First-passage competition with different speeds: positive density for both species is impossible

Abstract

Consider two epidemics whose expansions on $\mathbb{Z}^d$ are governed by two families of passage times that are distinct and stochastically comparable. We prove that when the weak infection survives, the space occupied by the strong one is almost impossible to detect: for instance, it could not be observed by a medium resolution satellite. We also recover the same fluctuations with respect to the asymptotic shape as in the case where the weak infection evolves alone. In dimension two, we prove that one species finally occupies a set with full density, while the other one only occupies a set of null density. We also prove that the Häggström-Pemantle non-coexistence result "except perhaps for a denumerable set" can be extended to families of stochastically comparable passage times indexed by a continuous parameter.