Percolation through Isoperimetry

Abstract

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. We further show that this condition is tight. More precisely, let $d=ω(1)$, let $ε>0$ be a small enough constant, and let $p \cdot d=1+ε$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique linear sized component, whose order is asymptotically $y(ε)n$, where $y(ε)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+ε)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + ε$, then $G_p$ typically contains a component of order $Ω(k)$.

Figures (2)

• Figure 1: Illustration of the sets utilised in the proof of Theorem \ref{['th: weak expander']} and their properties. By Lemma \ref{['l: |W_L(G(2))|']}, whp$|W_L(G(2))|=n-o(n/d)$ (appears in dark blue). $V_L(G(2))$ appears in purple, together with a partition of it into $A$ (red-dotted) and $B$ (green-dotted). This extends to $A'$ (red-dotted) and $B'$ (green-dotted), which cover $W_L(G(2))$. In the figure, the edge $xy$ between $A'$ and $B'$ appears in solid black. $X$, the neighbourhood of $x$ in $A$, appears in purple, and its size is lower-bounded by construction. Similarly, $Y$, the neighbourhood of $y$ in $B$, appears in purple with its size also lower-bounded by construction. When sprinkling with probability $p_3$, with very high probability there will be many black edges between $A'$ and $B'$ in $G_{p_3}$, which we can then extend to paths of length three between $A$ and $B$.
• Figure 2: An illustration of the \ref{['first construction']}

Theorems & Definitions (32)

• Theorem 1.1: ER60
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Theorem \ref{['th: general statement']}