Description
Melrose's geometric microlocal analysis involves the concept of blowing up submanifolds to resolve non-uniform operator kernel behaviour. This technique has been central to the analysis of singular spaces, but was previously disconnected from discrete geometry. We introduce a new framework that applies Melroses's blowup techniques to scaling limits of discrete surfaces to study the discrete heat kernel as the discretization mesh size goes to zero. This proves a full asymptotics of the determinant on discrete surfaces beyond the example of regular lattices that has been the best possible result so far.