Scaling limit and density conjecture for activated random walk on the complete graph

TL;DR

This paper investigates the behavior of activated random walk on complete graphs, revealing a Gumbel limit for sleeping particles, hyperuniformity, and phase transitions depending on parameters.

Contribution

It introduces new scaling limits and phase transition results for the stationary distribution of activated random walk on complete graphs.

Findings

Sleeping particles follow a Gumbel distribution under certain conditions.

Stationary configuration exhibits hyperuniformity and negative correlations.

Phase transition occurs at a critical density when no particles are allowed to sleep.

Abstract

We study driven-dissipative activated random walk with sleep probability $p$ on an $n$-vertex complete graph with a sink that traps jumping particles with probability $q_n$. We show that the number of sleeping particles $S_n$ left by the stationary distribution has a Gumbel scaling limit for $\exp(-n^{1/3}) \ll q_n \ll n^{-1/2}$. The particular scaling implies that $S_n$ is hyperuniform and thus the stationary configuration law has negative correlations and is not a product measure. We also prove that $S_n/n$ converges to $p$ if and only if $q_n = e^{-o(n)}$, and that, when $q_n=0$, the number of jumps to stabilization undergoes a phase transition at density $p$.