What is a $p$-adic Dyson Brownian motion?

TL;DR

This paper introduces a $p$-adic analogue of Dyson Brownian motion by analyzing singular numbers of a Markov process on $ ext{GL}_N(Q_p)$, revealing a Poisson jump process with reflection dynamics.

Contribution

It explicitly classifies the dynamics of singular numbers in a $p$-adic setting, establishing a novel $p$-adic Dyson Brownian motion model with explicit evolution rules.

Findings

Singular numbers evolve as a Poisson jump process on $Z^N$

Ordering is enforced by reflection off Weyl chamber walls

Provides a $p$-adic analogue to classical Dyson Brownian motion

Abstract

We consider the singular numbers of a certain explicit continuous-time Markov jump process on $\mathrm{GL}_N(\mathbb{Q}_p)$, which we argue gives the closest $p$-adic analogue of multiplicative Dyson Brownian motion. We do so by explicitly classifying the possible dynamics of singular numbers of processes on $\mathrm{GL}_N(\mathbb{Q}_p)$ satisfying natural properties possessed by Brownian motion on $\mathrm{GL}_N(\mathbb{C})$. Computing the evolution of singular numbers explicitly, we find that the $N$-tuple of singular numbers in decreasing order evolves as a Poisson jump process on $\mathbb{Z}^N$, with ordering enforced by reflection off the walls of the positive type $A$ Weyl chamber. This contrasts with -- and provides a $p$-adic analogue to -- the behavior of classical Dyson Brownian motion, where ordering is enforced by conditioning to avoid the Weyl chamber walls.