{
  "id": "2404.04581",
  "version": "v1",
  "published": "2024-04-06T10:31:42.000Z",
  "updated": "2024-04-06T10:31:42.000Z",
  "title": "Entropic curvature not comparable to other curvatures -- or is it?",
  "authors": [
    "Supanat Kamtue",
    "Shiping Liu",
    "Florentin Münch",
    "Norbert Peyerimhoff"
  ],
  "categories": [
    "math.DG",
    "math.MG",
    "math.PR"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-06T10:31:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "60J10",
      "05C81"
    ],
    "keywords": [
      "entropic curvature",
      "finite markov chains",
      "bakry-emery curvature",
      "universal lower curvature bounds",
      "finite abelian cayley graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}