Union of Random Trees and Applications

In 1986, Janson showed that the number of edges in the union of $k$ random trees in the complete graph $K_n$ is a shifted version of a Poisson distribution. Using results from the theory of electrical networks, we provide a new proof of this result, obtaining an explicit rate of convergence. This rate of convergence allows us to show a new upper tail bound on the number of trees in $G(n,p)$. As an application, we prove the law of the iterated logarithm for the number of spanning trees in $G(n,p)$. More precisely, consider the infinite random graph $G(\mathbb{N}, p)$, with vertex set $\mathbb{N}$ where each edge appears with probability $p$, a constant. By restricting to $\{1, 2, \dotsc, n\}$, we obtain a series of nested Erd\"{o}s-R\'{e}yni random graphs $G(n,p)$. We show that $X_n$, a scaled version of the number of spanning trees, satisfies the law of the iterated logarithm.