相关论文: Finite N Fluctuation Formulas for Random Matrices

We discuss an approach to compute the first and second moments of the number of eigenvalues $I_N$ that lie in an arbitrary interval of the real line for $N \times N$ Gaussian random matrices. The method combines the standard…

Let $\Psi_n$ be a product of $n$ independent, identically distributed random matrices $M$, with the properties that $\Psi_n$ is bounded in $n$, and that $M$ has a deterministic (constant) invariant vector. Assuming that the probability of…

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\la_{n,1},...,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law…

The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, S_{BGS} = - \int d{\bf H} [P({\bf H})] \ln [P({\bf H})], with suitable constraints. Here we construct and analyze…

We study the fluctuation of the eigenvalue number of any fixed interval $\Delta=[a,b]$ inside the spectrum for $\beta$- ensembles of random matrices in the case $\beta=1,2,4$. We assume that the potential $V$ is polynomial and consider the…

We present some applications of central limit theorems on mesoscopic scales for random matrices. When combined with the recent theory of "homogenization" for Dyson Brownian Motion, this yields the universality of quantities which depend on…

We investigate the probability density of rescaled sums of iterates of deterministic dynamical systems, a problem relevant for many complex physical systems consisting of dependent random variables. A Central Limit Theorem (CLT) is only…

We derive two probabilistic bounds for the relative forward error in the floating point summation of $n$ real numbers, by representing the roundoffs as independent, zero-mean, bounded random variables. The first probabilistic bound is based…

We study the fluctuations of the eigenvalues of real valued large centrosymmetric random matrices via its linear eigenvalue statistic. This is essentially a central limit theorem (CLT) for sums of dependent random variables. The dependence…

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the…

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are…

Consider a symmetric (finite) matrix ensemble, with a certain probability distribution. What is the probability that the spectrum belongs to a certain interval or union of intervals on the real line? In this paper, we show that, upon…

Invariant ensemble, which are characterised by the joint distribution of eigenvalues $P(\lambda_1,\ldots,\lambda_N)$, play a central role in random matrix theory. We consider the truncated linear statistics $L_K = \sum_{n=1}^K f(\lambda_n)$…

Inspired by a recent paper of I. Grama, E. Le Page and M. Peign\'e, we consider a sequence $(g_n)_{n \geq 1}$ of i.i.d. random $d\times d$-matrices with non-negative entries and study the fluctuations of the process $(\log \vert g_n\cdots…

We analyze the form of the probability distribution function P_{n}^{(\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \times n \beta-Gaussian random matrix, \beta…

The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree…

Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre…

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A.Lytova…

We present new explicit upper bounds for the smoothness of the distribution of the random diagonal sum $S_n=\sum_{j=1}^nX_{j,\pi(j)}$ of a random $n\times n$ matrix $X=(X_{j,r})$, where the $X_{j,r}$ are independent integer valued random…

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near…