Polynomial bound for a chip firing game on graphs
SIAM Journal on Discrete Mathematics
European Journal of Combinatorics
Games on line graphs and sand piles
Theoretical Computer Science
Chip-Firing and the Critical Group of a Graph
Journal of Algebraic Combinatorics: An International Journal
On the sandpile group of dual graphs
European Journal of Combinatorics
The structure of a linear chip firing game and related models
Theoretical Computer Science
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On the sandpile group of regular trees
European Journal of Combinatorics
Sandpile transience on the grid is polynomially bounded
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The Computational Complexity of One-Dimensional Sandpiles
Theory of Computing Systems
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Faster Generation of Random Spanning Trees
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Almost tight bounds for rumour spreading with conductance
Proceedings of the forty-second ACM symposium on Theory of computing
Rumour spreading and graph conductance
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs
Proceedings of the forty-third annual ACM symposium on Theory of computing
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The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar [14], Dhar et al. [15] 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 ([4]). We develop the theory of discrete diffusions in contrast to continuous harmonic functions on graphs and establish deep connections between standard results in the study of random walks on graphs and sandpiles on graphs. Using this connection and building other necessary machinery we improve the main result of Babai and Gorodezky (SODA 2007,[2]) of the bound on the transience class of an n x n grid, from O(n30) to O(n7). Proving that the transience class is small validates the general notion that for most natural phenomenon, the time during which the system is transient is small. In addition, we use the machinery developed to prove a number of auxiliary results. We exhibit an equivalence between two other tessellations of plane, the honeycomb and triangular lattices. We give general upper bounds on the transience class as a function of the number of edges to the sink. Further, for planar sandpiles we derive an explicit algebraic expression which provably approximates the transience class of G to within O(|E(G)|). This expression is based on the spectrum of the Laplacian of the dual of the graph G. We also show a lower bound of Ω(n3) on the transience class on the grid improving the obvious bound of Ω(n2).