Infinite and Giant Components in the Layers Percolation Model

In this work we continue the investigation lunched in [4] 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 Tk(G) denote the random vertexinduced subgraph of G, generated by ordering V according to Uniform[0, 1] i.i.d. clocks and including in Tk(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 l-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, T3(T ) a.s. has an infinite connected component. In contrast, we show that for any locally finite graph G, a.s. every connected component of T2(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 w.h.p. T3 has a giant component. Finally, we also consider Zd and show that if d is sufficiently large, then a.s. T4(Z d) contains an infinite cluster.