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. Particularly, 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. Furthermore, we observe the same fluctuations with respect to the asymptotic shape as for the weak infection evolving alone. By the way, we extend the Häggström-Pemantle non-coexistence result "except perhaps for a denumerable set" to families of stochastically comparable passage times indexed by a continuous parameter.