Cluster size and shape in random and correlated percolation

Abstract

A rigorous inequality between the pair correlation function and connectivity functions is proved for the Ising model (correlated percolation). This relation shows that large correlations imply large connectivity. Such inequality becomes equality in the random percolation problem (infinite temperature). Other relations among susceptibility cluster size and perimeter are also derived which give information on the shape of the cluster for the random and correlated percolation problems.