Magic squares, the symmetric group and M\"obius randomness

TL;DR

This paper provides a new combinatorial proof for moments of secular coefficients in the CUE ensemble, extends these results to symmetric powers, and explores their implications for number theory and matrix integrals.

Contribution

It introduces a novel combinatorial approach to moments in the CUE ensemble and extends the results to broader contexts, including symmetric powers and number theory conjectures.

Findings

New combinatorial proof of moments of secular coefficients

Extension of moments to traces of symmetric powers

A new formula for a matrix integral related to the divisor function

Abstract

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the M\"obius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the M\"obius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the $k$-fold divisor function.