Non-equilibrium multi-scale analysis and coexistence in competing first passage percolation

Abstract

The main contribution of this paper is the development of a novel approach to multi-scale analysis that we believe can be used to analyse processes with non-equilibrium dynamics. Our approach will be referred to as multi-scale analysis with non-equilibrium feedback and will be used to analyse a natural random growth process with competition on ${\mathbb{Z}}^d$ called first passage percolation in a hostile environment (FPPHE) that consists of two first passage percolation processes ${FPP}_1$ and ${FPP}_{\lambda }$ that compete for the occupancy of sites. Initially, ${FPP}_1$ occupies the origin and spreads through the edges of ${\mathbb{Z}}^d$ at rate 1, while ${FPP}_{\lambda }$ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter $p\in (0,1)$, where a seed remains dormant until ${FPP}_1$ or ${FPP}_{\lambda }$ attempts to occupy it before then spreading through the edges of ${\mathbb{Z}}^d$ at rate $\lambda >0$. Two fundamental challenges of FPPHE that our approach is able to handle are the absence of monotonicity (for instance, adding seeds could be beneficial to ${FPP}_1$ instead of ${FPP}_{\lambda }$) and its non-equilibrium dynamics; such characteristics, for example, prevent the application of a more standard multi-scale analysis. As a consequence of our main result for FPPHE, we establish a coexistence phase for $d\ge 3$, answering an open question of Sidoravicius and Stauffer (2019). This exhibits a rare situation where a natural random competition model on ${\mathbb{Z}}^d$ observes coexistence for processes with different speeds. Moreover, we are able to establish the stronger result that ${FPP}_1$ and ${FPP}_{\lambda }$ can both occupy a positive density of sites with positive probability, which is in stark contrast with other competition processes.