Infinite and Giant Components in the Layers Percolation Model

TL;DR

This paper investigates the structural properties of the Layers percolation model, proving the existence of infinite components in certain infinite graphs and conditions for giant components in random and lattice graphs.

Contribution

It establishes conditions under which the Layers model produces infinite or giant components in various graph classes, extending previous understanding of dependent percolation models.

Findings

Infinite components in infinite trees with sub-exponential degree growth.

Finite components in $T_2(G)$ for any locally finite graph.

Giant component existence in bounded degree random graphs with degree ≥ 3.

Abstract

In this work we continue the investigation launched in [FHR16] of the structural properties of the structural properties of the 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 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 ${\mathbb{Z}}^{d}$ and show that if $d$ is sufficiently large, then $\mathrm{a.s.}$ $T_4(\mathbb{Z}^d)$ contains an infinite cluster.