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

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 $Ω$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,λ)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λV^{1/3}$, where $λ$ 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 $|p-p_c|=Θ(\cn^{-1}V^{-1/3})$ is of size $Θ(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(ε^{-2}\log(Vε^3))$, with $ε=\cn(p_c-p)$. 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 $Θ(\cn(p-p_c))$. 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 $n>6$.

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