Eigenvalue distributions of random unitary matrices

2002 ◽  
Vol 123 (2) ◽  
pp. 202-224 ◽  
Author(s):  
K. Wieand
Keyword(s):  
Unitary Matrices ◽  
Random Unitary Matrices

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

scholarly journals Secular determinants of random unitary matrices

1996 ◽  
Vol 29 (13) ◽  
pp. 3641-3658 ◽  
Author(s):  
Fritz Haake ◽  
Marek Kus ◽  
Hans-Jürgen Sommers ◽  
Henning Schomerus ◽  
Karol Zyczkowski
Keyword(s):  
Unitary Matrices ◽  
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.

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