On strong sharp phase transition in the random connection model

TL;DR

This paper analyzes the phase transition in the random connection model, establishing bounds on cluster sizes and diameters below a critical intensity, and showing the equivalence of critical and percolation thresholds under certain conditions.

Contribution

It provides new exponential moment bounds for clusters and diameters, and proves the equality of critical and percolation thresholds in the stationary case, generalizing previous results.

Findings

Finite mean cluster size below critical intensity

Exponential decay of cluster diameters under certain conditions

Critical intensity equals percolation threshold in stationary case

Abstract

We consider a random connection model (RCM) $\xi$ driven by a Poisson process $\eta$. We derive exponential moment bounds for an arbitrary cluster, provided that the intensity $t$ of $\eta$ is below a certain critical intensity $t_T$. The associated subcritical regime is characterized by a finite mean cluster size, uniformly in space. Under an exponential decay assumption on the connection function, we also show that the cluster diameters are exponentially small as well. In the important stationary marked case and under a uniform moment bound on the connection function, we show that $t_T$ coincides with $t_c$, the largest $t$ for which $\xi$ does not percolate. In this case, we also derive some percolation mean field bounds. These findings generalize some of the recent results. Even in the classical unmarked case, our results are more general than what has been previously known. Our proofs are partially based on some stochastic monotonicity properties, which might be of interest in their own right.