arXiv 2009.054632020-09math.PRmath-phmath.MP

竞争首段渗流的非平衡多尺度分析与共存

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

Thomas Finn, Alexandre Stauffer

备注：: 48 pages, revised version with minor corrections

摘要：

本文的主要贡献是开发了一种新的多尺度分析方法，我们相信它可以用于分析非平衡动力学过程。我们的方法将被称为\emph{multi-scale analysis with non-equilibrium feedback}，并将被用来分析一个自然的随机增长过程与竞争的$\mathbb{Z}^d$称为\emph{first passage percolation in a hostile environment}，由两个第一次通过渗滤过程$FPP_1$和$FPP_λ$竞争的网站的占用。最初，$FPP_1$占据原点并以速率1通过$\mathbb{Z}^d$的边缘传播，而$FPP_λ$在称为\emph{seeds}的站点处初始化，这些站点根据参数$p\in(0,1)$的伯努利测度的乘积分布，其中种子保持休眠，直到$FPP_1$或$FPP_λ$尝试占据它，然后以速率$λ>0$通过$\mathbb{Z}^d$的边缘传播。FPPHE特别具有挑战性的方面是其非平衡动力学和缺乏单调性（例如，添加种子可能有利于$FPP_1$而不是$FPP_λ$）;例如，这些特性阻止了更标准的多尺度分析的应用。由于我们的主要结果FPPHE，我们建立了一个共存阶段的模型$d\geq3$，回答了一个悬而未决的问题，在\cite{sidoravicius2019multi}。这表现出一种罕见的情况下，自然的随机竞争模型$\mathbb{Z}^d$观察共存的进程与\emph{different}的速度。此外，我们能够建立更强的结果，$FPP_1$和$FPP_λ$都可以占据\emph{positive density}的网站与积极的概率，这是在鲜明的对比，与其他竞争过程。

英文摘要：

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 \emph{multi-scale analysis with non-equilibrium 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 $FPP_λ$ 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 $FPP_λ$ 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 $FPP_λ$ attempts to occupy it before then spreading through the edges of $\mathbb{Z}^d$ 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 $FPP_1$ instead of $FPP_λ$); 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 $d\geq3$, 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 $FPP_λ$ can both occupy a \emph{positive density} of sites with positive probability, which is in stark contrast with other competition processes.

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