Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables

TL;DR

This paper introduces a combinatorial approach using analytic combinatorics on permutation tables to simplify the calculation of moments of the determinant of random matrices with independent entries, extending previous results for specific moments.

Contribution

It presents a new method employing analytic combinatorics on permutation tables to analyze the moments of determinants of random matrices, simplifying prior complex recurrence-based analyses.

Findings

Simplified the calculation of determinant moments using combinatorial methods.

Extended analysis to moments beyond those previously studied, such as k=4 and k=6.

Provided a unified framework for understanding determinant moments for various distributions.

Abstract

We consider the following natural question. Given a matrix $A$ with i.i.d. random entries, what are the moments of the determinant of $A$? In other words, what is $\mathbb{E}[\det(A)^k]$? While there is a general expression for $\mathbb{E}[\det(A)^k]$ when the entries of $A$ are Gaussian, much less is known when the entries of $A$ have some other distribution. In two recent papers, we answered this question for $k = 4$ when the entries of $A$ are drawn from an arbitrary distribution and for $k = 6$ when the entries of $A$ are drawn from a distribution which has mean $0$. These analyses used recurrence relations and were highly intricate. In this paper, we show how these analyses can be simplified considerably by using analytic combinatorics on permutation tables.