The dimer model and dynamical incidence geometry

TL;DR

This paper introduces a geometric model related to the dimer model on bipartite graphs, leading to new dynamical incidence theorems with applications to integrable systems like pentagram maps.

Contribution

It establishes a geometric counterpart of the dimer model, defines coherent double circuit configurations, and develops a framework for dynamical incidence theorems in linear incidence geometry.

Findings

Model behaves consistently under local moves of the dimer model

Introduces new dynamical incidence theorems in geometry

Explores parametrization of configurations via spectral curves

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.