Papers:
A random walk on p-groups with a symmetric perfect pairing (
arxiv preprint)
The kernel of a random symmetric p-adic matrix is a random abelian group,
equipped with a symmetric pairing. If we consider not only the matrix but also
its top-left corners, we get a process valued in isomorphism classes of abelian
groups, equipped with such a pairing. We show that when the matrix is Haar
random, this process is a Markov chain, generated by an operator that we
explicitly describe. We will also prove that this operator is reversible with
respect to a Cohen-Lenstra type measure.
On the satisfaction frequency of spectral characterization conditions (joint with A. Van Werde) (
arxiv preprint)
We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial.
Universality results for random matrices over finite local rings (
arxiv preprint)
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.
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.
Recent Talks and Posters:
Talk: On the satisfaction frequency of spectral characterization conditions. IMS Annual Meeting 2026, Salzburg (2026) (
Slides)
Talk: Markov Chains in Random Matrix Theory. Mini-course on Cohen-Lenstra heuristics for random matrices, OSU (2026) (
Abstract)
Talk: Random Walks arising in Random Matrix Theory. Waterloo Number Theory Seminar (2026) (
Slides)
Talk: A random walk on the category of finite abelian p-groups. Mobius ANT Seminar, Université de Montréal (2025)
Talk: Universality results for random matrices over local rings. Maine-Quebec Number Theory Conference (2025) (
Slides)
Talk: A random walk on the category of finite abelian p-groups. Quebec-Maine Number Theory Conference (2024) (
Slides)
Poster: A random walk on the category of finite abelian p-groups. Bernoulli-IMS 11th World Congress in Probability in Statistics (2024) (
pdf)
Notes
A dynamic point of view on universality (
Preliminary version)
In this short note, we consider the corners process for an i.i.d. matrix. When the distribution of the entries is uniform, this process is a Markov chain, and hence the ergodic theorem for Markov chains can be applied. This implies, in particular, that for uniformly distributed p-adic random matrices, the cokernels of the corners are distributed according to the Cohen-Lenstra measure, almost surely. The purpose of this note is to show that the conclusion of the ergodic theorem also holds for i.i.d matrices, provided that the distribution of the entries is not concentrated on the translate of a subring, or the translate of an ideal.
Other documents
List of open questions related to Cohen-Lenstra heuristics for random matrices (
Questions)
This list of questions was prepared for the problem session at the 2026 workshop on Cohen-Lenstra heuristics for random matrices at Ohio State University.