$H$-percolation with a random $H$

TL;DR

The paper resolves BBM12'sProblem 1 for random base graphs by showing that for ${\mathcal G}_{k,\alpha}$ with $0<\alpha\le 1$, the percolation threshold exponent satisfies $\ell({\mathcal G}_{k,\alpha})=1/\lambda({\mathcal G}_{k,\alpha})$ with high probability as $k\to\infty$. In particular, when $\alpha=1/2$, the critical probability obeys $p_c(n,{\mathcal G}_{k,1/2})=n^{-1/\lambda({\mathcal G}_{k,1/2})+o(1)}$ with exponentially small failure probability, and almost surely for all large $k$ by Borel--Cantelli, under the condition $\alpha \ge A(\log k)/k$. The authors show that $G_{k,\alpha}$ are (strictly) balanced in this regime, enabling the use of existing bounds from BBM12 and BK23 to derive the threshold behavior; the results extend to $k$ growing with $n$ and provide insight into the role of the edge-vertex ratio $\lambda(H)$ in governing $H$-percolation dynamics. This establishes a stable, general mechanism for critical behavior when the target graph $H$ is random, with implications for understanding tipping points and network saturation in random-graph growth models.

Abstract

In $H$-percolation, we start with an Erdős--Rényi graph ${\mathcal G}_{n,p}$ and then iteratively add edges that complete copies of $H$. The process percolates if all edges missing from ${\mathcal G}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal G}_{k,1/2}$ is uniformly random, solving a problem of Balogh, Bollobás and Morris.

Figures (1)

• Figure 1: Suppose that edges in a "base" graph $B$ have already been added. Next, we add an edge $\{u,v\}$ using a copy of $H$. The "price" is $e_H-e_F-1$ edges and $v_H-v_F$ vertices, where $F=H\cap B$. Hence the edge per vertex "cost" is at least $\lambda_*$.

Theorems & Definitions (3)

• Theorem 1
• Proposition 2
• proof