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

Abstract: We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, 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 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. A seed remains dormant until FPP_1 or FPP_\lambda attempts to occupy it, after which it spreads through the edges of Z^d at rate \lambda. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on Z^d. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_\lambda could favor FPP_1. A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity. Based on joint works with Vladas Sidoravicius and Tom Finn.

mathematical physicscombinatoricsprobabilitystatistics theory

Audience: researchers in the topic

Abu Dhabi Stochastics Seminar