Sandpile avalanches in two dimensions are finite

Abstract

In the abelian sandpile model on $\mathbb{Z}^d$, a site with at least $2d$ grains topples by sending one grain to each neighbor. In a box $[-L,L]^d$ whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain, and Athreya and Járai (2004) constructed the stationary state on $\mathbb{Z}^d$ as the limit of its stationary distributions as $L\to\infty$. By a formula of Dhar (1990), when a grain is added at the origin of a sandpile in the stationary state, the expected number of topplings at a site in the resulting avalanche is proportional to the Green function of simple random walk from the origin. This Green function is finite for $d\geq3$, where Járai and Redig (2008) proved that avalanches are almost surely finite, and infinite for $d=2$. We prove that for $d=2$ avalanches are nevertheless almost surely finite: the probability that an avalanche reaches distance $R$ is at most a negative power of $\log R$. The proof follows the avalanche backward and reduces the question to an estimate for the uniform spanning tree, which we prove by a multiscale electrical argument.