5 November 2012
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.
- Geometry and Analysis Seminar