Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

International audienceThe purpose of this article is to study the eigenvalues u_{t_1} = e^{itθ_1} ,..., u_{t_N} = e^{itθ_N} of U^t where U is a large N × N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results