论文标题
超出关键时间的2D森林射击过程
A 2D forest fire process beyond the critical time
论文作者
论文摘要
我们在二维中研究森林火过程。在给定的平面晶格上，顶点以$ 1 $（最初都是空置的）独立切换到占用的，并且任何连接的组件“被烧毁”（其顶点即时变空），一旦其心脏跨度越过（通常大于典型的大型）阈值$ n，该模型的参数，该阈值$ n（通常很大）。 我们的分析提供了一个详细的描述，即$ n \ to \ infty $，其过程接近关键时间$ t_c $（在没有火灾的情况下会出现无限集群）。特别是我们证明了一个有点违反直觉的结果：存在$δ> 0 $，因此，如果概率很高，则原点不会在$ t_c +δ$之前燃烧。这为Van Den Berg和Brouwer的开放问题4.1提供了负面答案[COMM。数学。 Phys。，2006]。从非正式的话来说，结果可以用消防道的出现来解释，消防道的总密度可以忽略不计（如$ n \ to \ infty $），但对于恢复，它仍然足够强大。我们希望这种行为也适用于经典的Drossel-Schwabl模型。 本文的很大一部分致力于分析间隔$ [T_C，T_C +δ] $的回收率。这些恢复确实具有“显微镜”效应，但事实证明，它们对宏观尺度（实际上在相关的“中镜”量表上）的综合影响消失了，因为$ n \ to \ infty $。 为了证明这一点，我们使用Kiss，Manolescu和Sidoravicius的关键思想[Ann。 Probab。，2015]，引入了适当的归纳论点，以扩展和加强其结果。然后，我们用它来证明，浓缩会导致我们早期与KISS的联合工作，同时在森林火灾过程中也可以实现较淡的渗透。正如我们所解释的那样，这里出现了重大困难，因为恢复破坏了冷冻渗透的良好空间马尔可夫特性。
We study forest fire processes in two dimensions. On a given planar lattice, vertices independently switch from vacant to occupied at rate $1$ (initially they are all vacant), and any connected component "is burnt" (its vertices become instantaneously vacant) as soon as its cardinality crosses a (typically large) threshold $N$, the parameter of the model. Our analysis provides a detailed description, as $N \to \infty$, of the process near and beyond the critical time $t_c$ (at which an infinite cluster would arise in the absence of fires). In particular we prove a somewhat counterintuitive result: there exists $δ> 0$ such that with high probability, the origin does not burn before time $t_c + δ$. This provides a negative answer to Open Problem 4.1 of van den Berg and Brouwer [Comm. Math. Phys., 2006]. Informally speaking, the result can be explained in terms of the emergence of fire lanes, whose total density is negligible (as $N \to \infty$), but which nevertheless are sufficiently robust with respect to recoveries. We expect that such a behavior also holds for the classical Drossel-Schwabl model. A large part of this paper is devoted to analyzing recoveries during the interval $[t_c, t_c + δ]$. These recoveries do have a "microscopic" effect, but it turns out that their combined influence on macroscopic scales (and in fact on relevant "mesoscopic" scales) vanishes as $N \to \infty$. In order to prove this, we use key ideas of Kiss, Manolescu and Sidoravicius [Ann. Probab., 2015], introducing a suitable induction argument to extend and strengthen their results. We then use it to prove that a deconcentration result in our earlier joint work with Kiss on volume-frozen percolation also holds for the forest fire process. As we explain, significant additional difficulties arise here, since recoveries destroy the nice spatial Markov property of frozen percolation.
