Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twisted moments of characteristic polynomials of random matrices in the unitary group

Siegfred Baluyot, Brian Conrey

Abstract

Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $ζ(s)$. Their approach indicates how lower twisted moments of $ζ(s)$ may be used to evaluate higher moments. In this paper, we present a rigorous random matrix theory analogue of their heuristic. To do this, we develop a notion of "twisted moment" of characteristic polynomials of matrices in the unitary group $U(N)$, and we prove several identities involving Schur polynomials. Our results may be viewed as a proof of concept of the heuristic for $ζ(s)$.

Twisted moments of characteristic polynomials of random matrices in the unitary group

Abstract

Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function . Their approach indicates how lower twisted moments of may be used to evaluate higher moments. In this paper, we present a rigorous random matrix theory analogue of their heuristic. To do this, we develop a notion of "twisted moment" of characteristic polynomials of matrices in the unitary group , and we prove several identities involving Schur polynomials. Our results may be viewed as a proof of concept of the heuristic for .

Paper Structure

This paper contains 8 sections, 11 theorems, 125 equations.

Table of Contents

  1. Introduction and results
  2. Identities for sums of products of two Schur polynomials
  3. Proof of the BK Splitting
  4. Proof of Theorem \ref{['thm: twistedrecipe']}
  5. Proofs of the identities for sums of products of two Schur polynomials
  6. Proof of Theorem \ref{['thm: unitaryidentity']}
  7. Using lower twisted moments to evaluate higher untwisted moments
  8. Proofs of Lemmas \ref{['lem: schurorthogonal']} and \ref{['lem: bumpgamburdlemma']}

Key Result

Theorem 1.1

Let $N$ be a positive integer, and suppose that $A,B$ are finite multisets of complex numbers. Suppose that $A$ equals the multiset sum $A=A_1+ A_2$ and that $B$ equals the multiset sum $B=B_1 + B_2$. Let $\mathcal{M}_N$ be defined by eqn: twistedmomentdefinition. Then where $0$ denotes the zero partition $(0,0,\dots)$, the sum on the right hand side is over all dominant weights $\lambda$ of leng

Theorems & Definitions (20)

  • Theorem 1.1: The BK Splitting
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 10 more