Universality at the Edge for Unitary Matrix Models

TL;DR

This paper demonstrates that the local eigenvalue statistics in certain unitary matrix models are universal, unaffected by the specific potential, using advanced polynomial and eigenvalue analysis.

Contribution

It proves universality of local eigenvalue statistics for unitary matrix models with smooth potentials and single-interval support, extending previous results.

Findings

Eigenvalue statistics are independent of potential form.

Universality holds for four times differentiable potentials.

Results apply to models with single-interval support.

Abstract

Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.