Abstract
The main contribution of this paper is the development of a novel approach to multiscale analysis that we believe can be used to analyse processes with nonequilibrium dynamics. Our approach will be referred to as \emph{multiscale analysis with nonequilibrium feedback} and will be used to analyse a natural random growth process with competition on $\mathbb{Z}^d$ called \emph{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 $\mathbb{Z}^d$ at rate 1, while $FPP_{\lambda}$ is initialised at sites called \emph{seeds} that are distributed according to a product of Bernoulli measures of parameter $p\in(0,1)$, where a seed remains dormant until $FPP_1$ or $FPP_{\lambda}$ attempts to occupy it before then spreading through the edges of $\mathbb{Z}^d$ at rate $\lambda>0$. Particularly challenging aspects of FPPHE are its nonequilibrium dynamics and its lack of monotonicity (for instance, adding seeds could be benefitial to $FPP_1$ instead of $FPP_\lambda$); such characteristics, for example, prevent the application of a more standard multiscale analysis. As a consequence of our main result for FPPHE, we establish a coexistence phase for the model for $d\geq3$, answering an open question in \cite{sidoravicius2019multi}. This exhibits a rare situation where a natural random competition model on $\mathbb{Z}^d$ observes coexistence for processes with \emph{different} speeds. Moreover, we are able to establish the stronger result that $FPP_1$ and $FPP_{\lambda}$ can both occupy a \emph{positive density} of sites with positive probability, which is in stark contrast with other competition processes.
Original language  English 

Journal  Journal of the European Mathematical Society 
Publication status  Acceptance date  3 Jul 2022 
Keywords
 math.PR
 mathph
 math.MP
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Alexandre Stauffer
Person: Research & Teaching