Non-criticality criteria for Abelian sandpile models with sources and sinks

TL;DR

This paper investigates the non-critical behavior of Abelian sandpile models on various structures, including random trees and lattices with sources and sinks, providing conditions and connections to other models.

Contribution

It establishes non-criticality criteria for Abelian sandpile models on random trees and lattices with sources and sinks, linking to the parabolic Anderson and pinning models.

Findings

Sandpile on random binary trees is non-critical for all p<1.

Provides conditions for non-criticality in lattices with dissipative sites.

Connects sandpile behavior to the parabolic Anderson and pinning models.

Abstract

We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $\mathbb{Z}^d$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.