Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree
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2024
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| author | Diskin, Sahar Krivelevich, Michael |
| author_facet | Diskin, Sahar Krivelevich, Michael |
| contents | We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied graphs, such as (percolation on) the complete graph $K_n$, the binary hypercube $Q^d$, $d$-regular expanders, and random $d$-regular graphs. In particular, this serves as a unified proof for these (and other) cases.
Suppose that $G$ is a $d$-regular graph on $n$ vertices, with $d=ω(1)$. Let $ε>0$ be a small constant, and let $p=\frac{1+ε}{d}$. Let $y(ε)$ be the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. We show that if $G$ satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in $G$, then typically the percolated random subgraph $G_p$ contains a unique giant component of asymptotic order $y(ε)n$, and all the other components in $G_p$ are of order $O(\log n/ε^2)$.
We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in $G$, then there are $d$-regular graphs $G$ on $n$ vertices, where typically the second largest component is of order $Ω(d\log (n/d))=ω(\log n)$.
This is the first of a two-part sequence of papers. In the subsequent work, we consider supercritical percolation on regular graphs of constant degree, and establish similar sufficient (and essentially tight) conditions in that setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree Diskin, Sahar Krivelevich, Michael Combinatorics Probability We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied graphs, such as (percolation on) the complete graph $K_n$, the binary hypercube $Q^d$, $d$-regular expanders, and random $d$-regular graphs. In particular, this serves as a unified proof for these (and other) cases. Suppose that $G$ is a $d$-regular graph on $n$ vertices, with $d=ω(1)$. Let $ε>0$ be a small constant, and let $p=\frac{1+ε}{d}$. Let $y(ε)$ be the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. We show that if $G$ satisfies a (very) mild edge expansion requirement, and if one has fairly good control on the expansion of small sets in $G$, then typically the percolated random subgraph $G_p$ contains a unique giant component of asymptotic order $y(ε)n$, and all the other components in $G_p$ are of order $O(\log n/ε^2)$. We also show that this result is tight, in the sense that if one asks for a slightly weaker control on the expansion of small sets in $G$, then there are $d$-regular graphs $G$ on $n$ vertices, where typically the second largest component is of order $Ω(d\log (n/d))=ω(\log n)$. This is the first of a two-part sequence of papers. In the subsequent work, we consider supercritical percolation on regular graphs of constant degree, and establish similar sufficient (and essentially tight) conditions in that setting. |
| title | Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2408.04597 |