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

The purpose of this article is to study the eigenvalues $u_1^{, t}=e^{it\theta_1},\dots,u_N^{,t}=e^{it\theta_N}$of$U^t$where$U$isalarge$N\times N$randomunitarymatrixand$t>0$. 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. Numeric simulations are provided along the paper to illustrate the results.