  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.
