Spectral curves for dimers on a torus

I will review Kasteleyn’s method for planar graphs, its adaptation to graphs on a torus, the notion of slope of a tiling, and the spectral curve of a graph on a torus. I’ll explain how the spectral curve controls the asymptotic behavior of periodic tilings as the period domain expands. I’ll then state Kenyon and Okounkov’s theorem that the spectral curve is always a Harnack curve, and that all Harnack curves are spectral curves.

Date published: Wednesday, October 19, 2022