Finite free probability and $S$ transforms of Jacobi processes
Nizar Demni, Nicolas Gilliers, Tarek Hamdi
TL;DR
The paper develops a comprehensive finite and free probability analysis of Jacobi processes, deriving a PDE for the free Jacobi $S$-transform and studying stationary solutions tied to angle operators. It then analyzes the averaged characteristic polynomial of the Hermitian Jacobi process, establishing inverse Jacobi-heat dynamics, a Szegö-type Hermite unitary connection for specific parameters, and a Jacobi-polynomial expansion. In the high-dimensional limit, crystallization from the frozen Jacobi process to the free Jacobi process is established via the finite free $S$-transform, complemented by a general convergence result for finite differences of finite free transforms and a PDE obtained as a limit of finite-difference dynamics for the finite free $T$-transform. Together these results bridge deterministic root dynamics with free-probabilistic limits and provide tools to extract asymptotics for time-dependent finite free transforms. The work advances understanding of how finite free transforms encode the evolution and limiting behavior of Jacobi-type spectral measures.
Abstract
In this paper, we study the $S$ transforms of Jacobi processes in the frameworks of free and finite free probability theories. We begin by deriving a partial differential equation satisfied by the free $S$ transform of the free Jacobi process, and we provide a detailed analysis of its characteristic curves. We turn next our attention to the averaged characteristic polynomial of the Hermitian Jacobi process and to the dynamic of its roots, referred to as the \emph{frozen Jacobi process}. In particular, we prove, for a specific set of parameters, that the former aligns up to a Szegö variable transformation with the Hermite unitary polynomial. We also provide an expansion of the averaged characteristic polynomial of the Hermitian process in the basis of Jacobi polynomials. Finally, we establish the convergence of the frozen Jacobi process to the free Jacobi process in high dimensions by using the finite free S transform. In doing so, we prove a general result, interesting in its own, on the convergence of the finite differences of the finite free $S$ transform, which paves the way to obtain asymptotics of differential-difference equations satisfied by time-dependent finite free S-transforms of polynomial sequences with positive roots.