Improved bounds on the sandpile diffusions on Grid graphs
  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. 
Ayush Choure and Sundar Vishwanathan
math-ph cs.DM math.CO math.MP