scholarly journals Powers of large random unitary matrices and Toeplitz determinants

2009 ◽  
Vol 362 (03) ◽  
pp. 1169-1187 ◽  
Author(s):  
Maurice Duits ◽  
Kurt Johansson
Keyword(s):  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Random Unitary Matrices

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

2015 ◽  
Vol 04 (03) ◽  
pp. 1550011 ◽  
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.

scholarly journals Thinning and conditioning of the circular unitary ensemble

2017 ◽  
Vol 06 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Christophe Charlier ◽  
Tom Claeys
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Random Matrices ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Random Unitary Matrices ◽  
Gap Probabilities ◽  
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].

scholarly journals Mixed moments of characteristic polynomials of random unitary matrices

2019 ◽  
Vol 60 (8) ◽  
pp. 083509 ◽  
Author(s):  
E. C. Bailey ◽  
S. Bettin ◽  
G. Blower ◽  
J. B. Conrey ◽  
A. Prokhorov ◽  
...  
Keyword(s):  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Random Unitary Matrices

scholarly journals On the distribution of the zeros of the derivative of the Riemann zeta-function

2014 ◽  
Vol 157 (3) ◽  
pp. 425-442 ◽  
Author(s):  
STEPHEN LESTER
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Horizontal Distribution ◽  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Random Unitary Matrices

AbstractWe establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

Random unitary matrices

1994 ◽  
Vol 27 (12) ◽  
pp. 4235-4245 ◽  
Author(s):  
K Zyczkowski ◽  
M Kus
Keyword(s):  
Unitary Matrices ◽  
Random Unitary Matrices
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