Gaussian fluctuations for βEnsembles

Matrix Kernels for the Gaussian Orthogonal and Symplectic Ensembles

We derive the limiting matrix kernels for the the Gaussian Orthogonal and Symplectic ensembles scaled at the edge, with proofs of convergence in the operator norms that assure convergence of the determinants.

Ensemble dependence of fluctuations and the canonical/micro-canonical equivalence of ensembles

We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures. This is obtained by proving a local central limit theorem and a local large deviations principle. As an application we prove a formula due to Lebowitz-Percus-Verlet. It gives mean square fluctuations of an extensive observable, like the kinetic energy, in a classical micro canonical ensemble at fixed energy.

高斯张量族的特征化

Characterization of Gaussian Tensor Ensembles

本文基于Maxwell定理，定义了实对称、Hermitian和自偶Hermitian张量的高斯正交、酉和辛张量族，通过一阶和二阶情况恢复经典向量和矩阵高斯族，并证明了Maxwell型定理，统一并扩展了向量和矩阵的已有结果。

Global spectral fluctuations in the Gaussian Unitary Ensemble

We consider global fluctuations of the spectrum of the GUE. Using results on the linear statistics of such matrices as well as variance bounds on the eigenvalues, we show that under a suitable scaling, global fluctuations of the spectrum can be asymptotically described in terms of a logarithmically correlated Gaussian field. We also discuss briefly connections between different objects in RMT giving rise to log-correlated fields: the logarithm of the absolute value of the characteristic polynomial, the eigenvalue counting function, and the field of fluctuations of the eigenvalues around their …

稀疏衰减扰动下正交多项式系综的介观涨落稳定性

Stability of Mesoscopic Fluctuations of Orthogonal Polynomial Ensembles Under Sparse Decaying Perturbations

本研究利用正交多项式递推关系，证明实数轴上正交多项式系综的介观涨落在递推系数的稀疏衰减扰动下极限分布稳定，并推导出[-2,2]上奇异连续测度族的任意尺度介观中心极限定理。

Fluctuations of the process of moduli for the Ginibre and hyperbolic ensembles

We investigate the point process of moduli of the Ginibre and hyperbolic ensembles. We show that far from the origin and at an appropriate scale, these processes exhibit Gaussian and Poisson fluctuations. Among the possible Gaussian fluctuations, we can find white noise but also fluctuations with non-trivial covariance at a particular scale.

The Ginibre ensemble and Gaussian analytic functions

We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to Pólya's urn scheme. As a result, we get a random analytic function in the limit, which is given by a mixture of Gaussian analytic functions. This gives another reason why the zeros of Gaussian analytic functions and the Ginibre ensemble exhibit similar local repulsion, but different global behavior. Our approach gives new explicit formulas for the limiting analytic function.

从艾里线系综到高斯自由场

From the Airy line ensemble to the Gaussian free field

研究艾里线系综高度函数全局涨落，借助行列式结构和最速下降分析扩展艾里核，证明重标度后中心化高度函数收敛到高斯自由场的显式拉回，在特定意义下成立。

On the Rate of Convergence in the Central Limit Theorem for Linear Statistics of Gaussian, Laguerre, and Jacobi Ensembles

Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an estimate for the characteristic functions of the linear statistics; this estimate is uniform over the growing interval. The proof of the lemma relies on the Riemann--Hilbert approach.

The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source

In classical random matrix theory the Gaussian and chiral Gaussian random matrix models with a source are realized as shifted mean Gaussian, and chiral Gaussian, random matrices with real $(β= 1)$, complex ($β= 2)$ and real quaternion $(β= 4$) elements. We use the Dyson Brownian motion model to give a meaning for general $β> 0$. In the Gaussian case a further construction valid for $β> 0$ is given, as the eigenvalue PDF of a recursively defined random matrix ensemble. In the case of real or complex elements, a combinatorial argument is used to compute the averaged characteristic polynomial. Th…

CLT for fluctuations of $β$-ensembles with general potential

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $β$-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.

Mesoscopic fluctuations for the thinned Circular Unitary Ensemble

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the lim…

A Central Limit Theorem for Fluctuations in Polyanalytic Ginibre Ensembles

We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Virág. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure $q$-analytic kernels corresponding to different Landau levels. The fluctuations with r…

PDE's for the Gaussian ensemble with external source and the Pearcey distribution

The present paper studies a Gaussian Hermitian random matrix ensemble with external source, given by a fixed diagonal matrix with two eigenvalues a and -a. As a first result, the probability that the eigenvalues of the ensemble belong to a set satisfies a fourth order PDE with quartic non-linearity; the variables being the eigenvalue a and the boundary points of the set. This equation enables one to find a PDE for the Pearcey distribution. The latter describes the statistics of the eigenvalues near the closure of a gap; i.e., when the support of the equilibrium measure for large size random ma…

From Alternating Sign Matrices to the Gaussian Unitary Ensemble

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.

环面上的KPZ方程：高斯涨落

KPZ on torus: Gaussian fluctuations

本文研究了环面上的KPZ方程，推导并证明了其在长时间尺度下的高斯涨落性质。

Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $β$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion adapted for general $β$ Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.

Gaussian asymptotics of discrete $β$-ensembles

We introduce and study stochastic $N$-particle ensembles which are discretizations for general-$β$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. The covariance is universal and coincides with its counterpart in random matrix theory. Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work…

Toda系统的散射与高斯β系综

Scattering of the Toda system and the Gaussian $β$-ensemble

该研究利用作用于对称实三对角矩阵的经典Toda流，给出了Dumitriu-Edelman三对角模型谱服从高斯β系综的辛证明，拓展了Toda系统散射与对数气体矩阵模型的相关研究。

Gaussian fluctuations for linear spectral statistics of Wigner beta ensembles

As an important topic in Mathematical Physics and statistics, random matrices theory has found uses in many aspects of modern physics and multivariate analysis. This paper is to investigate the Gaussian fluctuations for linear spectral statistics (LSS) of Wigner beta ensembles. We first establish a central limit theorem (CLT) for LSS of Wigner quaternion matrices, then give a general CLT for Wigner $β$ ensembles.

On the distribution of the determinant of the Gaussian unitary ensemble matrix

The determinant of the Gaussian unitary ensemble matrix is show to be distributed as a product of independent chi random variables with parameters $1,3,3,5,5,\dots.$

Global spectrum fluctuations for Gaussian beta ensembles: a martingale approach

The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $β$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $nβ\to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

由与渐近自由随机矩阵系综共轭所诱导的涨落矩

Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles

本文研究常数矩阵经渐近自由随机矩阵系综共轭后产生的涨落矩与高阶矩，并确定了与离散傅里叶变换矩阵相关系综诱导的涨落矩。

Generalized $β$-Gaussian Ensemble Equilibrium measure method

We investigate $β$-Generalized random Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We investigate general method names as equilibrium measure method. When taking $n$ large limit we will see that the asymptotic density of eigenvalues generalize the Wigner semi-circle law.

Gaussian beta ensembles at high temperature: eigenvalue fluctuations and bulk statistics

We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.

Gaussian fluctuations and moderate deviations of eigenvalues in unitary invariant ensembles

We study the limiting behavior of the $k$-th eigenvalue $x_k$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both $k$ and $n-k$ tend to infinity, we obtain Gaussian fluctuations for $x_k$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel-Darboux kernels as well.

Extremes of Chi triangular array from the Gaussian $β$-Ensemble at high temperature

We study the extreme point process associated to the off-diagonal components in the matrix representation of the Gaussian $β$-Ensemble and prove its convergence to Poisson point process as $n\to +\infty$ when the inverse temperature $β$ scales with $n$ and tends to $0$. We consider two main high temperature regimes: $\displaystyle{β\ll \frac{1}{n}}$ and $\displaystyle{nβ= 2γ\geq 0}$. The normalizing sequences are explicitly given in each cases. As a consequence, we estimate the first order asymptotic of the largest eigenvalue of the Gaussian $β$-Ensemble.

高斯混合集成变换滤波器

A Gaussian mixture ensemble transform filter

本文提出一种基于高斯混合或高斯核密度估计先验分布的集成变换滤波器（EGMF），通过连续贝叶斯滤波分析步骤设计，并在三个测试问题中验证其对非高斯单峰和多模态分布系统的跟踪能力。

球面高斯系综游程集的巨分量：存在性、唯一性与体积集中

The giant component of excursion sets of spherical Gaussian ensembles: existence, uniqueness, and volume concentration

本文证明 Kostlan、带限和随机球面调和函数三类球面高斯场超临界游程集中存在唯一且体积集中的巨分量，并给出近最优或强定量界。

Large deviations principle for the largest eigenvalue of the Gaussian beta-ensemble at high temperature

We consider the Gaussian beta-ensemble when $β$ scales with $n$ the number of particles such that $\displaystyle{{n}^{-1}\ll β\ll 1}$. Under a certain regime for $β$, we show that the largest particle satisfies a large deviations principle in $\mathbb{R}$ with speed $nβ$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.

A convergent $1/n$-expansion for GSE and GOE

We generalize the results on the asymptotic expansion from Gaussian Unitary Ensembles case to all Gaussian Ensembles. We derive differential equations on densities and their moment generating functions for all Gaussian Ensembles. Also, we calculate explicit expressions for the moment generating functions for all Gaussian Ensembles.

Poisson statistics at the edge of Gaussian beta-ensembles at high temperature

We study the asymptotic edge statistics of the Gaussian $β$-ensemble, a collection of $n$ particles, as the inverse temperature $β$ tends to zero as $n$ tends to infinity. In a certain decay regime of $β$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.

The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles

An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$β$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N β$ is a constant.

圆β-系综迹的高斯极限

Gaussian limits for traces of circular beta-ensembles

本文利用单位圆上的正交多项式理论和Stein方法，直接证明了圆β-系综幂的迹的高斯极限，并借助Verblunsky系数的显式耦合获得了几乎必然和$L^2$收敛。

高斯正交系综的主子式

Principal minors of Gaussian orthogonal ensemble

本文研究了经典高斯正交系综（GOE）主子式最大特征值的极值过程，证明了其最大值在极限下服从Gumbel分布，并推导了最大值与相应特征向量的极限联合分布，表明二者渐近独立。

Local semicircle law at the spectral edge for Gaussian $β$-ensembles

We study the local semicircle law for Gaussian $β$-ensembles at the edge of the spectrum. We prove that at the almost optimal level of $n^{-2/3+ε}$, the local semicircle law holds for all $β\geq 1$ at the edge. The proof of the main theorem relies on the calculation of the moments of the tridiagonal model of Gaussian $β$-ensembles up to the $p_n$-moment where $p_n = O(n^{2/3-ε})$. The result is the analogous to the result of Sinai and Soshnikov for Wigner matrices, but the combinatorics involved in the calculations are different.

Generalized Gaussian Random Unitary Matrices Ensemble

We describe Generalized Hermitian matrices ensemble sometimes called Chiral ensemble. We give global asymptotic of the density of eigenvalues or the statistical density. We will calculate a Laplace transform of such a density for finite $n$, which will be expressed through an hypergeometric function. When the dimensional of the hermitian matrix begin large enough, we will prove that the statistical density of eigenvalues converge in the tight topology to some probability measure, which generalize the Wigner semi-circle law.

Uniqueness of the Gaussian Orthogonal Ensemble

A known result in random matrix theory states the following: Given a random Wigner matrix $X$ which belongs to the Gaussian Orthogonal Ensemble (GOE), then such matrix $X$ has an invariant distribution under orthogonal conjugations. The goal of this work is to prove the converse, that is, if $X$ is a symmetric random matrix such that it is invariant under orthogonal conjugations, then such matrix $X$ belongs to the GOE. We will prove this using some elementary properties of the characteristic function of random variables.

多变量集合卡尔曼滤波的波动与长时间稳定性

Fluctuations and Long-Time Stability of Multivariate Ensemble Kalman Filters

本文研究了多变量集合卡尔曼滤波的波动特性与长时间稳定性，通过建立非渐近稳定性理论，量化了集合大小、维度和观测噪声对误差传播的影响。

Global Fluctuations for Linear Statistics of β-Jacobi Ensembles

We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(λ_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $λ_1, ..., λ_n$ being the eigenvalues of a (general) $β$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean ($\sum_{i=1}^n f(λ_i) - \Exp \sum_{i=1}^n f(λ_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation …

Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information

A Random Geometric Graph (RGG) ensemble is defined by the disordered distribution of its node locations. We investigate how this randomness drives sample-to-sample fluctuations in the dynamical properties of these graphs. We study the distributional properties of the algebraic connectivity which is informative of diffusion and synchronization timescales in graphs. We use numerical simulations to provide the first characterisation of the algebraic connectivity distribution for RGG ensembles. We find that the algebraic connectivity can show fluctuations relative to its mean on the order of $30 \…

高温区域高斯β系综的大偏差原理

A large deviation principle for the Gaussian beta ensemble in the high temperature regime

本文证明了高温区域高斯β系综加权谱测度的大偏差原理，其速率函数仅对离散测度有限，且在极高温时额外惩罚多余支撑点。

Rectangular Young tableaux and the Jacobi ensemble

It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble.

Matrix measures, random moments and Gaussian ensembles

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n}, ..., S_{n,n})^* \sim \mathcal{U} (\mathcal{M}_n)$ are studied if $n \to \infty$. In particular, it is shown that an appropriately centered and standardized version of the vector $(S_{1,n}, ..., S_{k,n})^*$ converges weakly to a vector of $k$ independent $p \times p$ Gaussian ensembles. For the proof of our results we use some new relations between ordinary mo…

Universal Gaussian fluctuations of non-Hermitian matrix ensembles

We prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our techniques rely on a universality principle for the Gaussian Wiener chaos, recently proved by the authors together with Gesine Reinert, as well as on some combinatorial estimates. Unlike other related results in the probabilistic literature, we do not require that the law of the entries has a density with respect to the Lebesgue measure. In particular, our results apply to the ensemb…

Local semicircle law in the bulk for Gaussian $β$-ensemble

We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of $n^{-1+δ}$ for any $δ> 0$. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of $n^{-1/2+δ}$; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erdös, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.

Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[σ, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n\times n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-βθ(α)n^2]$ where the Dyson index $β$ characterizes the ensemble, $α$ is some extra parameter and the exponent $θ(α)$ is a function of $α$ which will be given explicitly. For $α=0$, $θ(0)= (\log 3)/4 = 0.274653...$ i…

$β$-ensembles 高斯消元法中的主元概率与范数效应

Pivot probabilities and norm effects in Gaussian elimination for $β$-ensembles

本文分析了 $2 \ imes 2$ 随机矩阵集在部分主元高斯消元法（GEPP）中的主元概率，通过推导标准 LAPACK 风格实现下的精确概率解决了 GUE 矩阵中理论与经验的差异，并证明 Dumitriu-Edelman 三对角 $β$-ensembles 符合理论预期。

与高斯β系综相关的对称三对角矩阵值过程的特征值过程

Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble

本文研究与高斯β系综相关的对称三对角矩阵值过程，推导其特征值过程满足的随机微分方程，并利用柯西交错论证给出特征值几乎必然不碰撞的充分条件。

Limit distributions of Gaussian block ensembles

It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial freeness, which is similar to freeness in free probability, but it also has some matricial features. Using this new concept of noncommutative independence, we described the asymptotics of blocks and symmetric blocks of certain classes of independent random matrices. In this paper, we present the main results obtained in this framework, concentrating on the ensem…

各向同性 Wishart 矩阵与 Gaussian unitary/orthogonal ensembles 混合迹矩的递归公式

A recursion formula for mixed trace moments of isotropic Wishart matrices and the Gaussian unitary/orthogonal ensembles

针对四类基本随机矩阵系综，论文给出混合迹矩的精确递归公式，并以无需多边形胶合背景的直接方式证明。

A Note on Functional Averages over Gaussian Ensembles

In this work we find a new formula for matrix averages over the Gaussian ensemble. Let ${\bf H}$ be an $n\times n$ Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an $n\times n$ positive definite matrix ${\bf A}$, and a continuous function $f:\R^{+}\to\R$ such that $\int_{0}^{\infty}{e^{-αt}|f(t)|^2\,dt}<\infty$ for every $α>0$, we find a new formula for the expectation $\E[\mathrm{Tr}(f({\bf HAH^{*}}))]$. Taking $f(x)=\log(1+x)$ gives another formula for the capacity of the MIMO communication channel, and taking $f(x)…