Ollivier-Ricci curvature and the discrete log-Sobolev inequality

Let $(P_t)_{t\geq 0}$ be the Markov semigroup on the discrete state space, let $d$ denote the relevant dimension parameter, and let ${\rm t}_{\textsc{ls}}$ denote the log-Sobolev time. The Ollivier-Ricci curvature $\tilde{\kappa}$ is characterized by

$${\mathrm{Lip}}(P_t f)\leq e^{-\tilde{\kappa}t}{\mathrm{Lip}}(f).$$

Ollivier-Ricci curvature and LSI. There exists a universal constant $c<\infty$ such that whenever the Ollivier-Ricci curvature $\tilde{\kappa}$ is positive, we have

$${\rm t}_{\textsc{ls}}\leq\frac{c\log d}{\tilde{\kappa}}.$$

The conjecture predicts a dimension-logarithmic bound on the discrete log-Sobolev time from positive Ollivier-Ricci curvature, analogous to the corresponding Bakry–Émery curvature estimate. The stated prediction was recently disproved, so it is refuted.