Florentin Münch: Log Sobolev inequality and discrete Ricci curvature

It was conjectured by Peres and Tetali that a positive lower Ollivier Ricci curvature bound implies a modified log-Sobolev inequality. We give a simple counter example via a weighted graph on three vertices. Moreover, we confirm the Peres Tetali conjecture in case of non-negative Ollivier sectional curvature. By this, we answer a recent open question by Pedrotti. The proof relies on the variational characterization of the modified log-Sobolev constant, and a characterization of non-negative Ollivier sectional curvature via logarithmic gradient estimates for the heat equation.