The hockey-stick conjecture for activated random walk

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.

💡 Research Summary

The paper provides the first rigorous proof of the “hockey‑stick” conjecture for the driven‑dissipative activated random walk (ARW) in one dimension, confirming the original Bak‑Tang‑Wittenfeld (BTW) picture of self‑organized criticality (SOC). In the driven‑dissipative setting particles are added uniformly at random to a finite interval with absorbing boundaries; after each addition the system is allowed to relax until all particles are either sleeping or have escaped. The authors show that the empirical density of sleeping particles after ⌈ρ n⌉ additions, denoted D₍ρ₎(n,λ), concentrates exponentially around min(ρ, ρ_FE(λ)), where ρ_FE(λ) is the critical density of the corresponding fixed‑energy ARW on ℤ. Theorem 1 gives a bound P(|D₍ρ₎−min(ρ, ρ_FE)|>ε)≤C e^{−c n} for any ε>0, and Corollary 2 extends this to a uniform sup‑norm convergence over a growing range of ρ.

The core technical contribution is Proposition 3, which bounds the number of particles lost to the sinks when stabilizing a sub‑critical or critical random configuration. To obtain this bound the authors develop a novel odometer construction: instead of a single odometer that must be stable at both ends, they build two separate stable odometers (one for each boundary) and combine them into an “extended” odometer defined on a larger interval J₀,n+1. This extended odometer is automatically non‑negative on the interior and provides sharp upper bounds on the loss at both boundaries.

A key tool is the layer‑percolation framework introduced in their earlier work