Beta Ensembles in the Freezing Regime and Finite Free Convolutions

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.