Existence of a Phase Transition in Harmonic Activation and Transport

Abstract

Harmonic activation and transport (HAT) is a stochastic process that rearranges finite subsets of $\mathbb{Z}^d$, one element at a time. Given a finite set $U \subset \mathbb{Z}^d$ with at least two elements, HAT removes $x$ from $U$ according to the harmonic measure of $x$ in $U$, and then adds $y$ according to the probability that simple random walk from $x$, conditioned to hit the remaining set, steps from $y$ when it first does so. In particular, HAT conserves the number of elements in $U$. We study the classification of HAT as recurrent or transient, as the dimension $d$ and number of elements $n$ in the initial set vary. It was recently proved that the stationary distribution of HAT (on sets viewed up to translation) exists when $d = 2$, for every number of elements $n \geq 2$. We prove that HAT exhibits a phase transition in both $d$ and $n$, in the sense that HAT is transient when $d \geq 5$ and $n \geq 4$. Remarkably, transience occurs in only one "way": The set splits into clusters of two or three elements, which then grow steadily, indefinitely separated. We call these clusters dimers and trimers. Underlying this characterization of transience is the fact that, from any set, HAT reaches a set consisting exclusively of dimers and trimers, in a number of steps and with at least a probability which depend on $d$ and $n$ only.