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

2014/12/09 by Olivier Marchal, Marchal, Olivier
Mathematics · #15B05 #15B52 #60B20 #Advanced Combinatorial Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1412.3085

openalex publication_date 2014/12/09 · openalex created_date 2022/11/05 · openalex updated_date 2026/07/28

Abstract

The purpose of this article is to study the eigenvalues u₁ ^,\nt=e^it\θ₁,\…,u_N ^,t=e^it\θ_N of U^t where U is a large\nN\× N random unitary matrix and t>0. In particular we are interested in\nthe typical times t for which all the eigenvalues are simultaneously close to\n1 in different ways thus corresponding to recurrence times in the issue of\nquantum measurements. Our strategy consists in rewriting the problem as a\nrandom matrix integral and use loop equations techniques to compute the first\norders of the large N asymptotic. We also connect the problem to the\ncomputation of a large Toeplitz determinant whose symbol is the characteristic\nfunction of several arc segments of the unit circle. In particular in the case\nof a single arc segment we recover Widom's formula. Eventually we explain why\nthe first return time is expected to converge towards an exponential\ndistribution when N is large. Numeric simulations are provided along the\npaper to illustrate the results.\n

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