Combinatorics of the double-dimer model

Abstract

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to Donaldson-Thomas and Pandharipande-Thomas theory which will be the subject of a forthcoming paper. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.