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

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 $p$.

22 pages; v2 adds convergence of variance and updates the discussion section