Local Convergence and Stability of Tight Bridge-addable Classes

A class of graphs is bridge-addable if given a graph in the class, any graph obtained by adding an edge between two connected components of is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if is bridge-addable and is a uniform-vertex graph from, then is connected with probability at least. The constant 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 is a bridge-addable class and the random graph is connected with probability close to e^-1/2, then is asymptotically close to a uniform-vertex random forest in a local sense. For example, if the probability converges to, then converges in the sense of Benjamini-Schramm to the uniformly infinite random forest. This result is reminiscent of so-called stability results in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.