Limit theorems for orthogonal polynomials related to circular ensembles

TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials associated with extended circular ensembles, revealing convergence properties, fluctuations, and large deviations as the ensemble size grows infinitely large.

Contribution

It extends the analysis of orthogonal polynomials in circular ensembles to a broader class, providing new limit theorems and fluctuation results for spectral measures.

Findings

Convergence of the rescaled process to a deterministic limit

Characterization of fluctuations around the limit

Large deviation principles for the spectral measure

Abstract

For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t in [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations.