On Resolvent Identities in Gaussian Ensembles at the Edge of the Spectrum

Moments of discrete orthogonal polynomial ensembles

We obtain factorial moment identities for the Charlier, Meixner and Krawtchouk orthogonal polynomial ensembles. Building on earlier results by Ledoux [Elect. J. Probab. 10, (2005)], we find hypergeometric representations for the factorial moments when the reference measure is Poisson (Charlier ensemble) and geometric (a particular case of the Meixner ensemble). In these cases, if the number of particles is suitably randomised, the factorial moments have a polynomial property, and satisfy three-term recurrence relations and differential equations. In particular, the normalised factorial moments…

Operator level hard-to-soft transition for $β$-ensembles

The soft and hard edge scaling limits of $β$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.

随机矩阵族与可积微分恒等式

Random matrix ensembles and integrable differential identities

本文研究了随机矩阵理论中单位和正交族的可积微分恒等式，通过构造连续极限下的动力学链，揭示了热力学极限下的可积系统特性。

Moments of the Gaussian $β$ Ensembles and the large-$N$ expansion of the densities

The loop equation formalism is used to compute the $1/N$ expansion of the resolvent for the Gaussian $β$ ensemble up to and including the term at $O(N^{-6})$. This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at $O(N^{-6})$. The latter contain non-integrable singularities at the endpoints of the support --- we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings $β= 1$, $2$ and $4$ there are characterisations of…

β-变形 WLZZ 模型的双 β-系综实现

Two $β$-ensemble realization of $β$-deformed WLZZ models

本文提出 β-变形 WLZZ 矩阵模型的双 β-系综实现，利用 β-变形 Harish-Chandra-Itzykson-Zuber 积分耦合外场，并用 Dunkl 算子构造相应 Ward 恒等式。

Loop Equation Analysis of the Circular $ β$ Ensembles

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,β>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(ζ)=(ζ+z)/(ζ-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its gener…

Differential identities for the structure function of some random matrix ensembles

The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 λ_j}$, $\sum_{j=1}^N e^{-i k_2 λ_j}$ for Hermitian matrices, and the same with the eigenvalues $λ_j$ replaced by the eigenangles $θ_j$ for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation $ρ_{(2)}$. For the circular $β$-ensemble of unitary matrices, and with $β$ even, we characterise the bulk scaling limit of $ρ_{(2)}$ as the solution of a linear differential equation of order $β+ 1$ -- a duality re…

Gurau的谱密度不是个体实对称张量的概率测度

Gurau's spectral density is not a probability measure for individual real symmetric tensors

该研究指出Gurau残差迹的谱密度在随机张量集合中平均有效，但对个体张量不成立，揭示了其在确定性张量上的局限性。

Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited

We derive expansions of the resolvent Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we give another proof of the derivation of an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn. We conclude with a brief discussion on the derivation of the probability distribution function of the corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and Gaussian Symplectic …

Covariant Symanzik identities

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of…

Eigenvector localization for random band matrices with power law band width

It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width $W$ raised to a power $μ$ remains smaller than the matrix size $N$. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width $W$, the estimate $μ\le 8$ holds.

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, thes…

一个双HCIZ高斯矩阵模型用于非相交布朗桥

A Two-HCIZ Gaussian Matrix Model for Non-intersecting Brownian Bridges

本文构建了一个酉不变的Hermitian矩阵集合，其固定时间的特征值规律与具有任意有限多重性的非相交布朗桥的Karlin-McGregor定律一致，推导了该构造的多个精确有限n结果，包括路径空间提升和分区函数的简化。

固定$q$的$q$-Krawtchouk系综的边缘极限

The Fixed-$q$ Edge Limit of the $q$-Krawtchouk Ensemble

本文研究$q$-Krawtchouk系综在固定$q$下的右边缘极限，通过有限多项式对偶性证明相关核收敛，并揭示极限算子的非唯一性及谱原子选择机制。

Recursive characterisations of random matrix ensembles and associated combinatorial objects

We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes: Established theory on Selberg correlation integrals is used to derive linear differential equations on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to $1$-point recursions, understood to be analogues of the Harer-Zagier recursion, for the expansion coefficients of the associated $1$-point cumulants, while loop equation anal…

All the lowest order PDE for spectral gaps of Gaussian matrices

Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent to a particular case of Schlesinger equations for isomonodromic deformations) are rewritten in a general form which allows one to derive all the lowest order equations (PDE) for spectral gap probabilities of UE without intermediate higher-order PDE. This is demonstrated on the example of Gaussian ensemble (GUE) for which all the third order PDE for gap probabilities are obtained explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities in the case of more than one spectral endpoint. This approach a…

Relations between moments for the Jacobi and Cauchy random matrix ensembles

We outline a relation between the densities for the $β$-ensembles with respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval $(-1,1)$ and the Cauchy weight $(1-\mathrm{i}x)^η(1+\mathrm{i}x)^{\barη}$ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for $β=2$, and of degree five for $β=1$ and $4$, analogues of which are already known for the Jacobi weight $x^a(1-x)^b$ supported on $(0,1)$. We concentrate on the case $a=b$ (Jacobi weight on $(-1,1)$) and $η$ real (Cauchy we…

Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials …

随机矩阵理论中的对偶性

Dualities in random matrix theory

本文综述随机矩阵理论中特征多项式乘积、幂次及矩的对偶恒等式，涵盖不同系综的对偶结构、研究工具及在标度极限计算等方面的应用。