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