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 $\exp(-n^{1/3}) \ll q_n \ll n^{-1/2}$. This implies that the stationary configuration law 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 $p$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• Remark 6
• Remark 7
• Remark 8
• Lemma 9
• Lemma 10