Powers of large random unitary matrices and Toeplitz determinants

Abstract

We study the limiting behavior of Tr ⁡ U k ( n ) \operatorname {Tr}U^{k(n)} , where U U is an n × n n\times n random unitary matrix and k ( n ) k(n) is a natural number that may vary with n n in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on n n in a particular way. As a consequence of this result, we find that for each fixed m ∈ N m\in \mathbb {N} , the random variables Tr ⁡ U k j ( n ) / min ( k j ( n ) , n ) \operatorname {Tr}U^{k_j(n)}/\sqrt {\min (k_j(n),n)} , j = 1 , … , m j=1,\ldots ,m , converge to independent standard complex normals.