Geometry-induced criticality in $p$-adic scaling limits of random walks

Abstract

An anisotropy parameter $h$ in $(0,1]$ induces on $\mathbb{Q}_p^2$ a duality-compatible, two-scale filtration that collapses to one scale at the right endpoint. This filtration defines shell-uniform transition laws for hierarchical random walks on a discrete group whose scaling limits are L\'{e}vy processes on $\mathbb{Q}_p^2$. The diffusion constants of the coordinate processes jump at the right endpoint, even though the radial jump law depends continuously on $h$. This instance of geometry-induced criticality isolates a structural mechanism that should extend to locally compact abelian groups and suggests a route to studying critical behavior in ultrametric models.