logo AFST
Discrete Ricci Curvature bounds for Bernoulli-Laplace and Random Transposition models
Matthias Erbar; Jan Maas; Prasad Tetali
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, p. 781-800

We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters.

Nous calculons une borne inférieure pour la courbure de Ricci pour quelques exemples classiques de marches aléatoires. Notamment nous considérons une marche sur une tranche du cube discret (dite modèle de Bernoulli-Laplace) et la marche sur le groupe symétrique des permutations obtenue par produits de transpositions indépendantes et uniformes.

Published online : 2016-01-21
@article{AFST_2015_6_24_4_781_0,
     author = {Matthias Erbar and Jan Maas and Prasad Tetali},
     title = {Discrete Ricci Curvature bounds for Bernoulli-Laplace and Random Transposition models},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {4},
     year = {2015},
     pages = {781-800},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2015_6_24_4_781_0}
}
Erbar, Matthias; Maas, Jan; Tetali, Prasad. Discrete Ricci Curvature bounds for Bernoulli-Laplace and Random Transposition models. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 24 (2015) no. 4, pp. 781-800. https://afst.centre-mersenne.org/item/AFST_2015_6_24_4_781_0/

[1] Ambrosio (L.), Gigli (N.), and Savaré (G.).— Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J., 163(7) p. 1405-1490 (2014). | MR 3205729 | Zbl 1304.35310

[2] Bakry (D.) and Émery (M.).— Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., pages 177-206. Springer, Berlin (1985). | Numdam | MR 889476 | Zbl 0561.60080

[3] Bobkov (S. G.) and Tetali (P.).— Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab., 19(2) p. 289-336 (2006). | MR 2283379 | Zbl 1113.60072

[4] Caputo (P.), Dai Pra (P.), and Posta (G.).— Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat., 45(3) p. 734-753, (2009). | Numdam | MR 2548501 | Zbl 1181.60142

[5] Chow (S.-N.), Huang (W.), Li (Y.), and Zhou (Z.).— Fokker-Planck equations for a free energy functional or Markov process on a graph. Arch. Ration. Mech. Anal., 203(3) p. 969-1008 (2012). | MR 2928139 | Zbl 1256.35173

[6] Diaconis (P.) and Saloff-Coste (L.).— Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab., 6(3) p. 695-750 (1996). | MR 1410112 | Zbl 0867.60043

[7] Diaconis (P.) and Shahshahani (M.).— Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete, 57(2) p. 159-179 (1981). | MR 626813 | Zbl 0485.60006

[8] Diaconis (P.) and Shahshahani (M.).— Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal., 18(1) p. 208-218 (1987). | MR 871832 | Zbl 0617.60009

[9] Erbar (M.) and Maas (J.).— Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal., 206(3) p. 997-1038 (2012). | MR 2989449 | Zbl 1256.53028

[10] Erbar (M.) and Maas (J.).— Gradient flow structures for discrete porous medium equations. Discrete Contin. Dyn. Syst., 34(4) p. 1355-1374 (2014). | MR 3117845 | Zbl 1275.49084

[11] Gao (F.) and Quastel (J.).— Exponential decay of entropy in the random transposition and Bernoulli-Laplace models. Ann. Appl. Probab., 13(4) p. 1591-1600 (2003). | MR 2023890 | Zbl 1046.60003

[12] Gigli (N.) and Maas (J.).— Gromov-Hausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal., 45(2) p. 879-899 (2013). | MR 3045651 | Zbl 1268.49054

[13] Goel (S.).— Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl., 114(1) p. 51-79 (2004). | MR 2094147 | Zbl 1074.60080

[14] Gozlan (N.), Melbourne (J.), Perkins (W.), Roberto (C.), Samson (P-M.), and Tetali (P.).— Working Group in New directions in mass transport: discrete versus continuous. AIM SQuaRE report, October (2013).

[15] Gozlan (N.), Roberto (C.), Samson (P.-M.), and Tetali (P.).— Displacement convexity of entropy and related inequalities on graphs. Probability Theory and Related Fields, 160, p. 47-94 (2014). | MR 3256809

[16] Lee (T.-Y.) and Yau (H.-T.).— Logarithmic Sobolev inequality for some models of random walks. Ann. Probab., 26(4) p. 1855-1873 (1998). | MR 1675008 | Zbl 0943.60062

[17] Lott (J.) and Villani (C.).— Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. (2), 169(3) p. 903-991 (2009). | MR 2480619 | Zbl 1178.53038

[18] Maas (J.).— Gradient flows of the entropy for finite Markov chains. J. Funct. Anal., 261(8) p. 2250-2292 (2011). | MR 2824578 | Zbl 1237.60058

[19] Mielke (A.).— A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity, 24(4) p. 1329-1346 (2011). | MR 2776123 | Zbl 1227.35161

[20] Mielke (A.).— Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differential Equations, 48(1) p. 1-31 (2013). | MR 3090532 | Zbl 1282.60072

[21] Ollivier (Y.).— Ricci curvature of Markov chains on metric spaces. J. Funct. Anal., 256(3) p. 810-864 (2009). | MR 2484937 | Zbl 1181.53015

[22] Ollivier (Y.).— A survey of Ricci curvature for metric spaces and Markov chains. Adv. Stud. Pure Math., 57, p. 343-381 (2010). | MR 2648269 | Zbl 1204.53035

[23] Sammer (M.D.).— Aspects of mass transportation in discrete concentration inequalities. PhD thesis, Georgia Institute of Technology (2005). | MR 2707115

[24] Sturm (K.-Th.).— On the geometry of metric measure spaces. I and II. Acta Math., 196(1) p. 65-177 (2006). | MR 2237206 | Zbl 1106.53032