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We study the interplay between chip-firing games and potential theory ongraphs, characterizing reduced divisors ($G$-parking functions) on graphs asthe solution to an energy (or potential) minimization problem and providing analgorithm to efficiently compute reduced divisors. Applications include an"efficient bijective" proof of Kirchhoff's matrix-tree theorem and a newalgorithm for finding random spanning trees. The running times of ouralgorithms are analyzed using potential theory, and we show that the boundsthus obtained generalize and improve upon several previous results in theliterature. We also extend some of these considerations to metric graphs.

Chip-firing games, potential theory on graphs, and spanning trees