Wigner- and Marchenko-Pastur-type limits for Jacobi processes

TL;DR

This paper investigates the asymptotic spectral distributions of Jacobi processes, deriving Wigner- and Marchenko-Pastur-type laws for large particle systems, connecting stochastic processes, random matrix theory, and polynomial zeros.

Contribution

It extends classical limit laws to Jacobi processes with arbitrary initial conditions, incorporating free convolution and generalizing known results for ensembles and polynomial zeros.

Findings

Almost sure Wigner semicircle law for Jacobi processes

Marchenko-Pastur law for empirical distributions as N→∞

Connection to zeros of Jacobi polynomials and free convolutions

Abstract

We study Jacobi processes $(X_{t})_{t\ge0}$ on the compact spaces $[-1,1]^N$ and on the noncompact spaces $[1,\infty[^N$ which are motivated by the Heckman-Opdam theory for the root systems of type BC and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for $t\to\infty$ to the distributions of the $\beta$-Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. Representing these processes by stochastic differential equations, we derive almost sure analogues of Wigner's semicircle and Marchenko-Pastur limit laws for $N\to\infty$ for the empirical distributions of the $N$ particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions These results generalize corresponding stationary limit results in the compact case for $\beta$-Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials by Dette and Studden. The results are also related to free limit theorems for multivariate Bessel processes, $\beta$-Hermite and $\beta$-Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $N\to\infty$.