Limiting shape for first-passage percolation models on random geometric graphs

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