On strong sharp phase transition in the random connection model
Mikhail Chebunin, Günter Last
TL;DR
The paper analyzes the random connection model (RCM) built on a Poisson point process with a symmetric connection function, establishing subcritical exponential moment bounds for cluster sizes and a strong sharp phase transition at the percolation threshold. It develops and leverages a Mecke-type equation and a spatial Markov property to derive monotonicity and coupling results, enabling subcritical and supercritical analyses. In the stationary marked setting, it shows t_T=t_c under natural integrability assumptions and provides susceptibility bounds, diameter tail estimates, and AB-differential inequalities that extend prior results to marked and non-Markovian contexts. Overall, the work unifies and extends sharp-transition results for continuum percolation models, offering practical criteria for finite cluster moments, diameter control, and mean-field-type bounds across general and stationary marked variants.
Abstract
We consider a random connection model (RCM) $ξ$ driven by a Poisson process $η$. We derive exponential moment bounds for an arbitrary cluster, provided that the intensity $t$ of $η$ 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 $ξ$ 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.