Nikita Lvov
Random Walks arising in Random Matrix Theory
The cokernel of a large p-adic random matrix M is a random abelian p-group. Friedman and Washington showed that its distribution asymptotically tends to the well-known Cohen-Lenstra distribution. 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 cokernels of corners of M. We show two surprising facts about this Markov chain. Firstly, it is reversible. Hence, one may regard it as 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. Finally, we show how these results generalize to random matrices over general pro-finite local rings.
MC 5403