induced by a probability distribution on the vertices of a fixed infinite graph. We say the [**random weak limit**]{} of $\Seq{\gh_n}$ is $\rtd$ if for any positive integer $R$ and any finite graph $H$, we have $\lim_{n \to\infty} p\big(R, H, U(\gh_n)\big) = p(R, H, \rtd)$. More generally, if $\gh_n$ are random finite graphs, then we say the [**random weak limit**]{} of $\Seq{\gh_n}$ is $\rtd$ if for any positive integer $R$, any finite graph $H$, and any $\epsilon > 0$, we have $\lim_{n \to\infty} \Pleft{\big|p\big(R, H, U(\gh_n)\big) - p(R, H, \rtd)\big| >
\epsilon} = 0$. Note that only the component of the root matters for convergence to $\rtd$. Thus, we may and shall assume that $\rtd$ is concentrated on connected graphs. If $\rtd$ is induced by a distribution on the vertices of a fixed transitive graph $\gh$, then the random weak limit depends only on $\gh$ and not on the root. In this case, we say that the [**random weak limit**]{} of $\Seq{\gh_n}$ is $\gh$.

Given $R > 0$ and a finite graph $\gh$, let $\nu_R(\gh)$ be the distribution of the number of edges in the ball of radius $R$ about a random vertex of $\gh$. Call a collection of finite 