Improved bounds on the sandpile diffusions on Grid graphs
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.
Cite
@article{arxiv.1210.4327,
title = {Improved bounds on the sandpile diffusions on Grid graphs},
author = {Ayush Choure and Sundar Vishwanathan},
journal= {arXiv preprint arXiv:1210.4327},
year = {2016}
}
Comments
This paper has been withdrawn because of typographical error in a proof, which lead to wrong bounds. the corrected version has been incorporated into the journal version of our paper, "random walks, electric networks, and transience class problem of sandpile"