Crosses-and-wrenches graphs have been introduced by David Speyer to explain the Laurent phenomenon (with positive coefficients) for the octahedron recurrence. Informally, these are the graphs obtained by performing spider moves and fusion of degree-2 vertices on an Aztec diamond, and contain generalized tower graphs and pinecone graphs.
In a joint work with Bishal Deb (Tsinghua University) and Béatrice de Tilière (Dauphine), we consider the dimer model on such a graph using Kasteleyn theory, under the assumption that the weights on edges are given by Fock's weights, expressed in terms of theta functions on a given Riemann surface. In this context, we provide a double integral formula over that surface for the inverse of the Kasteleyn matrix.
Using this formula, we study local statistics, and the limit shape phenomenon for a growing family of Speyer graphs. We prove in particular a holographic principle, noticed and proved by di Francesco and Vu in a few specific cases: once parameters of Fock's weights are fixed, for a well-positioned observer, the limit shape of any growing family of Speyer graphs look the same.