The Asymptotic Number of Spanning Trees in $d$-Dimensional Square Lattices

Abstract

We show that for every $d\ge 2$, the number of spanning trees of a $d$-dimensional grid with $N$ vertices grows like $C(d{)}^N$ for some constant $C(d)$. Moreover, we show that $C(d)=2d-\frac{1}{2}-\frac{5}{16d}+O(d^{-2})$ as $d$ goes to infinity.