Infinite and Giant Components in the Layers Percolation Model

21 Mar 2018

In this work we continue the investigation launched in \cite{feige2013layers} of the structural properties of the structural properties of the \emph{Layers model}, a dependent percolation model. Given an undirected graph $G=(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex-induced subgraph of $G$, generated by ordering $V$ according to Uniform$[0,1]$ $\mathrm{i.i.d.}$ clocks and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors having a faster clock. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} $k$. The layers model has found applications in the study of $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem and in the study of bootstrap percolation. We prove that every infinite locally finite tree $T$ with no leaves, satisfying that the degree of the vertices grow sub-exponentially in their distance from the root, $T_3(T)$ $\mathrm{a.s.}$ has an infinite connected component. In contrast, we show that for any locally finite graph $G$, $\mathrm{a.s.}$ every connected component of $T_2(G)$ is finite. We also consider random graphs with a given degree sequence and show that if the minimal degree is at least 3 and the maximal degree is bounded, then $\mathrm{w.h.p.}$ $T_3$ has a giant component. Finally, we also consider ${\Z}^{d}$ and show that if $d$ is sufficiently large, then $\mathrm{a.s.}$ $T_4(\Z^d)$ contains an infinite cluster.