Cohen-Lenstra heuristics and random matrix theory over finite fields

TL;DR

This paper explores the distribution of partitions associated with random elements of finite classical groups, showing they tend to a Cohen-Lenstra type measure as the group rank grows, with precise bounds on their divergence.

Contribution

It establishes sharp bounds on the total variation distance between the distribution of partitions from random group elements and the Cohen-Lenstra measure, linking finite group theory and random matrix theory.

Findings

Partition distribution converges to Cohen-Lenstra measure as rank increases

Sharp bounds on total variation distance are provided

Results connect finite classical groups with probabilistic partition models

Abstract

Let g be a random element of a finite classical group G, and let \lambda_{z-1}(g) denote the partition corresponding to the polynomial z-1 in the rational canonical form of g. As the rank of G tends to infinity, \lambda_{z-1}(g) tends to a partition distributed according to a Cohen-Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition \lambda_{z-1}(g) and the Cohen-Lenstra type measure.