Improved bounds on the sandpile diffusions on Grid graphs

TL;DR

This paper improves the upper bound on the transience class of the Abelian Sandpile Model on grid graphs from O(n^7) to O(n^7 log n), using combinatorial and harmonic analysis techniques.

Contribution

It presents a tighter bound on the transience class of sandpiles on grid graphs and introduces methods that could be useful for related graph resistance problems.

Findings

Bound on transience class improved to O(n^7 log n)

Tight bounds established on resistance ratios in grid networks

Tools developed may have broader applications in graph analysis

Abstract

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar [10], Dhar et al. [11]) 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 ([3]). Using elementary combinatorial arguments and symmetry properties, Babai and Gorodezky (SODA 2007,[2]) demonstrated a bound of O(n^30) on the transience class of an nxn grid. This was later improved by Choure and Vishwanathan (SODA 2012,[7]) to O(n^7) using techniques based on harmonic functions on graphs. We improve this bound to O(n^7 log n). We also demonstrate tight bounds on certain resistance ratios over grid networks. The tools used for deriving these bounds may be of independent interest.