A counterexample to a conjecture by Salez and Youssef
Florentin Münch
TL;DR
This paper addresses whether a positive Ollivier curvature lower bound $K$ implies a comparable lower bound on the log-Sobolev constant $\alpha_{LSI}$ with a $1/\log d$ factor, mirroring the Bakry–Emery case. The author constructs explicit birth–death chains with $3n$ states achieving $\kappa \ge \frac{1}{4n^2}$ and analyzes the isocapacitary bound via capacity between $A=\{1\}$ and $B=\{2n,\dots,3n\}$ to bound $\alpha_{LSI}$ by $O\left(\frac{K}{n \log d}\right)$. This leads to a counterexample showing the conjecture is false, highlighting the limitations of deriving $\alpha_{LSI}$ bounds from Ollivier curvature alone and motivating more refined curvature or chain-structure assumptions. The work underscores the utility of the isocapacitary framework in discrete settings and clarifies the relationship between Ollivier and Bakry–Emery curvature for functional inequalities.
Abstract
Remarkable progress has been made in recent years to establish log-Sobolev type inequalities under the assumption of discrete Ricci curvature bounds. More specfically, Salez and Youssef have proven that the log-Sobolev constant can be lower bounded by the Bakry Emery curvature lower bound divided by the logarithm of the sparsity parameter. They conjectured that the same holds true when replacing Bakry Emery by Ollivier curvature which is often times easier to compute in practice. In this paper, we show that this conjecture is wrong by giving a counter example on birth death chains of increasing length.