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

TL;DR

This paper proves that the critical points in independent long-range percolation models on vertex-transitive graphs change strictly monotonically with local modifications, using a coupling approach that improves upon previous methods.

Contribution

It introduces a coupling-based proof of strict monotonicity of critical points, working under minimal assumptions and applicable to both directed and undirected models.

Findings

Critical points are strictly monotonic with respect to local perturbations.

The method applies to both directed and undirected models.

Results improve upon classical enhancement techniques.

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.