Random walks in a field of soft traps and criticality for the dissipative Abelian Sandpile Model

TL;DR

This paper studies a one-dimensional random walk in a landscape of soft traps inspired by the dissipative abelian sandpile model, identifying a critical growth rate of trap intervals that marks a transition between critical and non-critical behavior.

Contribution

It precisely characterizes the critical growth rate of trap intervals that determines the transition from critical to non-critical states in the dissipative abelian sandpile model.

Findings

Identified a critical growth rate of trap intervals for phase transition.

Established the link between trap distribution and sandpile criticality.

Provided a mathematical criterion for dissipation levels affecting criticality.

Abstract

Motivated by the dissipative abelian sandpile model, we analyze the trajectories of a one-dimensional random walk in a landscape of soft traps. These traps, placed at increasing distances from each other, correspond to dissipative sites in the associated dissipative abelian sandpile model. We identify a critical growth rate of the sizes of intervals between successive traps where there is a transition between finiteness and non-finiteness of the expected survival time of the random walk. This corresponds to a transition between non-criticality and criticality of the associated dissipative abelian sandpile model. Therefore, in this setting, we thus identify precisely how much dissipation can be added to the original abelian sandpile model in order to disrupt its criticality.