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In in this paper we establish an explicit and sharp estimate of the spectralgap (Poincar\'{e} inequality) and the transportation inequality for Gibbsmeasures, under the Dobrushin uniqueness condition. Moreover, we give ageneralization of the Liggett's $M-\epsilon$ theorem for interacting particlesystems.

Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition