The dimer model and dynamical incidence geometry

Abstract

We propose a geometric counterpart of the dimer model on bipartite graphs. A state of our model consists of a choice of a point for each white vertex and hyperplane for each black vertex. This data is subject to certain conditions determined by the graph; the resulting configurations are called coherent double circuit configurations. We show that our model behaves consistently under standard local moves of the dimer model. On the geometric side, this gives rise to a new class of theorems in linear incidence geometry - dynamical incidence theorems. Examples include results on pentagram maps, pentagram spirals, and Q-nets. We also examine the problem of parametrizing coherent double circuit configurations. In particular, we study whether, once the white-vertex part of the data is fixed, one can recover the black-vertex data from a point on the spectral curve.