Poisson convergence for common edges in random spanning trees of general graphs

Determine general conditions on a large graph sequence under which the number of common edges in two independent uniformly random spanning trees approximately follows a Poisson distribution.

Background

The paper proves Poisson limit laws for the number of common edges in two independent uniformly random spanning trees of complete graphs, Erdős–Rényi random graphs with fixed edge probability, and complete multipartite graphs with fixed numbers of parts and proportionally growing part sizes. The authors then ask when analogous behavior holds for general graphs.

Poisson convergence cannot hold universally. For a tree, the two spanning trees coincide and therefore have all n−1 edges in common. For a complete bipartite graph K_{k,n−k} with fixed k, the common-edge count is approximately binomial and asymptotically normal. Graphs containing bridges or two dense components joined by a fixed number of disjoint edges also produce non-Poisson behavior. The unresolved problem is to characterize the graph-density or structural conditions that ensure Poisson behavior despite these obstructions.