Large dimensional limits of Jacobi processes of compact and non-compact type and related Pearson diffusions
Loading...
Date
2025
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Alternative Title(s)
Abstract
We study multivariate Jacobi processes of compact and non-compact type with arbitrary dimension of the corresponding state space. These processes are motivated by the Heckman-Opdam theory and dynamical random matrix theory. They 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 large times to the distributions of the β-Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We show that in the non-compact case and for certain parameters, these processes analogously tend for large times to the image of the β-Jacobi ensemble, respectively the ordered zeros of one-dimensional Jacobi polynomials, under the inversion map. We also provide a matrix model, build upon the Brownian motion on the positive definite matrices, whose eigenvalues form a non-compact Jacobi process. Then, we derive limit theorems for the empirical distributions of the N-dimensional Jacobi processes of compact and non-compact type for different parametrizations in the regime N to infinity. First, we establish a general theorem about the almost sure convergence of empirical measures of non-colliding Pearson diffusions. These diffusions include the Dyson Brownian motions, Laguerre processes, non-colliding geometric Brownian motions, Jacobi processes of compact and non-compact type, and Hua-Pickrell diffusions. We then move on to describe the limits obtained this way using free probability theory and free stochastic calculus. While some instances of these limits are well studied in the literature, the consideration of non-compact Jacobi processes and Hua-Pickrell diffusions is new. One key finding is the special role played by the free multiplicative Brownian motion, which can be thought of as an N=∞ variant of the Brownian motion on the general linear group of rank N. We show that one can use this free process to describe the limits of empirical measures of non-compact Jacobi processes and Hua-Pickrell diffusions. In particular, in two special cases we identify these limits using image measures of the distribution of the free positive multiplicative Brownian motion, where the initial limit distribution enters via free multiplicative convolutions. Moreover, we give a new proof of a formula which relates the distribution of the free positive multiplicative Brownian motion to the free additive convolution of the semicircle and uniform distribution on a compact interval. Lastly, we show that using higher order space-time scalings, the empirical measures of compact and non-compact Jacobi processes converge to the well-known semicircle and Marchenko-Pastur distributions, where the initial limit distributions enter via free additive convolutions. These results generalize corresponding results for β-Jacobi ensembles and the zeros of Jacobi-polynomials.
Description
Table of contents
Keywords
Stochastic processes, Interacting particle systems, Random matrix theory
Subjects based on RSWK
Stochastischer Prozess, Jacobi-Verfahren, Matrizentheorie