Addition of recurrent configurations in chip firing games: Finding minimal recurrent configurations with Markov Chains

Abstract

Generalizing the Abelian Sandpile Model by Bak, Tang and Wiesenfeld to general undirected graphs, one gets a variation of the Chip Firing Game intoduced by Chung and Ellis in 2002, which still contains most of the nice algebraic properties of the Abelian Sandpile Model. Particularly the group structure of the recurrent configurations is retained. Using a Markov Chain, we show how a pair consisting of one minimal recurrent configuration and one nearly minimal recurrent configuration can be constructed whose sum is the same as the sum of a given pair of recurrent configurations. Computer simulations of this Markov Chain for the Abelian Sandpile Model suggest that the number of steps needed to reach a final pair usually is proportional to the width of the grid, but can become proportional to the square of the width if one chooses particular configurations. © 2010 Springer-Verlag.