On graph parameters guaranteeing fast Sandpile diffusion

TL;DR

This paper identifies key graph properties such as volume growth, boundary regularity, and interior constraints that ensure polynomial bounds on the transience class of sandpile models, extending previous grid-based results.

Contribution

It generalizes polynomial bounds on sandpile transience classes from grids to broader classes of graphs using geometric and analytic properties.

Findings

Polynomially bounded transience classes for certain graph classes

Key graph properties influence diffusion speed in sandpile models

Extension of grid results to more general graphs

Abstract

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of self-organized criticality. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent (\cite{BT05}). We demonstrate a class of sandpile which have polynomially bound transience classes by identifying key graph properties that play a role in the rapid diffusion process. These are the volume growth parameters, boundary regularity type properties and non-empty interior type constraints. This generalizes a previous result by Babai and Gorodezky (SODA 2007,\cite{LB07}), in which they establish polynomial bounds on $n \times n$ grid. Indeed the properties we show are based on ideas extracted from their proof as well as the continuous analogs in complex analysis. We conclude with a discussion on the notion of degeneracy and dimensions in graphs.