2020/09/11 by Finn, Thomas, Stauffer, Alexandre
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2009.05463
The main contribution of this paper is the development of a novel approach to multi-scale analysis that we believe can be used to analyse processes with non-equilibrium dynamics. Our approach will be referred to as multi-scale analysis with non-equilibrium feedback and will be used to analyse 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 FPPλ that compete for the occupancy of sites. Initially, FPP1 occupies the origin and spreads through the edges of ℤd at rate 1, while FPPλ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p∈(0,1), where a seed remains dormant until FPP1 or FPPλ attempts to occupy it before then spreading through the edges of ℤd at rate λ>0. Particularly challenging aspects of FPPHE are its non-equilibrium dynamics and its lack of monotonicity (for instance, adding seeds could be benefitial to FPP1 instead of FPPλ); such characteristics, for example, prevent the application of a more standard multi-scale analysis. As a consequence of our main result for FPPHE, we establish a coexistence phase for the model for d≥3, answering an open question in \citesidoravicius2019multi. This exhibits a rare situation where a natural random competition model on ℤd observes coexistence for processes with different speeds. Moreover, we are able to establish the stronger result that FPP1 and FPPλ can both occupy a positive density of sites with positive probability, which is in stark contrast with other competition processes.