Research

Papers:

Universality results for random matrices over finite local rings (part 1, part 2, part 3)

Let R be a finite local ring. We prove a quantitative universality statement for the cokernel of random matrices with i.i.d. entries valued in R. Rather than use the moment method, we use the Lindeberg replacement technique. This approach also yields a universality result for several invariants that are finer than the cokernel, such as the span and the determinant.

We have separated the argument into three short articles, for ease of reading. The first contains generalities about measures on modules and may be of independent interest. The second is logically independent of the first; its purpose is to establish a technical estimate. The third article contains the proof of universality.

Markov chains arising in the study of random matrices over pro-finite local rings (preliminary version)

Recent work of the author investigates certain random processes, valued in abelian p-groups, that naturally arise in the study of Haar random matrices over the p-adic integers. Non-trivially, it was found that these processes are reversible Markov chains. In this short note, we give a simple alternative derivation of this fact. The new derivation also proves that this phenomenon is not specific to the p-adic integers, but generalizes to Haar random matrices over any profinite local ring.

A random walk on the category of finite abelian p-groups (arxiv preprint)

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.

Random walks on finite abelian p-groups with extra structure (in progress)

We define and study random walks on finite abelian p-groups with extra structure. We consider groups equipped with a perfect symmetric pairing, groups equipped with a perfect anti-symmetric pairing, and groups containing some fixed group B as a subgroup. We define an irreducible, reversible Markov chain on each class of objects, and study their properties. These Markov chains arise naturally in the context of random matrices.

Recent Talks and Posters:

Poster: A random walk on the category of finite abelian p-groups. Bernoulli-IMS 11th World Congress in Probability in Statistics (2024) (pdf)

Talk: A random walk on the category of finite abelian p-groups. Quebec-Maine Number Theory Seminar (2024) (Slides)