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arXiv · 1507.08652

Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices

Justine Louis
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Abstract

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 $\mathbb{Z}^2$ with a triangular boundary.

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Justine Louis. 2015-08-13. Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices. https://arxiv.org/abs/1507.08652

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