The Asymptotic Number of Spanning Trees in \(d\)-Dimensional Square Lattices

Abstract

We show that for every \(d \geq 2\), 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.