A random walk on the category of finite abelian $p$-groups

TL;DR

This paper investigates a reversible Markov chain on finite abelian p-groups linked to random matrices, revealing its spectral properties and explaining the Cohen-Lenstra distribution's emergence in this context.

Contribution

It introduces a reversible Markov chain on finite abelian p-groups and explicitly determines its spectrum, connecting random matrix cokernels to Cohen-Lenstra distribution.

Findings

The Markov chain is reversible.

The spectrum of the transition matrix is explicitly determined.

The Cohen-Lenstra distribution arises naturally in this setting.

Abstract

We study an irreducible Markov chain on the category of finite abelian $p$-groups, whose stationary measure is the Cohen-Lenstra distribution. This Markov chain arises when one studies the cokernel of a random matrix $M$, after conditioning on a submatrix of $M$. We show two surprising facts about this Markov chain. Firstly, it is reversible. Hence, one may regard it is a random walk on finite abelian $p$-groups. The proof of reversibility also explains the appearance of the Cohen-Lenstra distribution in the context of random matrices. Secondly, we can explicitly determine the spectrum of the infinite transition matrix associated to this Markov chain.