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

TL;DR

This work addresses strict monotonicity of the critical value in independent long-range percolation under local perturbations of the connectivity function on transitive graphs. It replaces differential-inequality techniques with a stochastic domination framework built on a local exploration algorithm and a coupling that ties $G_J$, $G_{J'}$, and $G_{pJ}$. The authors prove that strong criticality coincides with standard criticality, and derive subcritical and supercritical behavior under downward and upward perturbations, including finite-susceptibility and exponential decay in the finitely supported case. The results extend to directed/oriented percolation and to a broad class of transitive and beyond-transitive graphs, offering robust insights into how local edge perturbations affect global percolation thresholds.

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.

Figures (1)

• Figure 1: Schematic depiction of part of the exploration scheme: when the active edge $x_ty_t$ (red) is explored, a passed (F)-check ensures that the only unexplored vertices that can be reached from $y_t$ are in $\mathcal{N}_{\Delta }( y_{t})$ (blue rectangle). In the coupling of exploration to random graphs, these edges correspond to edges in $G_J$ that are potentially all removed in $G_{J'}$, in which case $x_ty_t$ becomes irrelevant for the percolation event in $G_{J'}$.

Theorems & Definitions (11)

• Remark 2.1
• Theorem 2.2: Characterisation of critical parameter set
• Theorem 2.3: Well-behaviour under upward perturbation
• Theorem 2.4: Well-behaviour under downward perturbation
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['thm:main2']}
• proof : Proof of Theorem \ref{['thm:main']}