  The combinatorial mutation of polygons, which transforms a given lattice
polygon into another one, is an important operation to understand mirror
partners for two-dimensional Fano manifolds, and the mutation-equivalent
polygons give ${\mathbb Q}$-Gorenstein deformation-equivalent toric varieties.
On the other hand, for a dimer model, which is a bipartite graph described on
the real two-torus, one can assign a lattice polygon called the perfect
matching polygon. It is known that for each lattice polygon $P$ there exists a
dimer model having $P$ as the perfect matching polygon and satisfying certain
consistency conditions. Moreover, a dimer model has rich information regarding
toric geometry associated with the perfect matching polygon. In this paper, we
introduce a set of operations which we call deformations of consistent dimer
models, and show that the deformations of consistent dimer models realize the
combinatorial mutations of the associated perfect matching polygons.
