Union of Random Trees and Applications

In 1986, Janson showed that the number of edges in the union of \(k\) random spanning trees in the complete graph \(K_n\) is a shifted Poisson distribution. Using results from the theory of electrical networks, we provide a new proof of this result, and we obtain an explicit rate of convergence. This rate of convergence allows us to show a new upper tail bound on the number of trees in \(G(n,p)\), for \(p\) a constant not depending on \(n\). The number of edges in the union of \(k\) random trees is related to moments of the number of spanning 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 \(\mathbb{N}\) and where each edge appears independently with constant probability \(p\). 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 a scaled version of the number of spanning trees satisfies the law of the iterated logarithm.