Connection probabilities in Poisson random graphs with uniformly bounded edges

2017/12/19 by Faggionato, Alessandra, Mimun, Hlafo Alfie
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1712.07016

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 (λ-λ_c), λ being the density of the Poisson point process and λ_c being the critical density.

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