Entropic curvature on graphs along Schrödinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete spaces by C. Léonard by replacing W_2-Wasserstein geodesics by Schrödinger bridges in the definition of entropic curvature [Léo]. As a remarkable fact, as a temperature parameter goes to zero, these Schrödinger bridges are supported by geodesic of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete graphs. We give a method that provides lower bound for the entropic curvature for several examples of graphs spaces.

As opposed to Erbar-Maas strategy about entropic curvature on graphs [EM], our approach deals with curvature terms that provide new Prékopa-Leindler type of inequalities, and new transport-entropy inequalities. Moreover, these transport-entropy inequalities imply refined concentration properties. In particular, it extends to other graphs the Talagrand's concentration results on the discrete cube related to the so-called convex-hull method.

[EM] M. Erbar and J. Maas. Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal., 206(3):997–1038, 2012.

[Léo] C. Léonard. Lazy random walks and optimal transport on graphs. Ann. Probab., 44(3):1864–1915, 2016.