Random walks and the Potts model

Abstract

The random-walk problem is shown to be related to the $q$-component Potts model in a field in the limit $q\ensuremath{\rightarrow}0$. On a regular lattice when the random-walk problem is easily solved, this gives the solution of the Potts model in a field in all dimensions in the limit $q\ensuremath{\rightarrow}0$. Random walks on percolation clusters are equivalent to a Potts model with random exchange interactions. This relation is used to develop a mean-field theory for random walks at the percolation threshold. This model provides an interesting application of the replica method, which leads naturally to the introduction of distribution functions.