Electronic Journal of Probability

Ricci curvature bounds for weakly interacting Markov chains

Matthias Erbar, Christopher Henderson, Georg Menz, and Prasad Tetali

Full-text: Open access


We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in several examples, namely: Glauber dynamics for a class of spin systems including the Ising and Curie–Weiss models, a class of hard-core models and random walks on groups induced by a conjugacy invariant set of generators.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 40, 23 pp.

Received: 23 August 2016
Accepted: 14 March 2017
First available in Project Euclid: 28 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J22: Computational methods in Markov chains [See also 65C40] 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Ricci curvature modified logarithmic Sobolev inequality Markov chain Ising model hardcore model random walk Cayley graph

Creative Commons Attribution 4.0 International License.


Erbar, Matthias; Henderson, Christopher; Menz, Georg; Tetali, Prasad. Ricci curvature bounds for weakly interacting Markov chains. Electron. J. Probab. 22 (2017), paper no. 40, 23 pp. doi:10.1214/17-EJP49. https://projecteuclid.org/euclid.ejp/1493345027

Export citation


  • [1] D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206.
  • [2] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math. 84 (2000), no. 3, 375–393.
  • [3] A.-I. Bonciocat and K.-Th. Sturm, Mass transportation and rough curvature bounds for discrete spaces, J. Funct. Anal. 256 (2009), no. 9, 2944–2966.
  • [4] P. Caputo, P. Dai Pra, and G. Posta, Convex entropy decay via the Bochner-Bakry-Emery approach, Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 734–753.
  • [5] P. Dai Pra and G. Posta, Entropy decay for interacting systems via the Bochner-Bakry-Émery approach, Electron. J. Probab. 18 (2013), no. 52, 21.
  • [6] M. Disertori and A. Giuliani, The nematic phase of a system of long hard rods, Comm. Math. Phys. 323 (2013), no. 1, 143–175.
  • [7] M. Dyer and C. Greenhill, On Markov chains for independent sets, J. Algorithms, 35 (2000), 17–49.
  • [8] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal. 206 (2012), no. 3, 997–1038.
  • [9] M. Erbar, J. Maas, and P. Tetali, Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 4, 781–800.
  • [10] M. Fathi and J. Maas, Entropic Ricci curvature bounds for discrete interacting systems, Ann. Appl. Probab. 26 (2016), no. 3, 1774–1806.
  • [11] N. Gozlan, C. Roberto, P.-M. Samson, and P. Tetali, Displacement convexity of entropy and related inequalities on graphs, Probab. Theory Related Fields 160 (2014), no. 1–2, 47–94.
  • [12] R. Holley and D. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models, J. Statist. Phys. 46 (1987), no. 5–6, 1159–1194.
  • [13] D. Levin, Y. Peres, and E. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009, With a chapter by James G. Propp and David B. Wilson.
  • [14] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. (2) 169 (2009), no. 3, 903–991.
  • [15] M. Luby and E. Vigoda, Fast convergence of the Glauber dynamics for sampling independent sets, Random Structures Algorithms 15 (1999), no. 3–4, 229–241, Statistical physics methods in discrete probability, combinatorics, and theoretical computer science (Princeton, NJ, 1997).
  • [16] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal. 261 (2011), no. 8, 2250–2292.
  • [17] K. Marton, Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance, arXiv:1507.02803 (2015).
  • [18] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity 24 (2011), no. 4, 1329–1346.
  • [19] A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations 48 (2013), no. 1–2, 1–31.
  • [20] Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009), no. 3, 810–864.
  • [21] Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains, Probabilistic approach to geometry, Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010, pp. 343–381.
  • [22] K.-Th. Sturm, On the geometry of metric measure spaces. I and II, Acta Math. 196 (2006), no. 1, 65–177.
  • [23] E. Vigoda, A note on the Glauber dynamics for sampling independent sets, Electron. J. Combin. 8 (2001), no. 1, Research Paper 8, 8 pp. (electronic).