Beta Ensembles in the Freezing Regime and Finite Free Convolutions

TL;DR

This paper investigates the behavior of eigenvalues in Gaussian beta ensembles as temperature decreases, revealing convergence to deterministic processes described by finite free convolutions and establishing Gaussian fluctuations.

Contribution

It introduces a dynamical perspective on eigenvalue convergence in the freezing regime, linking finite free convolutions to eigenvalue processes and extending prior static results.

Findings

Eigenvalues converge to zeros of Hermite polynomials in the freezing regime.

Eigenvalue processes converge to deterministic limits via finite free convolution.

Gaussian fluctuations around the limit are established.

Abstract

In the freezing regime where the system size N is fixed and the inverse temperature beta tends to infinity, the eigenvalues of Gaussian beta ensembles converge to zeros of the Nth Hermite polynomial. That law of large numbers has been proved by analyzing the joint density or reading off the random matrix model. This paper studies its dynamical version of this phenomenon. We show that in the freezing regime the eigenvalue processes called beta Dyson Brownian motions converge to deterministic limiting processes which can be written as the finite free convolution of the initial data and the zeros of Hermite polynomials. This result is a counterpart of those in the random matrix regime where N tends to infinity with fixed beta, as well as to the high temperature regime where N tends to infinity while beta N remains bounded. We also establish Gaussian fluctuations around the limit and deal with the Laguerre case.