Components, large and small, are as they should be I: supercritical percolation on regular graphs of growing degree

TL;DR

The paper identifies mild global and local expansion conditions on a growing-degree $d$-regular graph $G$ that guarantee a phase transition in the percolated subgraph $G_p$ around $p=rac{1+ ext{ε}}{d}$, yielding a unique giant of size $(1+o(1))y( ext{ε})n$ and small components of size $O( ext{log} n)$. The authors develop a unified two-stage proof using a double-exposure (sprinkling) approach, showing large components become everywhere dense, merge into a single giant, and concentrate around the predicted size. They establish tightness by constructing graphs with near-optimal local expansion that fail to exhibit the full ERCP, and they present two variants covering high- and low-degree regimes. The results recover classical ER results for $G(n,p)$, the hypercube, pseudo-random graphs, and random $d$-regular graphs, while providing a cohesive framework for a broad class of regular graphs with growing degree. This work lays groundwork for a companion paper addressing constant-degree graphs and further tightness results.

Abstract

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.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• proof