Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

Abstract

For a natural extension of the circular unitary ensemble of order n, we study as \(n\rightarrow \infty \) the asymptotic behavior of the sequence of monic orthogonal polynomials \((\varPhi _{k,n}, k=0, \ldots , n)\) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, as \(n\rightarrow \infty \), the sequence of processes \((\log \varPhi _{\lfloor nt\rfloor ,n}(1), t \in [0,1])\) converges to a deterministic limit, and we describe the fluctuations and the large deviations.