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- W4300815383 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." @default.
- W4300815383 created "2022-10-04" @default.
- W4300815383 creator A5047575690 @default.
- W4300815383 date "2016-11-05" @default.
- W4300815383 modified "2023-09-25" @default.
- W4300815383 title "Infinite and Giant Components in the Layers Percolation Model" @default.
- W4300815383 doi "https://doi.org/10.48550/arxiv.1611.01693" @default.
- W4300815383 hasPublicationYear "2016" @default.
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