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arXiv · 1207.0421

On graph parameters guaranteeing fast Sandpile diffusion

Ayush Choure·Sundar Vishwanathan
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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.

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Ayush Choure, Sundar Vishwanathan. 2012-11-01. On graph parameters guaranteeing fast Sandpile diffusion. https://arxiv.org/abs/1207.0421

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