The Sixth Moment of Random Determinants for Arbitrarily Distributed Random Entries

Dominik Beck; Zelin Lv; Aaron Potechin

combinatorics

The Sixth Moment of Random Determinants for Arbitrarily Distributed Random Entries

arXiv:2607.26857

summary

The paper derives a closed‑form expression for the sixth moment of the determinant of an n×n matrix with i.i.d. entries of any distribution, using a combinatorial decomposition of marked permutation tables and showing the associated exponential generating function is D‑finite.

Abstract

Via the method of marked permutation tables presented in this paper, we generalize the formula for the sixth moment of a random determinant to account for entries with arbitrary distribution. That is, let $f_{6} (n) = E (det A)^{6}$ , where $A$ is an $n$ by $n$ random matrix with independent and identically distributed entries. We show that the exponential generating function $F_{6} (t) = \sum_{n = 0}^{\infty} f_{6} (n) t^{n} / (n!)^{2}$ is D-finite and we present it in a closed form. Our method relies on carefully decomposing marked permutation tables into a shell, a core, and a floating component, each of which has a separate contribution to $F_{6} (t)$ . After this decomposition, it is sufficient to enumerate over a finite number of possible shells, which we did using a highly intricate computer program. We verified our result up to $n = 7$ in the general case and up to $n = 9$ for random matrices whose entries only take two values by using a different method for computing $f_{6} (n)$ for these cases.

94 pages, 9 figures, 3 tables

Topics & keywords

#random matrices #determinant moments #generating functions #combinatorial enumeration #D-finite seriessixth momentrandom determinantmarked permutation tablesexponential generating functioni.i.d. entries