Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions

This note is motivated by results in arXiv:math/0608132 and arXiv:0806.2425 about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on tree…

Exponential growth of ponds in invasion percolation on regular trees

In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.

Limit theorems for 2D invasion percolation

We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence $({\mathbf{O}}(n))$ of outlet variables, the $n$th of which gives the number of outlets in the box centered at the origin of side length $2^n$. The most important of these properties describes the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers for $({\mathbf{O}}(n))$. We then show consequences of these limit theorems for the pon…

Relations between invasion percolation and critical percolation in two dimensions

We study invasion percolation in two dimensions. We compare connectivity properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any $k\geq1$, the $k$-point function of the first so-called pond has the same asymptotic behavior as the probability that $k$ points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint $p_c$-open clusters. Further, for $k>1$, …

Invasion percolation on the Poisson-weighted infinite tree

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $σ\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary"…

正规树上的入侵渗流

Invasion percolation on regular trees

我们考虑一个有根的正则树上的入侵渗流。对于从根侵入的无限簇，我们确定其$r$ -点函数的缩放行为的任何$r\geq2$和它的体积都在给定的高度和低于给定的高度。我们发现，虽然幂律的标度是相同的初始无限集群的普通渗流，标度函数不同。因此，有点令人惊讶的是，这两个集群的行为不同，事实上，我们证明，他们的法律是相互奇异的。此外，我们推导出简单的随机游动的规模估计的集群从根开始。我们表明，入侵渗流集团是随机主导的初始无限集团。远远高于根，这两个集群具有相同的法律局部，但不是全球。证明中的一个关键因素是沿着入侵渗流簇的主干的向前最大权重的分析。这些权重衰减到普通渗流的临界值，但只是缓慢的，这种缓慢的衰减导致标度行为不同于初始的无限集群。