The size of a pond in 2D invasion percolation

We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.

Ladder Sandpiles

We study Abelian sandpiles on graphs of the form $G times I$, where $G$ is an arbitrary finite connected graph, and $I subset Z$ is a finite interval. We show that for any fixed $G$ with at least two vertices, the stationary measures $mu_I = mu_{G times I}$ have two extremal weak limit points as $I uparrow Z$. The extremal limits are the only ergodic measures of maximum entropy on the set of infinite recurrent configurations. We show that under any of the limiting measures, one can add finitely many grains in such a way that almost surely all sites topple infinitely often. We also show that the extremal limiting measures admit a Markovian coding.