Local convergence and stability of tight bridge-addable graph classes

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if $\mathcal{G}$ is bridge-addable and $G_n$ is a uniform $n$-vertex graph from $\mathcal{G}$, then $G_n$ is connected with probability at least $(1+o_n(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible since it is reached for the class of all forests. In this paper we prove a form of uniqueness in this statement: if $\mathcal{G}$ is a bridge-addable class and the random graph $G_n$ is connected with probability close to $e^{-1/2}$, then $G_n$ is asymptotically close to a uniform $n$-vertex random forest in some local sense. For example, if the probability converges to $e^{-1/2}$, then $G_n$ converges in the sense of Benjamini-Schramm to the uniform infinite random forest $F_\infty$. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the stable extremum is not a graph but a graph class.