Percolation Models and Animals

We first define site- and bond-percolation models on a general graph. We underline the link between the percolation probability and the enumeration of animals. Next, we focus on certain polynomials introduced as approximants to the percolation probability for several specific graphs. We show that such polynomials can be defined for any graph and explain why they often converge in the algebra of formal power series. fference between these polynomials—denotedPn(q)—and their formal limitP∞(q)is characterized by a doubly indexed family of correction termsdn,k. These correction terms play an important role, since they permit to extend the expansion ofP∞(q)whenPn(q)can no longer be computed. For a givenk, thedn,kʹs are related to ‘compact’ animals, i.e. animals with small site-perimeter with respect to their height. These animals are often easier to enumerate than general animals, and this is why some researchers were able to conjecture formulas fordn,k, forksmall, over several specified graphs. We prove some of these formulas: for site and bond directed percolation on the square lattice, and for bond directed percolation on the honeycomb lattice.