On the intersection of pairs of trees

We consider the number of common edges in two independent random spanning trees of a graph $G$. For complete graphs $K_n$, we give a new proof of the fact, originally obtained by Moon, that the distribution converges to a Poisson distribution with expected value $2$. This is applied to show a Poisson limit law for the number of common edges in two independent random spanning trees of an Erdős--Rényi random graph $G(n,p)$ for constant~$p$, as well as a central limit theorem in the case where $p\to 0$ and $p\geq n^{-2/3+\gep}$. We also use the same method to prove an analogous result for complete multipartite graphs.