Related papers: An Exposition on Wigner's Semicircular Law

In this article, we consider random Wigner matrices, that is symmetric matrices such that the subdiagonal entries of Xn are independent, centered, and with variance one except on the diagonal where the entries have variance two. We prove…

We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change…

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between…

We apply Lindeberg's method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation principles for martingales as well as for random…

We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors $T\in \otimes^p \mathbb{R}^N$ of order $p\ge 3$. The tensor resolvent yields an integral representation for a class of tensor invariants…

Hochst\"attler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the…

We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes…

We extend our recent result [Cipolloni, Erd\H{o}s, Schr\"oder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed complex entries to the real…

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence…

This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable…

We establish local laws for sample covariance matrices $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ where the random vectors $\g_1, \ldots, \g_N \in \R^n$ are independent with common covariance $\Sigma$. Previous work has largely focused on the…

We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We…

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le…

We compute the limiting statistical distribution of the eigenvalues of sequences of matrices whose entries satisfy what we call a vanishing mean variation condition and are $\mu$-distributed for some probability measure. As an application…

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the…

This article provides tools for the study of the Dirichlet random walk in $\mathbb{R}^d$. By this we mean the random variable $W=X_1\Theta_1+\cdots+X_n\Theta_n$ where $X=(X_1,\ldots,X_n) \sim \mathcal{D}(q_1,\ldots,q_n)$ is Dirichlet…

We study the number of occurrences of any fixed vincular permutation pattern. We show that this statistics on uniform random permutations is asymptotically normal and describe the speed of convergence. To prove this central limit theorem,…

A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of…

We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_n$ be a Hermitian random matrix of size $n\times n$ that can be split into independent blocks…

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion…