Weighted graphs, tetrahedron equation and loop quantum gravity

Edge-weighted graphs are a combinatorial object parametrizing several archetypal models of statistical physics: the dimer model, the Ising model and the model of electrical networks. Each of these models describes some combinatorial problem, is associated with some solution of the tetrahedron equation, is associated with one of the versions of the completely positive Grassmannian. I will talk about these phenomena and how these problems are related to loop quantum gravity, namely spin foam evaluations.