Salez–Youssef conjecture on Ollivier curvature and log-Sobolev inequalities

Let $(X,p)$ be a reversible Markov chain. Write $d:=\max\{1/p(x,y):p(x,y)>0\}$ for its sparsity constant, and let $\operatorname{Lip}$ denote the Lipschitz constant with respect to the combinatorial distance, also called the hop-count distance. Suppose that the Ollivier curvature is bounded below by $K>0$, in the sense that

$$\operatorname{Lip}(P_t f)\leq e^{-Kt}\operatorname{Lip}(f)\quad\text{for all }f\in\mathbb{R}^X\text{ and }t>0.$$

Salez–Youssef conjecture. There is a universal constant $c>0$ such that

$$\alpha_{LSI}\geq c\cdot\frac{K}{\log d}.$$

Salez and Youssef established the analogous estimate under a Bakry–Émery curvature lower bound, while the Ollivier-curvature version would avoid the stronger non-negative sectional-curvature assumption and is often easier to compute in practice. The conjecture is presented here as the subject of the paper; the supplied text gives no resolution status.