Related papers: Counting integral matrices with a given characteri…

The joint moments of the derivatives of the characteristic polynomial of a random unitary matrix, and also a variant of the characteristic polynomial that is real on the unit circle, in the large matrix size limit, have been studied…

We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a…

This paper contains a re-evaluation of the spectral approach and factorizability for regular matrix polynomials. In addition, solvent theory is extended from the monic and comonic cases to the regular case. The classification of extended…

We construct approximate transport maps for perturbative several-matrix models. As a consequence, we deduce that local statistics have the same asymptotic as in the case of independent GUE or GOE matrices, i.e., they are given by the…

Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…

We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of…

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…

We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic…

In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from…

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 count mxn non-negative integer matrices (contingency tables) with prescribed row and column sums (margins). For a wide class of smooth margins we establish a computationally efficient asymptotic formula approximating the number of…

We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to…

Motivated by finding analogues of elliptic curve point counting techniques, we introduce one deterministic and two new Monte Carlo randomized algorithms to compute the characteristic polynomial of a finite rank-two Drinfeld module. We…

We represent the number of mxn non-negative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a non-negative random matrix with exponentially distributed entries. We…

The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been…

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…

Given a large real symmetric, positive semidefinite m-by-m matrix, the goal of this paper is to show how a numerical approximation of the entropy, given by the sum of the entropies of the individual eigenvalues, can be computed in an…