Wigner- and Marchenko–Pastur-Type Limit Theorems for Jacobi Processes

Abstract

We study Jacobi processes \((X_{t})_{t\ge 0}\) on \([-1,1]^N\) and \([1,\infty [^N\) which are motivated by the Heckman–Opdam theory 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 \(t\rightarrow \infty \) to the distributions of the \(\beta \)-Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for \(N\rightarrow \infty \) 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. 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 \(N\rightarrow \infty \).