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Given a flat torus, we consider certain discrete graph approximations of
it and determine the asymptotics of the number of spanning trees
("complexity") of these graphs as the mesh gets finer. The constants in the
asymptotics involve various notions of determinants such as the
determinant of the Laplacian ("height") of the torus. The analogy between
the complexity of graphs and the height of manifolds was previously
commented on by Sarnak and Kenyon. In dimension two, similar asymptotics
were established earlier by Barber and Duplantier-David in the context of
statistical physics.
Our proofs rely on heat kernel analysis involving Bessel functions, which
in the torus case leads into modular forms and Epstein zeta functions. In
view of a folklore conjecture it also suggests that tori corresponding to
densest regular sphere packings should have approximating graphs with the
largest number of spanning trees, a desirable property in network theory.
Joint work with G. Chinta and J. Jorgenson.