Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles

Abstract

We study CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients. Under mild assumptions, we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay, we obtain the $\beta$-ensembles of random matrix theory. The temperature ${\beta }^{-1}$ appears as the square of the coupling constant