Asymptotics of Selberg-like integrals by lattice path counting

TL;DR

This paper derives explicit formulas for moments of eigenvalue densities in large-dimensional Jacobi and Laguerre ensembles using combinatorial lattice path counting, connecting random matrix theory with combinatorics.

Contribution

It introduces a combinatorial method to evaluate asymptotics of Selberg-like integrals through lattice path enumeration, providing explicit moment formulas.

Findings

Explicit expressions for moments of eigenvalue densities in large matrices

Connection established between random matrix integrals and lattice path counting

Method offers a new combinatorial approach to asymptotic analysis

Abstract

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 the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations.