Peres–Tetali conjecture on coarse Ricci curvature and modified log-Sobolev inequalities

Let $\Omega$ be a finite unweighted graph and consider the stochastic matrix $P$ associated with the simple random walk on this graph, $\tilde{P} = \frac{P+I}{2}$ associated to the lazy simple random walk and the generator $L= P-I$. The notation $\operatorname{Ric_{dc,1}}$ denotes the coarse Ricci curvature of a stochastic matrix, and $\operatorname{Ric_{cc,1}}$ denotes the corresponding curvature for a generator; $\operatorname{MLSI}(c)$ denotes the modified log-Sobolev inequality with constant $c$. Peres–Tetali conjecture. There exists a universal constant $\alpha >0$ such that, if $\operatorname{Ric_{dc,1}} \geq K >0$ for the lazy stochastic matrix $\tilde{P}$ (or, equivalently, if $\operatorname{Ric_{cc,1}}\geq 2K$ for $L$), then $\operatorname{MLSI}(\alpha K)$ holds. This conjecture connects coarse Ricci curvature with functional inequalities for lazy simple random walks on finite graphs. It remains an important open problem in the theory of discrete curvature and functional inequalities for Markov chains.