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Entropic Curvature Not Comparable to Other Curvatures -- or Is It? by Kamtue, Supanat; Liu, Shiping; Münch, Florentin; Peyerimhoff, Norbert is a scholarly article available to read on EtoBox.

What is Entropic Curvature Not Comparable to Other Curvatures -- or Is It? about?

In this paper we consider global $\theta$-curvatures of finite Markov chains with associated means $\theta$ in the spirit of the entropic curvature (based on the logarithmic mean) by Erbar-Maas and Mielke. As in the case of Bakry-\'Emery curvature, we also allow for a finite dimension parameter by making use of an adapted $\Gamma$ calculus for $\theta$-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-\'Emery curvatures). Finally, we present examples where entropic curvature differs significantly from other curvature notions like Bakry-\'Emery curvature or Ollivier Ricci and sectional curvatures.

Author

Kamtue, Supanat; Liu, Shiping; Münch, Florentin; Peyerimhoff, Norbert

Published

2024

Language

EN