Complexity and height of tori

I will explain detailed asymptotics for the number of spanning trees, called complexity, of discrete tori as they grow. One constant that appears here is the height of an associated limiting real torus. The height is the derivative at 0 of the spectral zeta function and its minimum conjecturally occurs for lattices giving optimal regular sphere packings. This type of asymptotics has been studied particularly in the statistical physics literature. Our lead term was already found in Sokal-Starinets (2001) and we extend the 2-dimensional expansion of Barber-Duplantier-David (1970s) to all dimensions. Heat kernel analysis is used in the proof. Joint work with G. Chinta and J. Jorgenson.