Cluster sizes in subcritical soft Boolean models

TL;DR

This paper analyzes the tail behavior of cluster sizes in the subcritical phase of the soft Boolean model, revealing how the tail exponents of radii and edge weights influence connectivity and component size distributions.

Contribution

It provides a sharp criterion based on tail exponents that distinguishes regimes where either radius or edge weight dominates the cluster size behavior.

Findings

Identifies the tail behavior of the Euclidean diameter of components.

Determines the distribution of the number of points in a typical component.

Establishes a criterion based on tail exponents for different connectivity regimes.

Abstract

We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected.