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In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of d -dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the number of spanning trees by the well-known matrix tree theorem. Asymptotics for 2 and 3 component rooted spanning forests in these graphs are also derived. Moreover, we express the number of spanning trees in a 2-dimensional square lattice in terms of the one in a 2-dimensional discrete torus and also in the quartered Aztec diamond. As a consequence, we find an asymptotic expansion of the number of spanning trees in a subgraph of Z 2 with a triangular boundary. [ABSTRACT FROM AUTHOR]

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators. [ABSTRACT FROM PUBLISHER]