Strict monotonicity of critical points in independent long-range percolation models

Abstract

We consider independent long-range percolation models on locally finite vertex-transitive graphs. Using coupling ideas we prove strict monotonicity of the critical points with respect to local perturbations in the connection function, thereby improving upon previous results obtained via the classical essential enhancement method of Aizenman and Grimmett in several ways. In particular, our approach allows us to work under minimal assumptions, namely shift-invariance and summability of the connection function, and it applies to both undirected and directed bond percolation models.

arXiv commentv2: The manuscript is unchanged; however there is a mistake in the proof of the key technical result Prop. 3.2. The authors are in the process of preparing a revised version