Percolation theory on pairs of matching lattices

Publisher Website
Google Scholar
Abstract
An important magnitude in percolation theory is the critical probability, which is defined as the supremum of those values of the occupation‐probability p, for which only finite clusters occur. In 1964 Sykes and Essam obtained the relation P(s)c(L) +P(s)c(L*) =1, where L and L* are a pair of matching lattices and P(s)c denotes the critical probability (site‐case). The proof was not complete, but based on certain assumptions about the mean number of clusters. Though Sykes and Essam suggested that the above relation holds for all mosaics (i.e., multiply‐connected planar graphs) and decorated mosaics, we have constructed a counterexample. Subsequently, for a more restricted class of graphs, an alternative derivation of the Sykes–Essam relation is given, this time based on the usual assumption that below the critical probability the mean cluster size is finite. The latter assumption is also used to prove for some nontrivial subgraphs of the simple quadratic lattice S, that their critical probability is equal to P(s)c(S). Finally, for a certain class of lattices, sequences of numbers are constructed, which converge to the critical probability. In the case of the site process on S, the number with highest index we found, is 0.5925±0.0002, which seems to be a reasonable estimate of P(s)c(S).

This publication has 7 references indexed in Scilit: