Reflecting Poisson walks and dynamical universality in $p$-adic random matrix theory
Abstract
We prove dynamical local limits for the singular numbers of -adic random matrix products at both the bulk and edge. The limit object which we construct, the reflecting Poisson sea, may thus be viewed as a -adic analogue of line ensembles appearing in classical random matrix theory. However, in contrast to those it is a discrete space Poisson-type particle system with only local reflection interactions and no obvious determinantal structure. The limits hold for any -invariant matrix distributions under weak universality hypotheses, with no spatial rescaling.
Cite
@article{arxiv.2312.11702,
title = {Reflecting Poisson walks and dynamical universality in $p$-adic random matrix theory},
author = {Roger Van Peski},
journal= {arXiv preprint arXiv:2312.11702},
year = {2026}
}
Comments
v1: 49 pages, 3 figures. First version, comments welcome! v2: updates in response to referee comments, in particular a mis-statement in Theorem 1.4 and several errors in technical lemmas in Section 6 are corrected since the previous version. Published version, to appear in Annals of Probability