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

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 Z ^d called first passage percolation in a hostile environment (FPPHE) that consists of two first passage percolation processes FPP ₁ and FPP that compete for the occupancy of sites. Initially, FPP ₁ occupies the origin and spreads through the edges of Z ^d at rate 1, while FPP is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p 2 .0; 1/, where a seed remains dormant until FPP ₁ or FPP attempts to occupy it before then spreading through the edges of Z ^d at rate > 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) 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 3, answering an open question of Sidoravicius and Stauffer (2019). This exhibits a rare situation where a natural random competition model on Z ^d observes coexistence for processes with different speeds. Moreover, we are able to establish the stronger result that FPP ₁ and FPP can both occupy a positive density of sites with positive probability, which is in stark contrast with other competition processes.