In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett’s M−ɛ theorem for interacting particle systems.
Liming Wu. "Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition." Ann. Probab. 34 (5) 1960 - 1989, September 2006. https://doi.org/10.1214/009117906000000368