scholarly journals Random Matrices, Magic Squares and Matching Polynomials

10.37236/1859 ◽  
2004 ◽  
Vol 11 (2) ◽  
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.

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.

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

Pair dependent linear statistics for CβE

2020 ◽  
pp. 2150035
Author(s):  
Ander Aguirre ◽  
Alexander Soshnikov ◽  
Joshua Sumpter
Keyword(s):  
Zeta Function ◽  
Random Matrices ◽  
Riemann Zeta Function ◽  
Classical Result ◽  
Pair Correlation ◽  
Test Function ◽  
Linear Statistics ◽  
Smooth Test ◽  
Riemann Zeta ◽  
Counting Statistics

We study the limiting distribution of a pair counting statistics of the form [Formula: see text] for the circular [Formula: see text]-ensemble (C[Formula: see text]E) of random matrices for sufficiently smooth test function [Formula: see text] and [Formula: see text] For [Formula: see text] and [Formula: see text] our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

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 Moments of moments of characteristic polynomials of random unitary matrices and lattice point counts

2020 ◽  
pp. 2150019
Author(s):  
Theodoros Assiotis ◽  
Jonathan P. Keating
Keyword(s):  
Explicit Expression ◽  
Lattice Point ◽  
Characteristic Polynomials ◽  
Leading Order ◽  
Unitary Matrices ◽  
Point Count ◽  
Point Counts ◽  
Order Coefficient ◽  
The Matrix ◽  
Random Unitary Matrices

In this note, we give a combinatorial and noncomputational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand–Tsetlin patterns with constraints.

scholarly journals Moments of Moments and Branching Random Walks

2021 ◽  
Vol 182 (1) ◽  
Author(s):  
E. C. Bailey ◽  
J. P. Keating
Keyword(s):  
Random Fields ◽  
Binary Tree ◽  
Random Variable ◽  
Gaussian Random Fields ◽  
Branching Random Walk ◽  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Branching Random Walks ◽  
Explicit Formulae ◽  
Random Unitary Matrices

AbstractWe calculate, for a branching random walk $$X_n(l)$$ X n ( l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable $$\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}$$ 1 2 n ∑ l = 1 2 n e 2 β X n ( l ) , for $$\beta \in {\mathbb {R}}$$ β ∈ R . We obtain explicit formulae for the first few moments for finite n. In the limit $$n\rightarrow \infty $$ n → ∞ , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

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