Uniform Hyperbolicity for Szeg\H{o} Cocycles and Applications to Random CMV Matrices and the Ising Model

TL;DR

This paper proves uniform hyperbolicity for Szeg ext{"o} cocycles and applies it to determine spectra of random CMV matrices, show absence of phase transitions in 1D Ising models, and analyze Lee-Yang zeros.

Contribution

It establishes uniform hyperbolicity for Szeg ext{"o} cocycles and applies this to spectral analysis, phase transition absence, and Lee-Yang zero characterization.

Findings

Explicit spectrum of random CMV matrices identified.

No phase transition in 1D Ising models.

Gap labeling for Lee-Yang zeros in thermodynamic limit.

Abstract

We consider products of the matrices associated with the Szeg\H{o} recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with non-negative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transition. Also, in the case of dynamically generated interaction couplings, we describe a gap labeling theorem for the Lee-Yang zeros in the thermodynamic limit.