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.