Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

TL;DR

This paper studies the eigenvalue and singular value distributions of random convolution operators on large finite groups, revealing their convergence to known laws and connections to free probability theory.

Contribution

It extends prior results to non-Abelian groups, characterizes limiting distributions, and proves convergence to free circular elements for certain group sequences.

Findings

Eigenvalue distributions converge to circular laws for symmetric groups.

Singular value distributions follow quarter circular laws.

Convergence to free circular elements under specific conditions.

Abstract

We analyse the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by M. Meckes for the Abelian case. We show that for regular sequences of groups the limiting distribution of eigenvalues (resp. singular values) is a mixture of eigenvalue (resp. singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular for symmetric groups of increasing order) families of stochastically independent random projection operators converge in moments to free circular elements. Finally, in the Gaussian case we provide Central Limit Theorems for linear eigenvalue statistics.