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. 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.
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