论文标题
非平衡多尺度分析和在竞争的第一通道渗透方面的共存
Non-equilibrium multi-scale analysis and coexistence in competing first passage percolation
论文作者
论文摘要
本文的主要贡献是开发了一种新的多尺度分析方法，我们认为可以用来分析具有非平衡动力学的过程。我们的方法将被称为\ emph {多尺度分析，具有非平衡反馈}，并将用于分析自然的随机增长过程，并在$ \ mathbb {z}^d $上进行竞争，称为\ emph {emph {first passage {first vassage percolation in to ant taptage Environment}，在敌对的环境中均由两次通用pppece $ fpppece $ fppe $ fppp y ppp_ 1站点。最初，$ fpp_1 $占据起源，并通过$ \ mathbb {z}^d $以1的价格传播，而$fpp_λ$是在称为\ emph {seeds}的网站初始化的，这些网站是根据$ p fpp的$ p \ in（$ fpp）的$ p \ in（$ fpp）的$ fpp f ppp y dormant dormant dormant dormant dormant dormant $ formant $ formant $ formant $ formant $ formant dormant domp doMe的产物在此之前占用它，然后通过$ \ Mathbb {z}^d $的边缘散布，价格$λ> 0 $。 FPPHE特别具有挑战性的方面是其非平衡动力学和缺乏单调性（例如，添加种子可能对$ fpp_1 $而不是$fpp_λ$）有益）；例如，这样的特征阻止了应用更标准的多尺度分析。由于我们对FPPHE的主要结果，我们为$ d \ geq3 $建立了模型的共存阶段，在\ cite {sidoravicius2019multi}中回答了一个空旷的问题。这表现出一种罕见的情况，在$ \ mathbb {z}^d $上的自然随机竞争模型观察\ emph {不同}速度的过程共存。此外，我们能够确定$ fpp_1 $和$fpp_λ$都可以占据积极概率的站点的\ emph {正密度}，这与其他竞争过程形成鲜明对比。
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.
