Limiting shape for First-Passage Percolation models on Random Geometric Graphs

TL;DR

This paper studies the asymptotic shape in first-passage percolation on supercritical random geometric graphs, showing it is an Euclidean ball and analyzing convergence properties in specific models.

Contribution

It establishes conditions for the existence of an asymptotic shape and proves it is an Euclidean ball, with applications to Bernoulli percolation and the Richardson model.

Findings

Asymptotic shape exists and is an Euclidean ball.

Convergence to a nonstandard branching process in the Richardson model.

Applicable to Bernoulli percolation and Richardson models.

Abstract

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.