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The hockey-stick conjecture for activated random walk

Christopher Hoffman, Tobias Johnson, Matthew Junge

Abstract

We prove a conjecture of Levine and Silvestri that the driven-dissipative activated random walk model on an interval drives itself directly to and then sustains a critical density. This marks the first rigorous confirmation of a sandpile model behaving as in Bak, Tang, and Wiesenfeld's original vision of self-organized criticality.

The hockey-stick conjecture for activated random walk

Abstract

We prove a conjecture of Levine and Silvestri that the driven-dissipative activated random walk model on an interval drives itself directly to and then sustains a critical density. This marks the first rigorous confirmation of a sandpile model behaving as in Bak, Tang, and Wiesenfeld's original vision of self-organized criticality.

Paper Structure

This paper contains 5 sections, 11 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of the hockey-stick conjecture
  3. Proof of Proposition \ref{['thm:lower.bound']}
  4. Selected notation and definitions from [HJJ24]
  5. Selected results from [HJJ24]

Key Result

Theorem 1

For any $\lambda,\rho,\epsilon >0$, for constants $c,C>0$ depending only on $\lambda$ and $\epsilon$.

Figures (2)

  • Figure 1: The limiting empirical density profile for the abelian sandpile on the flower graph as proven in fey2010approach. Though it appears at first glance to have the shape established in Theorem \ref{['thm:hockey']}, it in fact grows to a constant approximately equal to $1.6689$ before decreasing toward its asymptotic limit $5/3$. The numerical data given by Fey, Levine, and Wilson indicate similar behavior for the abelian sandpile on the two-dimensional lattice.
  • Figure 2: A sample path of $D_\rho(n,\lambda)$ with $n=2000$ and $\lambda=.8$. The graph shows the empirical density as $2500$ particles are added uniformly at random one at a time, stabilizing after each addition, on an interval of length $2000$. The critical density $\rho_{\mathtt{FE}}$ appears to be approximately $.889$. The sample path is a near-perfect match to the limiting empirical density profile $\rho\mapsto\min(\rho,\rho_{\mathtt{FE}})$. See levine2023universality for a similar simulation in dimension two.

Theorems & Definitions (15)

  • Theorem 1
  • Corollary 2
  • Proposition 3
  • proof : Proof of \ref{['thm:hockey']}
  • proof : Proof of \ref{['cor:sup.norm']}
  • Lemma 4
  • proof
  • proof : Proof of \ref{['thm:lower.bound']}
  • Theorem 5: Theorem 8.4
  • Proposition 6: Proposition 8.5
  • ...and 5 more