random graph

Sharp results of the above type become infeasible when the graph is random. The Erdős-Rényi random graph is the result of performing bond percolation on the complete graph, and is a toy model of a graph with a random geometry. Let $\mathrm{ER}_n(p)$ denotes the resulting random graph on $n$ vertices with percolation parameter $p=f(n)/n$ for some $f(n)$ satisfying $\lim_{n\to\infty} f(n)
=\infty$, the so-called *dense* Erdős-Rényi random graph. Then, as shown in the appendix, metastable behaviour occurs for any $h,J>0$, and $$\lim_{n\to\infty} \frac{\Gamma_n^\star}{\tfrac14 Jnf(n)} = 1 \quad
\text{ in distribution under the law of }  \mathrm{ER}_n(f(n)/n),
\label{eq:er1}$$ which is accurate up to leading order. The computation of ${\mathcal{C}}_n^\star$ and $K_n^\star$, however, is a *formidable task*. The reason for this is that, while allows for a small error in the energy, the set ${\mathcal{C}}_n^\star$ is made up of configurations that have *exactly* the critical energy $\Gamma_n^\star$.

When $f(n)=\lambda$ for some constant $\lambda>1$, the *sparse case*, an analysis similar to the one carried out in this paper can be used to obtain lower and upper bounds on the communication height. However, we have been unable to prove a convergence of the form in .

Configuration Model {#configurationmodel}
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In this section 