Random matrix theory over finite fields: a survey

TL;DR

This survey explores the application of generating function methods and probabilistic models to analyze random matrices over finite fields, connecting diverse mathematical theories and providing a comprehensive overview.

Contribution

It offers a unified survey of generating function techniques and probabilistic frameworks for finite field matrices, linking multiple mathematical disciplines.

Findings

Probabilistic models of conjugacy classes are coherent and elegant.

Connections between random matrices and symmetric functions are established.

The survey highlights interdisciplinary links with Markov chains, potential theory, and identities.

Abstract

First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful. Connections are made with symmetric function theory, Markov chains, potential theory, Rogers-Ramanujan type identities, quivers, and various measures on partitions.