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arXiv · 2507.03651

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

Dominik Beck·Zelin Lv·Aaron Potechin
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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.

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Dominik Beck, Zelin Lv, Aaron Potechin. 2025-07-04. Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables. https://arxiv.org/abs/2507.03651

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