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

**Speaker:** Alexandre Stauffer (Università Roma Tre and University of Bath)

**Date:** September 14, 2020 - at 3 p.m

**Google Meet:** https://meet.google.com/nxh-

**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. A crucial step in our analysis is the addition of some non-local events to the multi-scale framework, and interplaying the non-local events with a by now "standard" multi-scale renormalization.

Based on a joint work with Tom Finn (Univ. of Bath).

All the talks are held in English.