Friedrich Götze, Mikhail Gordin
Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace ensemble.
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In the present paper, fixed trace $β$-Hermite ensembles generalizing the fixed trace Gaussian Hermite ensemble are considered. For all $β$, we prove the Wigner semicircle law for these ensembles by using two different methods: one is the moment equivalence method with the help of the matrix model for general $β$, the other is to use asymptotic analysis tools. At the edge of the density, we prove that the edge scaling limit for $β$-HE implies the same limit for fixed trace $β$-Hermite ensembles. Consequently, explicit limit can be given for fixed trace GOE, GUE and GSE. Furthermore, for even $β…
Gaussian limits for traces of circular beta-ensembles
本文利用单位圆上的正交多项式理论和Stein方法，直接证明了圆β-系综幂的迹的高斯极限，并借助Verblunsky系数的显式耦合获得了几乎必然和$L^2$收敛。
We obtain the recursive identities for the joint moments of the traces of the powers of the resolvent for Gaussian ensembles of random matrices at the soft and hard edges of the spectrum. We also discuss the possible ways to extend these results to the non-Gaussian case.
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。
Fluctuations of the spectrum in rotationally invariant random matrix ensembles
本文利用 Stein 方法研究旋转不变随机矩阵系综中矩阵幂的迹，建立了带收敛速率的多变量中心极限定理，并发现非正规矩阵比 Hermite 矩阵更易处理。
Law of Large Numbers for continuous $N$-particle ensembles at fixed temperature
本文基于Bessel生成函数的渐进行为，研究了固定温度下N粒子集合平均经验测度的大数定律条件，并应用Dunkl算子和Chapuy--Dolega公式证明了θ-求和和θ-角的自由卷积与自由投影。
We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $β$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^α}$, for some constant $c>0$ and with $α\in (0,2)$, the case of Gaussian Wigner matrices, and the case of $β$-ensembles associated with a convex potential with polynomial growth.
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。
We investigate the linear statistics of random matrices with purely imaginary Bernoulli entries of the form $H_{pq} = \overline{H}_{qp} = \pm i$, that are either independently distributed or exhibit global correlations imposed by the condition $\sum_{q} H_{pq} = 0$. These are related to ensembles of so-called random tournaments and random regular tournaments respectively. Specifically, we construct a random walk within the space of matrices and show that the induced motion of the first $k$ traces in a Chebyshev basis converges to a suitable Ornstein-Uhlenbeck process. Coupling this with Stein'…
Smooth Multi-Trace Statistics of Classical Ensembles: Large $N$ Expansions, Cumulants, and Matrix Integrals
本文研究了经典集合中光滑多迹统计的大 $N$ 展开，推导了累积量的渐近消失、中心极限定理以及矩阵积分自由能和可观测量的渐近展开。
Thermal Concentration and Poisson-Dirichlet Edge Statistics for Random-Lattice Gibbs Ensembles
研究高维Haar随机幺模格上的吉布斯测度，证明在边缘窗口内吉布斯质量的泊松点过程极限，并建立原始方向吉布斯系综的固定因子能见度曲线。
Macroscopic asymptotics in discrete beta-ensembles and random tilings
研究离散β集合体的宏观渐近行为，通过分析粒子分布的线性统计和分区函数，揭示随机铺砌和高斯自由场的波动特性。
A family of m independent identically distributed random variables indexed by a chemical potential ϕ\in[0,γ] represents piles of particles. As ϕincreases to γ, the mean number of particles per site converges to a maximal density ρ_c<\infty. The distribution of particles conditioned on the total number of particles equal to n does not depend on ϕ(canonical ensemble). For fixed m, as n goes to infinity the canonical ensemble measure behave as follows: removing the site with the maximal number of particles, the distribution of particles in the remaining sites converges to the grand canonical meas…
We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications …
Let $θ_1,\ldots,θ_n$ be random variables from Dyson's circular $β$-ensemble with probability density function $\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{iθ_j}-e^{iθ_k}|^β$. For each $n\geq2$ and $β>0$, we obtain some inequalities on $\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]$, where $Z_n=(e^{iθ_1},\ldots,e^{iθ_n})$ and $p_μ$ is the power-sum symmetric function for partition $μ$. When $β=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $ \lim_{n\to\infty}\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]= δ_{μν}(\frac{2}β)^{…
We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erdös-Renyi random graph with $N$ vertices of average degree $Z$, inserting a real symmetric $d \times d$ random block at each non-vanishing entry. We consider some ensembles of random block matrices with rank $r < d$ and with maximal rank, $r=d$. The spectral moments of the sparse random block matrix are evaluated for $N \to \infty$, $d$ finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probabilit…
We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP):…
Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble
本文研究复椭圆 Ginibre 系综，证明其归一化特征多项式在谱椭圆外依分布收敛于高斯解析函数的指数，并借助 Hermite 核控制与矩量法回答可积情形的插值问题。
We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the Batalin-Vilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are described in terms of ribbon graph Feynman diagrams that then lead to a counting problem for the corresponding surfaces. The canonical relations provided by this homological setup determine a recurrence relation for these correlation functions. Using this recurrence relation and properties of the Catalan numbers, we determine the leading order behavior of the corre…
Bessel processes associated with the root systems $A_{N-1}$ and $B_N$ describe interacting particle systems with $N$ particles on $\mathbb R$; they form dynamic versions of the classical $β$-Hermite and Laguerre ensembles. In this paper we study corresponding Cauchy processes constructed via some subordination. This leads to $β$-Cauchy ensembles in both cases with explicit distributions. For these distributions we derive central limit theorems for fixed $N$ in the freezing regime, i.e., when the parameters tend to infinity. The results are closely related to corresponding known freezing result…
In this note, we prove that the normalized trace of the resolvent of the beta-Laguerre ensemble eigenvalues is close to the Stieltjes transform of the Marchenko-Pastur (MP) distribution with very high probability, for values of the imaginary part greater than m^{-1+ε}. As an immediate corollary, we obtain convergence of the one-point density to the MP law on short scales. The proof serves to illustrate some simplifications of the method introduced in our previous work to prove a local semi-circle law for Gaussian beta-ensembles.
We compute the expected normalized trace norm (matrix/graph energy) of random symmetric band circulant matrices and graphs in the limit of large sizes, and obtain explicit bounds on the rate of convergence to the limit, and on the probabilities of large deviations. We also show that random symmetric band Toeplitz matrices have the same limit norm assuming that their band widths remain small relative to their sizes. We compare the limit norms across a range of related random matrix and graph ensembles.
Mesoscopic Rates of Convergence for Hermitian Unitary Ensembles
本文研究了三种赫米特正交族在不同边缘的介观收敛速率，通过控制积分算子的trace类范数来证明收敛性。
This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrix. Under quite general assumptions, we prove that the traces are approximately normal distributed. Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth-death Markov kernel, the random birth-death $Q$ matrix and the $β$-Hermite ensemble. Furthermore, under some independent and identically distributed condition, we also prove the large deviation…
The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.
A countable group Γis called shift-minimal if every non-trivial measure preserving action of Γweakly contained in the Bernoulli shift of Γon ([0,1]^Γ,λ^Γ) is free. We show that any group Γwhose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group Γadmitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that Γ/N is shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal groups. We also ex…
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