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

Speaker: Thomas Finn (University of Bath)

Lunedi' 12 Ottobre 2020 ORE 14:00

Dipartimento di Matematica e Fisica Università degli Studi Roma Tre Largo San Leonardo Murialdo,1 - Pal.C - Aula 211

Abstract We consider a natural random growth process with competition on Z^d called first passage percolation in a hostile environment (FPPHE), in which two processes compete for the occupancy of sites. The first starts at the origin and spreads through the edges of Z^d at rate 1, while the second is initially dormant in seeds, which are distributed as a product of Bernoulli random variables of parameter p on Z^d, and grows from a seed at rate lambda>0 once either process attempts to occupy that seed. Depending on the value of p and lamba, the model observes a rich behaviour with multiple phase transitions. We prove that there is a coexistence phase in which both processes occupy a positive density of sites, which is a stark contrast to the behaviour of other competition models on Z^d.

In studying FPPHE, we introduce a novel approach to multi-scale analysis that can be used to analyse processes with non-equilibrium dynamics. Indeed, FPPHE lacks both equilibrium dynamics and monotonicity (adding a seed could benefit the first process), preventing the use of a more standard multi-scale approach. In this talk, we outline our main results for FPPHE and provide an overview for how this new multi-scale approach can be applied.