Connection probabilities in Poisson random graphs with uniformly bounded edges

TL;DR

This paper analyzes connection probabilities in Poisson random graphs with bounded edges, demonstrating exponential decay in the subcritical phase and positive lower bounds in the supercritical phase, relevant for disordered systems and conductivity models.

Contribution

It applies randomized algorithm methods to establish phase transition behaviors in connection probabilities for various bounded-edge Poisson graph models.

Findings

Exponential decay of connection probability in subcritical phase

Positive lower bound for infinite connectivity in supercritical phase

Application to conductivity models in disordered systems

Abstract

We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with lower-bounded conductances. The latter is relevant for the analysis of conductivity by Mott variable range hopping in strongly disordered systems. By using the method of randomized algorithms developed by Duminil-Copin et al. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to $ (\lambda-\lambda_c)$, $\lambda$ being the density of the Poisson point process and $\lambda_c$ being the critical density.