Criteria for entropic curvature on graph spaces

Abstract

In this paper we establish new simple local geometric criteria for discrete entropic curvature introduced in [47] that are powerful enough to capture many geometric properties of complex models arising in mathematical physics. These results are robust in the sense that they apply to any discrete graph equipped with a Markov reversible generator. Our definitions of entropic curvature differ from the one of the pioneering works of Erbar-Maas [19,20] (which is already a discrete analog of the Lott-Sturm-Villani entropic curvature in the continuous setting). Singularly, our results provide refined concentration properties related to the celebrated convex-hull method by Talagrand [50,51] for a large class of probability measures that cannot be captured from Erbar- Maas entropic definition of curvature. Our approach gives also a new insight of the convex hull method, without being related to induction arguments. We illustrate the power of our results, as well as the general entropic strategy developed in this paper, to tackle challenging models studied in mathematical physics, including Gibbs measures with interaction potentials such as Ising models on the discrete hypercube and measures with interaction potential on the lattice $\mathbb{Z}^n$. For instance, we significantly improve the constant of the refined convex concentration properties obtained in [2] for Ising models. Moreover, when dealing with the antiferromagnetic Curie-Weiss model, we improve the previously known bound for entropic curvature by a factor of $\sqrt{n}$. Our simple criteria also provides the expected right order of magnitude $C/\sqrt n$ for the lower-bound on the entropic curvature for the renowned Sherrington-Kirkpatrick model from the spin glass theory. This last result is consistant with the recent works [6,17] on the modified logarithmic Sobolev and Poincar\'e inequalities for the Sherrington-Kirkpatrick model.