UPPER BOUND ON THE TRUNCATED CONNECTIVITY IN ONE-DIMENSIONAL β/|x–y|2 PERCOLATION MODELS AT β>1

Abstract

We consider one-dimensional Fortuin-Kasteleyn percolation models generated by the bond occupation probabilities [Formula: see text] and weighting factor κ. For any β>1 and κ≥1 the percolation density M is known to be strictly positive provided p is sufficiently close to 1. We prove that, under these assumptions, the following upper bound for the truncated connectivity [Formula: see text] holds with [Formula: see text] where [Formula: see text] as [Formula: see text]. This result extends Imbrie-Newman’s upper bound to independent percolation models (κ=1) and random-cluster models with non-integer values of κ.