Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Circular Beta Ensembles

Abstract

We study CMV matrices (a discrete one-dimensional Dirac-type operator) 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 circular beta ensembles of random matrix theory. The temperature β^{-1} appears as the square of the coupling constant.

References in corpus (5)

• Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
• From Random Matrices to Stochastic Operators
• Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures
• Interpretations of some parameter dependent generalizations of classical matrix ensembles
• Random discrete Schrödinger operators from Random Matrix Theory

Cited by in corpus (6)

• Continuum limits of random matrices and the Brownian carousel
• The Canopy Graph and Level Statistics for Random Operators on Trees
• Diffusion at the random matrix hard edge
• Poisson statistics of eigenvalues in the hierarchical Anderson model
• A random matrix decimation procedure relating $\beta =2/(r+1)$ to $\beta =2(r+1)$
• Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities