Papers:
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.
A dynamic perspective on the universality phenonomenon for random p-adic matrices (in progress)
We use the random walk defined in a previous paper to give a new perspective on the universality phenomenon for random p-adic matrices.
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)