Limiting shape for first-passage percolation models on random geometric graphs
- Coletti, Cristian F.
- de Lima, Lucas R.
- Hinsen, Alexander
- Jahnel, Benedikt
- Valesin, Daniel R.
2020 Mathematics Subject Classification
- 52A22 60F15 60K35
- Poisson--Gilbert graph, Richardson model, Bernoulli bond percolation, subergodicity, high-density limit, isotropic shapes
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 branching process in the joint limit of large intensities and slow passing times.