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 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.

scholarly journals A Closer Look at Lattice Points in Rational Simplices

10.37236/1469 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Matthias Beck
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Lattice Points ◽  
Point Count ◽  
Point Counts ◽  
Reciprocity Law ◽  
Dilation Factor

We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an $n$-dimensional polytope with $n+1$ rational vertices, we use its description as the intersection of $n+1$ halfspaces, which determine the facets of the simplex. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We give an elementary proof that the lattice point counts in the interior and closure of such a vector-dilated simplex are quasipolynomials satisfying an Ehrhart-type reciprocity law. This generalizes the classical reciprocity law for rational polytopes. As an example, we derive a lattice point count formula for a rectangular rational triangle, which enables us to compute the number of lattice points inside any rational polygon.

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 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.

scholarly journals Moments of Characteristic Polynomials Enumerate Two-Rowed Lexicographic Arrays

10.37236/1717 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
E. Strahov
Keyword(s):  
Characteristic Polynomials ◽  
Combinatorial Interpretation ◽  
Unitary Matrices ◽  
Last Passage Percolation ◽  
Percolation Problem ◽  
Unexpected Consequence ◽  
Random Unitary Matrices

A combinatorial interpretation is provided for the moments of characteristic polynomials of random unitary matrices. This leads to a rather unexpected consequence of the Keating and Snaith conjecture: the moments of $|\zeta(1/2+it)|$ turn out to be connected with some increasing subsequence problems (such as the last passage percolation problem).

scholarly journals On the use of 10-minute point counts and 10-species lists for surveying birds in lowland Atlantic Forests in southeastern Brazil

2012 ◽  
Vol 52 (28) ◽  
pp. 333-340 ◽  
Author(s):  
Vagner Cavarzere ◽  
Thiago Vernaschi Vieira da Costa ◽  
Luís Fábio Silveira
Keyword(s):  
Rapid Assessment ◽  
Southeastern Brazil ◽  
Point Counting ◽  
Point Count ◽  
Point Counts ◽  
Time Period ◽  
Minute Point ◽  
Species Lists ◽  
Point Count Method ◽  
Breeding Seasons

Due to rapid and continuous deforestation, recent bird surveys in the Atlantic Forest are following rapid assessment programs to accumulate significant amounts of data during short periods of time. During this study, two surveying methods were used to evaluate which technique rapidly accumulated most species (> 90% of the estimated empirical value) at lowland Atlantic Forests in the state of São Paulo, southeastern Brazil. Birds were counted during the 2008-2010 breeding seasons using 10-minute point counts and 10-species lists. Overall, point counting detected as many species as lists (79 vs. 83, respectively), and 88 points (14.7 h) detected 90% of the estimated species richness. Forty-one lists were insufficient to detect 90% of all species. However, lists accumulated species faster in a shorter time period, probably due to the nature of the point count method in which species detected while moving between points are not considered. Rapid assessment programs in these forests will rapidly detect more species using 10-species lists. Both methods shared 63% of all forest species, but this may be due to spatial and temporal mismatch between samplings of each method.

scholarly journals The matrix element method at next-to-leading order

2012 ◽  
Vol 2012 (11) ◽  
Author(s):  
John M. Campbell ◽  
Walter T. Giele ◽  
Ciaran Williams
Keyword(s):  
Matrix Element ◽  
Leading Order ◽  
Matrix Element Method ◽  
The Matrix ◽  
Element Method
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