相关论文: Characteristic polynomials of random matrices

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $\beta$-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In…

A t by n random matrix A is formed by sampling n independent random column vectors, each containing t components. The random Gram matrix of size n, G_n, contains the dot products between all pairs of column vectors in the randomly generated…

We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a…

We derive analytic expressions for infinite products of random 2x2 matrices. The determinant of the target matrix is log-normally distributed, whereas the remainder is a surprisingly complicated function of a parameter characterizing the…

It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an…

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s \in…

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of…

We investigate the average characteristic polynomial $\mathbb E\big[\prod_{i=1}^N(z-x_i)\big] $ where the $x_i$'s are real random variables which form a determinantal point process associated to a bounded projection operator. For a subclass…

A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected consequence of the Keating and Snaith conjecture: the moments of $\mid\xi(1/2+it)\mid$…

We introduce a family of random matrices where correlations between matrix elements are induced via interaction-derived Boltzmann factors. Varying these yields access to different ensembles. We find a universal scaling behavior of the…

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure,…

For random matrix ensembles with unitary symmetry, there is interest in the large $N$ form of the moments of the absolute value of the characteristic polynomial for their relevance to the Riemann zeta function on the critical line, and to…

We conjecture results about the moments of mixed derivatives of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this in two different ways, both giving us the same conjecture. In the first,…

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 $(\beta = 1)$, complex ($\beta = 2)$ and real…

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite…

We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of…

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written…

We reconsider the problem of calculating arbitrary negative integer moments of the (regularized) characteristic polynomial for $N\times N$ random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient…

In this article, we provide a unified framework for studying the convergence of rescaled characteristic polynomials of random matrices from various classical ensembles as well as functional convergence results for the Riemann zeta function.…