2015/07/30 by Justine Louis, Louis, Justine
Mathematics · #05C05 (Primary) #05C30 #11M99 #58J52 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math.CO #msc:05C05 #msc:05C30 #msc:11M99 #msc:58J52
paper · pdf · doi:10.48550/arxiv.1507.08652
20 pages, 3 figures
openalex publication_date 2015/07/30 · arxiv created 2015/08/13 · arxiv updated 2015/08/14 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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 ℤ2 with a triangular boundary.