2024/10/21 by Faipeur, Corentin
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2410.15722
In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and vertices are said to be active independently with probability p ∈ [0, 1]. We consider W to be the reunion of random balls with an active center. In certain circumstances, the model does not exhibit a phase transition, in the sense that W almost surely contains an infinite component for all p > 0, or even W covers the entire graph. In this paper, we give a sufficient condition on the radius distribution for the existence of a subcritical phase, namely a regime such that almost surely all the connected components of W are finite. Additionally, we provide a sufficient condition for the exponential decay of the size of a typical component.