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We study functional inequalities for Markov chains on discrete spaces withentropic Ricci curvature bounded from below. Our main results are that whencurvature is non-negative, but not necessarily positive, the spectral gap, theCheeger isoperimetric constant and the modified logarithmic Sobolev constant ofthe chain can be bounded from below by a constant that only depends on thediameter of the space, with respect to a suitable metric. These estimates arediscrete analogues of classical results of Riemannian geometry obtained by Liand Yau, Buser and Wang.

Poincar\'e, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature