Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks

TL;DR

This paper proves the existence of an absorbing-state phase transition in stochastic sandpiles and activated random walks across all dimensions, providing quantitative bounds and a versatile multi-scale analysis method.

Contribution

It establishes the phase transition rigorously in any dimension and introduces a multi-scale analysis approach applicable to other non-equilibrium systems.

Findings

Confirmed phase transition existence in all dimensions.

Derived bounds on activity cessation speed.

Developed a multi-scale analysis technique.

Abstract

We study the dynamics of two conservative lattice gas models on the infinite d-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the Stochastic Sandpiles Model (SSM), introduced in the physics literature in the early nineties. Theoretical arguments and numerical analysis predicted that the ARW and SSM undergo a phase transition between an absorbing phase and an active phase as the initial density crosses a critical threshold. However a rigorous proof of the existence of an absorbing phase was known only for one-dimensional systems. In the present work we establish the existence of such phase transition in any dimension. Moreover, we obtain several quantitative bounds for how fast the activity ceases at a given site or on a finite system. The multi-scale analysis developed here can be extended to other contexts providing an efficient tool to study non-equilibrium phase transitions.