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Supercritical Site Percolation on Regular Graphs

Sahar Diskin, Michael Krivelevich, Itay Markbreit

Abstract

We consider site (vertex) percolation on $d$-regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erdős-Rényi component phenomenon" in the supercritical regime $p=\frac{1+ε}{d-1}$: namely, the appearance of a unique giant component of order $n/d$ in the percolated subgraph, with all other components being of size $O(\log n)$. Our main results apply both to the $d$-dimensional hypercube and to pseudo-random graphs, and resolve two open questions in these cases. We further discuss differences (and similarities) between bond (edge) percolation setting and site percolation setting.

Supercritical Site Percolation on Regular Graphs

Abstract

We consider site (vertex) percolation on -regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erdős-Rényi component phenomenon" in the supercritical regime : namely, the appearance of a unique giant component of order in the percolated subgraph, with all other components being of size . Our main results apply both to the -dimensional hypercube and to pseudo-random graphs, and resolve two open questions in these cases. We further discuss differences (and similarities) between bond (edge) percolation setting and site percolation setting.
Paper Structure (16 sections, 21 theorems, 63 equations)

This paper contains 16 sections, 21 theorems, 63 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. A modified Breadth First Search process
  4. Auxiliary Lemmas
  5. Some General Lemmas on the Percolated Subgraph
  6. Proof of Theorem \ref{['general']}
  7. Components are Big or Small
  8. Big components merge after sprinkling
  9. Sprinkling does not create new large components
  10. Modification of the proof of Theorem \ref{['general']} for $0<\alpha<1$
  11. Proof of Theorem \ref{['n,d,lambda']}
  12. Proof of Theorem \ref{['constant']}
  13. Proof of Theorem \ref{['no_second']}
  14. The Construction
  15. The absence of ERCP
  16. ...and 1 more sections

Key Result

Theorem 1

Let $c_1,c_2>0$ and let $0<\alpha\le 1$ be constants. Let $\epsilon>0$ be a sufficiently small constant. Then, there exist $c_3\coloneqq (c_2,\epsilon,\alpha)>0$ and $C\coloneqq C(\alpha)>0$ constants such that the following holds. Let $n$ be a sufficiently large integer, and let $d$ be an integer s Set $p=\frac{1+\epsilon}{d}$. Then, whp$G[V_p]$ contains a unique giant component $L_1$, satisfying

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Claim 2.1
  • Claim 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 37 more