Largest component and sharpness in continuum percolation

TL;DR

This paper analyzes the size and sharpness of large connected components in the Poisson Random Connection model, revealing logarithmic growth in subcritical and linear growth in supercritical regimes.

Contribution

It provides new asymptotic growth results for the largest component and establishes a sharpness property with exponential tail decay in the subcritical regime.

Findings

Largest component grows logarithmically in subcritical regime

Largest component grows linearly in supercritical regime

Cluster size at the origin has exponential tail decay in subcritical regime

Abstract

We investigate the behavior of large connected components in the Poisson Random Connection model in non-critical regimes with any bounded connection function. We show that the asymptotic size of the largest component restricted to a window grows logarithmically in the volume of that window in the subcritical case, and linearly in the supercritical case. We also prove a sharpness result saying that the order of the cluster at the origin has an exponentially decaying tail in the subcritical regime.