# Moments of the Hermitian Matrix Jacobi process

## TL;DR

This paper derives explicit formulas for the moments of the Hermitian matrix Jacobi process by analyzing eigenvalue distributions, expanding in symmetric polynomials, and studying asymptotic behavior as matrix size grows.

## Contribution

It introduces a novel method to compute moments of the Hermitian matrix Jacobi process using eigenvalue density expansions and Schur polynomial techniques.

## Key findings

- Explicit formulas for moments of the Hermitian matrix Jacobi process.
- Asymptotic behavior of moments as matrix size tends to infinity.
- Special parameter cases simplify to Cauchy determinants.

## Abstract

In this paper, we compute the expectation of traces of powers of the hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy-Binet formula in order to determine the partitions having non zero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized Beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.