Related papers: Characteristic polynomials of random matrices
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we…
Moments of secular and inverse secular coefficients, averaged over random matrices from classical groups, are related to the enumeration of non-negative matrices with prescribed row and column sums. Similar random matrix averages are…
Properties of universality have essential relevance for the theory of random matrices usually called the Wigner ensemble. The issue was analysed up to recent years with detailed and relevant results. We present a slightly different view and…
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external…
In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials…
The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of…
Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…
We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show…
We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation…
We express the averages of products of characteristic polynomials for random matrix ensembles associated with compact symmetric spaces in terms of Jack polynomials or Heckman and Opdam's Jacobi polynomials depending on the root system of…
Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$…
We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the…
In this article, we study microscopic properties of a two-dimensional eigenvalue ensemble near a conical singularity arising from insertion of a point charge in the bulk of the support of eigenvalues. In particular, we characterize all…
In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a…
We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor…
We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the…
We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of…
We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried…
In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity.…
Explicit expressions are proven for derivatives of the ratio of a determinant or Pfaffian determinant and a Vandermonde determinant. Such ratios appear for example in general group integrals of Harish-Chandra--Itzykson--Zuber type and in…