ASYMMETRIC ABELIAN SANDPILE MODELS

In the Abelian sandpile models introduced by Dhar, long-time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system. Dhar propos ed a simple procedure, the burning algorithm, as a possible test of wh ether a configuration is recurrent, and later with Majumdar verified t he correctness of this test when the toppling rules of the sandpile ar e symmetric. We observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles; we als o obtain necessary and sufficient conditions for the validity of the o riginal test. The results are applied to a family of deterministic one -dimensional sandpile models originally studied by Lee, Liang, and Tze ng.