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Cutoff for activated random walk

Christopher Hoffman, Tobias Johnson, Matthew Junge, Josh Meisel

TL;DR

This work resolves the mixing-time behavior of driven-dissipative activated random walk on an interval by proving a cutoff at $(\rho_{\mathrm{FE}}+o(1))n$ under uniform or central driving. The authors introduce a general graph-level mixing criterion: once activity likely visits every site, the chain mixes to its stationary distribution, regardless of initial conditions. They develop a street-sweeper lemma and a preemptive abelian property within a sitewise ARW framework, then specialize to dimension one by proving that a density slightly above the fixed-energy critical density suffices to visit all sites with high probability, yielding the cutoff. The results support universality of ARW as a self-organized criticality model, connect driven-dissipative behavior to fixed-energy criticality via layer percolation, and lay groundwork for extensions to higher dimensions.

Abstract

We prove that the mixing time of driven-dissipative activated random walk on an interval of length $n$ with uniform or central driving exhibits cutoff at $n$ times the critical density for activated random walk on the integers. The proof uses a new result for arbitrary graphs showing that the chain is mixed once activity is likely at every site.

Cutoff for activated random walk

TL;DR

This work resolves the mixing-time behavior of driven-dissipative activated random walk on an interval by proving a cutoff at under uniform or central driving. The authors introduce a general graph-level mixing criterion: once activity likely visits every site, the chain mixes to its stationary distribution, regardless of initial conditions. They develop a street-sweeper lemma and a preemptive abelian property within a sitewise ARW framework, then specialize to dimension one by proving that a density slightly above the fixed-energy critical density suffices to visit all sites with high probability, yielding the cutoff. The results support universality of ARW as a self-organized criticality model, connect driven-dissipative behavior to fixed-energy criticality via layer percolation, and lay groundwork for extensions to higher dimensions.

Abstract

We prove that the mixing time of driven-dissipative activated random walk on an interval of length with uniform or central driving exhibits cutoff at times the critical density for activated random walk on the integers. The proof uses a new result for arbitrary graphs showing that the chain is mixed once activity is likely at every site.

Paper Structure

This paper contains 10 sections, 11 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Self-organized criticality and sandpiles
  3. Mixing time for ARW: prior results and methods
  4. Methods of our proof
  5. Organization
  6. Notation for the sitewise construction
  7. Bounds on general graphs
  8. Mixing in dimension one
  9. Proof of Theorem \ref{['thm:mix']} assuming Proposition \ref{['prop:exit.criterion']}
  10. Proof of Proposition \ref{['prop:exit.criterion']}

Key Result

Theorem 1.1

Let $(\sigma_t)_{t\geq 1}$ denote the driven-dissipative ARW chain on $\llbracket1,n\rrbracket$ with uniform or central driving. For any sleep rate $\lambda\in(0,\infty)$ and $\epsilon>0$, there exist constants $c,C>0$ such that for all $n \geq 1$

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1: Preemptive abelian property
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Street-sweeper lemma
  • proof
  • ...and 12 more