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.