the nearest-neighbor graph on $\Z^2$. In this case, one has that $$\lim_{n \to\infty} |\verts(\gh_n)|^{-1} \log \cp(\gh_n) 
=
\int_0^1 \int_0^1 \log (4 - 2 \cos 2\pi x - 2 \cos 2\pi y) \,dx \,dy
=
4\cat/\pi
\,,$$ where $\cat := \sum_{k=0}^\infty (-1)^k / (2k+1)^2 = 0.9160^- $ is [**Catalan’s constant**]{}. See, e.g., \[b\].BurPem/ and \[b\].ShrockWu/ and the references therein for this and several other such examples. We shall reformulate the Matrix-Tree Theorem as an infinite series whose terms have a probabilistic meaning (see \[p\].finite/). In fact, the terms of the series are “local" quantities. This will permit us to find the asymptotic complexity in a very general setting (see \[t\].asym/). Other than assuming a bound on the average degree of the finite graphs of whose complexity we find the asymptotics, our results are essentially the most general possible. In the simplest case, where the finite graphs $\gh_n$ tend to a fixed transitive infinite graph $\gh$, we shall prove that $$\lim_{n \to\infty} |\verts(\gh_n)|^{-1} \log \cp(\gh_n)
=
\asym(\gh)
:=
\log \deg_\gh(o) - \sum_{k \ge 1} p_k(o;\gh)/k
\,,$$ where $o$ is a fixed vertex of $\gh$ and $p_k(o;\gh)$ is the probability that simple random walk started at $o$ on $\gh$ is again at $o$ 