  The Jacobian algebra arising from a consistent dimer model is a bimodule
$3$-Calabi-Yau algebra, and its center is a $3$-dimensional Gorenstein toric
singularity. A perfect matching of a dimer model gives the degree making the
Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part
of such an algebra is finite dimensional, then it is a $2$-representation
infinite algebra which is a generalization of a representation infinite
hereditary algebra. Internal perfect matchings, which correspond to toric
exceptional divisors on a crepant resolution of a $3$-dimensional Gorenstein
toric singularity, characterize the property that the degree zero part of the
Jacobian algebra is finite dimensional. Combining this characterization with
the theorems due to Amiot-Iyama-Reiten, we show that the stable category of
graded maximal Cohen-Macaulay modules admits a tilting object for any
$3$-dimensional Gorenstein toric isolated singularity. We then show that all
internal perfect matchings corresponding to the same toric exceptional divisor
are transformed into each other using the mutations of perfect matchings, and
this induces derived equivalences of $2$-representation infinite algebras.
