Alexandre Stauffer

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

We consider a natural random growth process with competition on ℤᵈ called first−passage percolation in a hostile environment, that consists of two first−passage percolation processes FPP₁ and FPP_λ that compete for the occupancy of sites. Initially FPP₁ occupies the origin and spreads through the edges of ℤᵈ at rate 1, while FPP_λ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP₁ or FPP_λ attempts to occupy it, after which it spreads through the edges of ℤᵈ at rate λ. 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 ℤᵈ. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_λ could favor FPP₁. 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 a joint work with Tom Finn (Univ. of Bath).