  A dimer model is a bipartite graph described on the real two-torus, and it
gives the quiver as the dual graph. It is known that for any three-dimensional
Gorenstein toric singularity, there exists a dimer model such that a GIT
quotient parametrizing stable representations of the associated quiver is a
projective crepant resolution of this singularity for some stability parameter.
It is also known that the space of stability parameters has the
wall-and-chamber structure, and for any projective crepant resolution of a
three-dimensional Gorenstein toric singularity can be realized as the GIT
quotient associated to a stability parameter contained in some chamber. In this
paper, we consider dimer models giving rise to projective crepant resolutions
of a toric compound Du Val singularity. We show that sequences of zigzag paths,
which are special paths on a dimer model, determine the wall-and-chamber
structure of the space of stability parameters. Moreover, we can track the
variations of stable representations under wall-crossing using the sequences of
zigzag paths.
