The Kashaev Equation and Related Recurrences

TL;DR

This paper explores the Kashaev equation and the hexahedron recurrence, revealing their connections to principal minors of symmetric matrices, cluster algebras, and models in statistical mechanics and the Ising model.

Contribution

It characterizes solutions of the Kashaev equation derived from the hexahedron recurrence and introduces related recurrences with similar properties.

Findings

Characterization of Kashaev equation solutions from the hexahedron recurrence

New insights into principal minors of symmetric matrices

Identification of recurrences related to cluster algebras and s-holomorphicity

Abstract

The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.