Entropic curvature not comparable to other curvatures -- or is it?

Abstract

In this paper we consider global $θ$-curvatures of finite Markov chains with associated means $θ$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-Émery curvature, we also allow for a finite dimension parameter by making use of an adapted $Γ$ calculus for $θ$-curvatures. We prove explicit positive lower curvature bounds (both finite- and infinite-dimensional) for finite abelian Cayley graphs. In the case of cycles, we provide also an upper curvature bound which shows that our lower bounds are asymptotically sharp (up to a logarithmic factor). Moreover, we prove new universal lower curvature bounds for finite Markov chains as well as curvature perturbation results (allowing, in particular, to compare entropic and Bakry-Émery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-Émery curvature or Ollivier Ricci and sectional curvatures.