Thinning and conditioning of the circular unitary ensemble

We apply the operation of random independent thinning on the eigenvalues of [Formula: see text] Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as [Formula: see text].

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TENSOR PRODUCTS OF RANDOM UNITARY MATRICES

Random Matrices Theory and Application ◽

10.1142/s2010326312500098 ◽

2012 ◽

Vol 01 (04) ◽

pp. 1250009 ◽

Cited By ~ 6

Author(s):

TOMASZ TKOCZ ◽

MAREK SMACZYŃSKI ◽

MAREK KUŚ ◽

OFER ZEITOUNI ◽

KAROL ŻYCZKOWSKI

Keyword(s):

Tensor Product ◽

Random Matrices ◽

Tensor Products ◽

Unitary Matrices ◽

Product Of Random Matrices ◽

Spectral Statistics ◽

Random Unitary Matrices ◽

Unitary Ensemble

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M = 2, N become large or M become large and N = 2.

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

Random Matrices Theory and Application ◽

10.1142/s2010326315500112 ◽

2015 ◽

Vol 04 (03) ◽

pp. 1550011 ◽

Cited By ~ 1

Author(s):

O. Marchal

Keyword(s):

Return Time ◽

Unitary Matrix ◽

Unitary Matrices ◽

Toeplitz Determinants ◽

Recurrence Times ◽

Toeplitz Determinant ◽

Large N ◽

Single Arc ◽

Random Unitary Matrices ◽

First Return Time

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut 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 toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

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Random Matrices, Magic Squares and Matching Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/1859 ◽

2004 ◽

Vol 11 (2) ◽

Cited By ~ 16

Author(s):

Persi Diaconis ◽

Alex Gamburd

Keyword(s):

Random Matrices ◽

Riemann Zeta Function ◽

Quantum Chaos ◽

Higher Moments ◽

Characteristic Polynomials ◽

Stochastic Matrices ◽

Unitary Matrices ◽

Magic Squares ◽

Riemann Zeta ◽

Random Unitary Matrices

Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta-function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed.

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Asymptotics for Averages over Classical Orthogonal Ensembles

International Mathematics Research Notices ◽

10.1093/imrn/rnaa354 ◽

2021 ◽

Author(s):

Tom Claeys ◽

Gabriel Glesner ◽

Alexander Minakov ◽

Meng Yang

Keyword(s):

Orthogonal Polynomials ◽

Unit Circle ◽

Upper Bound ◽

Unitary Group ◽

Hankel Determinants ◽

Eigenvalue Statistics ◽

Orthogonal And Symplectic Ensembles ◽

Gap Probabilities ◽

Orthogonal Ensembles

Abstract We study the averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar-distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher–Hartwig singularities in cases where some of the singularities merge together and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the circular orthogonal and symplectic ensembles and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.

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Eigenvector Convergence for Minors of Unitarily Invariant Infinite Random Matrices

International Mathematics Research Notices ◽

10.1093/imrn/rnz330 ◽

2020 ◽

Author(s):

Joseph Najnudel

Keyword(s):

Weak Convergence ◽

Random Matrices ◽

Invariant Measures ◽

Alternative Proof ◽

Probability Measures ◽

Cayley Transform ◽

Hermitian Matrices ◽

Unitary Matrices ◽

Extreme Eigenvalues ◽

Unitary Ensemble

Abstract In [10], Pickrell fully characterizes the unitarily invariant probability measures on infinite Hermitian matrices. An alternative proof of this classification is given by Olshanski and Vershik in [9], and in [3] Borodin and Olshanski deduce from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense that is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali [6], for which a coupling of the circular unitary ensemble of all dimensions is considered.

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