)^{-1}}\left(\frac{1}{N\theta},p(a),q(\vec{x}),\{(-N)^{-k}\}_{k\geq1};\theta\right)\right]_{\leq n}=\left[B_a(\vec{x};\theta)\right]_{\leq n}. \tag*{(60)} [ F ( 1 + z / N ) − 1 ( N θ 1 , p ( a ) , q ( x ) , {( − N ) − k } k ≥ 1 ; θ ) ] ≤ n = [ B a ( x ; θ ) ] ≤ n . (60) The brackets denote the terms of total degree at most n n n x ⃗ \vec{x} x n < N n<N n < N N N N B a ( 0 N ; θ ) = 1 B_a(0^N;\theta)=1 B a ( 0 N ; θ ) = 1 x ⃗ = 0 \vec{x}=0 x = 0 0 0 0
By (58) , we have the formal expansion
ln B a ( x ⃗ ; θ ) = ∑ d = 1 n α d − 1 d ∑ ( M , c ) ∈ C d , ρ ( ∞ ) M n o r m a l ( α − 1 ) ν ρ ( M , c ) ∏ f ∈ F ( M ) p deg ( f ) ( a / N ) N ∏ v ∈ V 0 ( M ) p deg ( v ) ( x ⃗ ) N N ℓ ( μ 1 ( M ) ) + ℓ ( μ 2 ( M ) ) − ∣ η ( M ) ∣ f η ( M ) ( { ( − 1 ) − k } k ≥ 1 ) + O ( ∥ x ⃗ ∥ n + 1 ) . (61) \ln B_a(\vec{x};\theta)=\sum_{d=1}^{n}\frac{\alpha^{d-1}}{d}\sum_{\substack{(\mathrm{M},c)\in\mathcal{C}_{d,\rho}^{(\infty)}\\ \mathrm{M}\ \mathrm{normal}}}(\alpha-1)^{\nu_\rho(\mathrm{M},c)}\prod_{f\in F(\mathrm{M})}\frac{p_{\deg(f)}(a/N)}{N}\prod_{v\in V_0(\mathrm{M})}\frac{p_{\deg(v)}(\vec{x})}{N}N^{\ell(\mu_1(\mathrm{M}))+\ell(\mu_2(\mathrm{M}))-|\eta(\mathrm{M})|}f_{\eta(\mathrm{M})}\left(\{(-1)^{-k}\}_{k\geq1}\right)+O(\|\vec{x}\|^{n+1}). \tag*{(61)} ln B a ( x ; θ ) = d = 1 ∑ n d α d − 1 ( M , c ) ∈ C d , ρ ( ∞ ) M normal ∑ ( α − 1 ) ν ρ ( M , c ) f ∈ F ( M ) ∏ N p d e g ( f ) ( a / N ) v ∈ V 0 ( M ) ∏ N p d e g ( v ) ( x ) N ℓ ( μ 1 ( M )) + ℓ ( μ 2 ( M )) − ∣ η ( M ) ∣ f η ( M ) ( {( − 1 ) − k } k ≥ 1 ) + O ( ∥ x ∥ n + 1 ) . (61) Here we have used weighted homogeneity: f η ( { ( − N ) − k } k ≥ 1 ) = N − ∣ η ∣ f η ( { ( − 1 ) k } k ≥ 1 ) f_\eta(\{(-N)^{-k}\}_{k\geq1})=N^{-|\eta|}f_\eta(\{(-1)^k\}_{k\geq1}) f η ({( − N ) − k } k ≥ 1 ) = N − ∣ η ∣ f η ({( − 1 ) k } k ≥ 1 ) N N N (61) is 2 − 2 g ( M ) 2-2g(\mathrm{M}) 2 − 2 g ( M ) g ( M ) = g g(\mathrm{M})=g g ( M ) = g
ℓ ( η ( M ) ) ≤ ∣ η ( M ) ∣ = ℓ ( μ 1 ( M ) ) + ℓ ( μ 2 ( M ) ) + 2 g − 2 < ∣ μ 1 ( M ) ∣ + ∣ μ 2 ( M ) ∣ + 2 g = 2 d + 2 g ≤ 2 n + 2 g . \begin{aligned}
\ell(\eta(\mathrm{M}))\leq|\eta(\mathrm{M})|=\ell(\mu_1(\mathrm{M}))+\ell(\mu_2(\mathrm{M}))+2g-2 \\
&<|\mu_1(\mathrm{M})|+|\mu_2(\mathrm{M})|+2g=2d+2g\leq2n+2g.
\end{aligned} ℓ ( η ( M )) ≤ ∣ η ( M ) ∣ = ℓ ( μ 1 ( M )) + ℓ ( μ 2 ( M )) + 2 g − 2 < ∣ μ 1 ( M ) ∣ + ∣ μ 2 ( M ) ∣ + 2 g = 2 d + 2 g ≤ 2 n + 2 g . As a result, if we set k : = 2 n + 2 g k:=2n+2g k := 2 n + 2 g M \mathrm{M} M k k k [6 Theorem 3.1] , the total number of k k k d d d (61) can be rewritten as (4) , with a finite inner sum for every fixed g ∈ 1 2 Z ≥ 0 g\in\frac{1}{2}\mathbb{Z}_{\geq0} g ∈ 2 1 Z ≥ 0
It remains to check that when N > max ( α , 1 ) n N>\max(\alpha,1)n N > max ( α , 1 ) n g ∈ 1 2 Z ≥ 0 g\in\frac{1}{2}\mathbb{Z}_{\geq0} g ∈ 2 1 Z ≥ 0 (4) converges absolutely. Using (59) and the fact that B a ( 0 N ; θ ) = 1 B_a(0^N;\theta)=1 B a ( 0 N ; θ ) = 1 ln B a ( x ⃗ ; θ ) \ln B_a(\vec{x};\theta) ln B a ( x ; θ ) ∏ j = 1 m J λ j ( a ; θ ) J λ j ( x ⃗ ; θ ) \prod_{j=1}^{m}J_{\lambda_j}(a;\theta)J_{\lambda_j}(\vec{x};\theta) ∏ j = 1 m J λ j ( a ; θ ) J λ j ( x ; θ ) λ 1 , … , λ m \lambda_1,\ldots,\lambda_m λ 1 , … , λ m p μ 1 ( a ) p μ 2 ( x ⃗ ) p_{\mu_1}(a)p_{\mu_2}(\vec{x}) p μ 1 ( a ) p μ 2 ( x ) ∣ μ 1 ∣ = ∣ μ 2 ∣ |\mu_1|=|\mu_2| ∣ μ 1 ∣ = ∣ μ 2 ∣ n n n N N N N = ∞ N=\infty N = ∞ N > max ( α , 1 ) n N>\max(\alpha,1)n N > max ( α , 1 ) n N N N θ \theta θ N + c θ ( i , j ) N+c_\theta(i,j) N + c θ ( i , j ) ∣ λ ∣ ≤ n |\lambda|\leq n ∣ λ ∣ ≤ n ∣ c θ ( i , j ) ∣ < max ( α , 1 ) n |c_\theta(i,j)|<\max(\alpha,1)n ∣ c θ ( i , j ) ∣ < max ( α , 1 ) n (60) , the expansion of the coefficient of p μ 1 ( a ) p μ 2 ( x ⃗ ) p_{\mu_1}(a)p_{\mu_2}(\vec{x}) p μ 1 ( a ) p μ 2 ( x ) ∣ μ 1 ∣ = ∣ μ 2 ∣ = d |\mu_1|=|\mu_2|=d ∣ μ 1 ∣ = ∣ μ 2 ∣ = d
α d − 1 d ∑ g ∈ 1 2 Z ≥ 0 ∑ ( M , c ) ∈ C d , ρ ( ∞ ) , M n o r m a l g ( M ) = g , μ 1 ( M ) = μ 1 , μ 2 ( M ) = μ 2 ( α − 1 ) ν ρ ( M , c ) N 2 − 2 g − ℓ ( μ 1 ) − ℓ ( μ 2 ) − d f η ( M ) ( { ( − 1 ) − k } k ≥ 1 ) . (62) \frac{\alpha^{d-1}}{d}\sum_{g\in\frac{1}{2}\mathbb{Z}_{\geq0}}\sum_{\substack{(\mathrm{M},c)\in\mathcal{C}_{d,\rho}^{(\infty)},\ \mathrm{M}\ \mathrm{normal}\\ g(\mathrm{M})=g,\ \mu_1(\mathrm{M})=\mu_1,\ \mu_2(\mathrm{M})=\mu_2}}(\alpha-1)^{\nu_\rho(\mathrm{M},c)}N^{2-2g-\ell(\mu_1)-\ell(\mu_2)-d}f_{\eta(\mathrm{M})}\left(\{(-1)^{-k}\}_{k\geq1}\right). \tag*{(62)} d α d − 1 g ∈ 2 1 Z ≥ 0 ∑ ( M , c ) ∈ C d , ρ ( ∞ ) , M normal g ( M ) = g , μ 1 ( M ) = μ 1 , μ 2 ( M ) = μ 2 ∑ ( α − 1 ) ν ρ ( M , c ) N 2 − 2 g − ℓ ( μ 1 ) − ℓ ( μ 2 ) − d f η ( M ) ( {( − 1 ) − k } k ≥ 1 ) . (62) Hence this genus expansion converges absolutely as well, with the finite sum at each fixed genus performed first. Since there are only finitely many pairs of partitions of degrees at most n n n
