  We explore various aspects of the correspondence between dimer models and
integrable systems recently introduced by Goncharov and Kenyon. Dimer models
give rise to relativistic integrable systems that match those arising from 5d
N=1 gauge theories studied by Nekrasov. We apply the correspondence to dimer
models associated to the Y^{p,0} geometries, showing that they give rise to the
relativistic generalization of the periodic Toda chain originally studied by
Ruijsenaars. The correspondence reduces the calculation of all conserved
charges to a straightforward combinatorial problem of enumerating
non-intersecting paths in the dimer model. We show how the usual periodic Toda
chain emerges in the non-relativistic limit and how the Lax operator
corresponds to the Kasteleyn matrix of the dimer model. We discuss how the
dimer models for general Y^{p,q} manifolds give rise to other relativistic
integrable systems, generalizing the periodic Toda chain and construct the
integrable systems for general Y^{p,p} explicitly. The impurities introduced in
the construction of Y^{p,q} quivers are identified with impurities in twisted
sl(2) XXZ spin chains. Finally we discuss how the physical concept of higgsing
a dimer model provides an efficient method for producing new integrable systems
starting from known ones. We illustrate this idea by constructing the
integrable systems for higgsings of Y^{4,0}.
