Loop covering probabilities

Given a sequence of undirected connected graphs $G_n$ of $n$ vertices with bounded degrees and weights and an additional sequence of killing parameters $c_n$, one can define a sequence ${\mu }_n$ of natural non-normalized measures on non-trivial loops of $G_n$. Denote by $Q_n$ the transition matrix associated with $G_n$. Assuming that the empirical distributions ${\nu }_n$ of the eigenvalues of $Q_n$ converge, we determine the limit of the ${\mu }_n-$proportion of \{loops which cover every vertex\} as $n\to \mathrm{\infty }$. Let ${\mathcal{L}}_n$ be the Poissonian "loop soup" with intensity $\frac{{\mu }_n}{n}$. A a corollary, we determine the limit law of the number of the loops covering the whole graph. As two concrete examples, we consider the closed balls in a regular graph and the discrete tori.