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Moments of Characteristic Polynomials of Non-Symmetric Random Matrices

Pax Kivimae

Abstract

We study the moments of the absolute characteristic polynomial of the real elliptic ensemble, including the case of the real Ginibre ensemble. We obtain asymptotics for all integral moments inside the real bulk to order 1 + o(1). In particular, for the real Ginibre ensemble, this extends known computations for even moments, and confirms a recent conjecture of Serebryakov and Simm [48] in the integral case. For the elliptic case, this generalizes computations of first two moments by Fyodorov [25] and Fyodorov and Tarnowski [31]. We additionally find uniform asymptotics for the multi-point correlations of the absolute characteristic polynomial. Our proof relies on a relation between expectations for the absolute characteristic polynomial and the real correlation functions, as well as an algebraic method of obtaining asymptotics for the behavior of these correlation functions near the diagonal.

Moments of Characteristic Polynomials of Non-Symmetric Random Matrices

Abstract

We study the moments of the absolute characteristic polynomial of the real elliptic ensemble, including the case of the real Ginibre ensemble. We obtain asymptotics for all integral moments inside the real bulk to order 1 + o(1). In particular, for the real Ginibre ensemble, this extends known computations for even moments, and confirms a recent conjecture of Serebryakov and Simm [48] in the integral case. For the elliptic case, this generalizes computations of first two moments by Fyodorov [25] and Fyodorov and Tarnowski [31]. We additionally find uniform asymptotics for the multi-point correlations of the absolute characteristic polynomial. Our proof relies on a relation between expectations for the absolute characteristic polynomial and the real correlation functions, as well as an algebraic method of obtaining asymptotics for the behavior of these correlation functions near the diagonal.

Paper Structure

This paper contains 8 sections, 13 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Result
  3. Related Results
  4. Method
  5. Characteristic Polynomials and Correlations
  6. Proof of Theorems \ref{['theorem: moments of real bulk']} and \ref{['theorem: real bulk multi point']}
  7. The Pfaffian Kernel and The Proof of Lemma \ref{['lem:error estimate for Pfaffian kernel']}
  8. Conjecture \ref{['conjecture: moment conjecture']} and Fractional Moments

Key Result

Theorem 1.1

Let us fix $\tau\in [0,1)$, $\mu\in (-(1+\tau),1+\tau)$, and $\ell \in \mathbb{N}$. Then we have that where here $C_{\tau}(\ell)$ is the constant Moreover, this estimate is uniform in $\mu \in (-(1+\tau)+\epsilon,1+\tau-\epsilon)$ for any fixed $\epsilon>0$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Corollary 3.2
  • proof
  • ...and 15 more