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 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. In this paper, we show that 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. Moreover,
combining this result 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.Comment: 28 pages, v2: improved some proof