Random subgraphs of finite graphs: I. The scaling window under the triangle condition

TL;DR

This paper investigates the phase transition and scaling window in percolation on finite transitive graphs, establishing conditions under which the largest cluster size exhibits mean-field behavior similar to classical random graphs.

Contribution

It introduces a general framework for analyzing the scaling window and phase transition in percolation on finite graphs using the triangle condition, extending known results to new models.

Findings

Largest cluster size in the scaling window is of order V^{2/3}

Below the window, the largest cluster size is logarithmic in V

Above the window, the expected largest cluster size is proportional to V(p-p_c)

Abstract

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,\lambda)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $\lambda V^{1/3}$, where $\lambda$ is fixed and positive. We show that for any such model, there is a phase transition at $p_c$ analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold $p_c$. In particular, we show that the largest cluster inside a scaling window of size $|p-p_c|=\Theta(\cn^{-1}V^{-1/3})$ is of size $\Theta(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(\epsilon^{-2}\log(V\epsilon^3))$, with $\epsilon=\cn(p_c-p)$. We also obtain an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order $\Theta(\cn(p-p_c))$. Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the $n$-cube and certain Hamming cubes, as well as the spread-out $n$-dimensional torus for $n>6$.