previous :: next
Logarithmic Sobolev inequalities for finite Markov chains
P. Diaconis and L. Saloff-Coste
Source: Ann. Appl. Probab. Volume 6, Number 3
(1996), 695-750.
Abstract
This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric two-point space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1034968224
Mathematical Reviews number (MathSciNet): MR1410112
Digital Object Identifier: doi:10.1214/aoap/1034968224
Zentralblatt MATH identifier: 0867.60043
References
1 ALDOUS, D. and DIACONIS, P. 1987. Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69 97.
Mathematical Reviews (MathSciNet): MR88d:60175
Zentralblatt MATH: 0631.60065
Digital Object Identifier: doi:10.1016/0196-8858(87)90006-6
2 ALON, N. 1987. Eigenvalues and expanders. Combinatorica 5 83 96.
Mathematical Reviews (MathSciNet): MR875835
Digital Object Identifier: doi:10.1007/BF02579166
3 ALON, N. and MILMAN, V. 1985., isoperimetric inequalities for graphs and superconcen1 trators. J. Combin. Theory Ser. B 38 78 88.
Mathematical Reviews (MathSciNet): MR782626
Zentralblatt MATH: 0549.05051
Digital Object Identifier: doi:10.1016/0095-8956(85)90092-9
4 BAKRY, D. and EMERY, M. 1985. Diffusions hy percontractive. Seminaire de Probabilite ´ ´ XIX. Lecture Notes in Math. 1123 179 206. Springer, Berlin.
5 BAKRY, D. 1994. L'hy percontractivite et son utilisation en theorie des semigroups. Ecole ´ ´ ´ d'Ete de Saint Flour 1992. Lecture Notes in Math. 1581. Springer, Berlin. ´
6 BOLLOBAS, B. 1980. A probabilistic proof of an asy mptotic formula for the number of labelled regular graphs. European J. Combin. 1 311 316.
Mathematical Reviews (MathSciNet): MR82i:05045
7 DAVIES, E. B. 1989. Heat Kernels and Spectral Theory. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR990239
Zentralblatt MATH: 0699.35006
8 DEUSCHEL, J-D. and STROOCK, D. 1989. Large Deviations. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR90h:60026
Zentralblatt MATH: 0675.60086
9 DIACONIS, P. 1988. Group Representations in Probability and Statistics. IMS, Hay ward, CA.
Mathematical Reviews (MathSciNet): MR90a:60001
Zentralblatt MATH: 0695.60012
10 DIACONIS, P. and SALOFF-COSTE, L. 1996. Nash's inequalities for finite Markov chains. Journal of Theoretical Probability 9 459 510.
Mathematical Reviews (MathSciNet): MR97d:60114
Zentralblatt MATH: 0870.60064
Digital Object Identifier: doi:10.1007/BF02214660
11 DIACONIS, P. and SALOFF-COSTE, L. 1993. Comparison techniques for random walk on finite groups. Ann. Probab. 21 2131 2156.
Mathematical Reviews (MathSciNet): MR95a:60009
Zentralblatt MATH: 0790.60011
Digital Object Identifier: doi:10.1214/aop/1176989013
Project Euclid: euclid.aop/1176989013
12 DIACONIS, P. and SALOFF-COSTE, L. 1993. Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696 730.
Mathematical Reviews (MathSciNet): MR94i:60074
Zentralblatt MATH: 0799.60058
Digital Object Identifier: doi:10.1214/aoap/1177005359
Project Euclid: euclid.aoap/1177005359
13 DIACONIS, P. and SALOFF-COSTE, L. 1994. Moderate growth and random walk on finite groups. Geometry and Functional Analy sis 4 1 34.
Mathematical Reviews (MathSciNet): MR95d:60118
Zentralblatt MATH: 0795.60005
Digital Object Identifier: doi:10.1007/BF01898359
14 DIACONIS, P. and SALOFF-COSTE, L. 1995. Random walks on contingency tables with fixed row and column sums. Unpublished manuscript.
15 DIACONIS, P. and SALOFF-COSTE, L. 1995. An application of Harnack inequalities to random walk on nilpotent quotients. Journal of Fourier Analy sis and Its Applications Z. Kahane special issue 189 207.
Mathematical Reviews (MathSciNet): MR97d:60113
Zentralblatt MATH: 0889.60008
16 DIACONIS, P. and SALOFF-COSTE, L. 1995. Random walks on finite groups: a survey of analytic techniques. In Probability Measures on Groups and Related Structures 11 Z. H. Hey er, ed. 46 75. World Scientific, Singapore.
Mathematical Reviews (MathSciNet): MR97k:60013
Zentralblatt MATH: 0918.60059
17 DIACONIS, P. and SALOFF-COSTE, L. 1996. Random walks on generating sets of finite Abelian groups. Probab. Theory Related Fields. To appear.
Mathematical Reviews (MathSciNet): MR98a:60093
Zentralblatt MATH: 0847.60081
Digital Object Identifier: doi:10.1007/BF01192214
18 DIACONIS, P. and SALOFF-COSTE, L. 1996. What do we know about the Metropolis algorithm J. Comput. Sy stem Sci. To appear.
Mathematical Reviews (MathSciNet): MR1649805
Zentralblatt MATH: 0920.68054
Digital Object Identifier: doi:10.1006/jcss.1998.1576
19 DIACONIS, P. and SHAHSHAHANI, M. 1981. Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159 179.
Mathematical Reviews (MathSciNet): MR82h:60024
Zentralblatt MATH: 0485.60006
Digital Object Identifier: doi:10.1007/BF00535487
20 DIACONIS, P. and SHAHSHAHANI, M. 1987. Time to reach stationarity in the Bernoulli Laplace diffusion model. SIAM J. Math. Anal. 18 208 218.
Mathematical Reviews (MathSciNet): MR88e:60014
Zentralblatt MATH: 0617.60009
Digital Object Identifier: doi:10.1137/0518016
21 DIACONIS, P. and STROOCK, D. 1991. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36 61.
Mathematical Reviews (MathSciNet): MR92h:60103
Zentralblatt MATH: 0731.60061
Digital Object Identifier: doi:10.1214/aoap/1177005980
Project Euclid: euclid.aoap/1177005980
22 FELLER, W. 1968. An Introduction to Probability Theory and Its Applications, 3rd ed., 1. Wiley, New York.
Mathematical Reviews (MathSciNet): MR228020
23 FILL, J. 1991. Eigenvalue bounds on convergence to stationarity for nonversible Markov chains, with application to the exclusion process. Ann. Appl. Probab. 1 62 87.
Mathematical Reviews (MathSciNet): MR1097464
Zentralblatt MATH: 0726.60069
Digital Object Identifier: doi:10.1214/aoap/1177005981
Project Euclid: euclid.aoap/1177005981
24 GROSS, L. 1976. Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061 1083.
Zentralblatt MATH: 0318.46049
Mathematical Reviews (MathSciNet): MR420249
Digital Object Identifier: doi:10.2307/2373688
JSTOR: links.jstor.org
25 GROSS, L. 1993. Logarithmic Sobolev inequalities and contractivity properties of semigroups. Lecture Notes in Math. 1563. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1292277
Zentralblatt MATH: 0812.47037
Digital Object Identifier: doi:10.1007/BFb0074091
26 HIGUCHI, Y. and YOSHIDA, N. 1995. Analy tic conditions and phase transition for Ising Z. models. Unpublished lecture notes in Japanese.
27 HOLLEY, R. and STROOCK, D. 1987. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phy s. 46 1159 1194.
Mathematical Reviews (MathSciNet): MR89e:82013
Zentralblatt MATH: 0682.60109
Digital Object Identifier: doi:10.1007/BF01011161
28 HORN, P. and JOHNSON, C. 1985. Matrix Analy sis. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR87e:15001
29 HORN, P. and JOHNSON, C. 1990. Topics in Matrix Analy sis. Cambridge Univ. Press.
30 KORZENIOWSKI, A. and STROOCK, D. 1985. An example in the theory of hy percontractive semigroups. Proc. Amer. Math. Soc. 94 87 90.
Mathematical Reviews (MathSciNet): MR86i:47057
Zentralblatt MATH: 0577.47043
Digital Object Identifier: doi:10.2307/2044957
31 LIGGETT, T. 1985. Interacting Particle Sy stems. Springer, New York.
Mathematical Reviews (MathSciNet): MR86e:60089
32 LU, S. L. and YAU, H. T. 1993. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dy namics. Comm. Math. Phy s. 161 399 433.
Mathematical Reviews (MathSciNet): MR1233852
Zentralblatt MATH: 0779.60078
Digital Object Identifier: doi:10.1007/BF02098489
Project Euclid: euclid.cmp/1104253633
33 LUBOTZKY, A. 1994. Discrete Groups, Expanding Graphs and Invariant Measures. Birkhauser, Boston. ¨
Mathematical Reviews (MathSciNet): MR96g:22018
34 MARGULIS, G. 1973. Explicit construction of concentrators. Problemy Peredachi Informatsii
35 MICLO, L. 1995a. Sur les problemes de sortie discrets inhomogenes. Preprint.
Mathematical Reviews (MathSciNet): MR1422980
Digital Object Identifier: doi:10.1214/aoap/1035463326
Project Euclid: euclid.aoap/1035463326
36 MICLO, L. 1995b. Remarques sur l'hy percontractivite et l'evolution de l'entropie pour des ´ ´ chaines de Markov finies. Preprint.
37 QUASTEL, J. 1992. Diffusion of colour in the simple exclusion process. Comm. Pure Appl. Math. 45 623 679.
Mathematical Reviews (MathSciNet): MR1162368
Zentralblatt MATH: 0769.60097
Digital Object Identifier: doi:10.1002/cpa.3160450602
38 ROTHAUS, O. 1980. Logarithmic Sobolev inequalities and the spectrum of Sturm Liouville operators. J. Funct. Anal. 39 42 56.
Mathematical Reviews (MathSciNet): MR81m:34025
Zentralblatt MATH: 0472.47024
Digital Object Identifier: doi:10.1016/0022-1236(80)90018-X
39 ROTHAUS, O. 1981. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42 102 109.
Mathematical Reviews (MathSciNet): MR83f:58080a
Zentralblatt MATH: 0471.58027
Digital Object Identifier: doi:10.1016/0022-1236(81)90049-5
40 ROTHAUS, O. 1981. Logarithmic Sobolev inequalities and the spectrum of Schrodinger ¨ operators. J. Funct. Anal. 42 110 120.
Mathematical Reviews (MathSciNet): MR83f:58080b
Digital Object Identifier: doi:10.1016/0022-1236(81)90050-1
42 SARNACK, P. 1990. Some Applications of Modular Forms. Cambridge Univ. Press.
43 SINCLAIR, A. 1992. Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability and Computing 1 351 370.
Mathematical Reviews (MathSciNet): MR94f:60087
Zentralblatt MATH: 0801.90039
Digital Object Identifier: doi:10.1017/S0963548300000390
44 SINCLAIR, A. 1993. Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhauser, Boston. ¨
45 STEIN, E. 1993. Harmonic Analy sis. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR1232192
Zentralblatt MATH: 0821.42001
46 STEIN, E. and WEISS, G. 1971. Introduction to Fourier Analy sis on Euclidean Spaces. Princeton Univ. Press.
Mathematical Reviews (MathSciNet): MR304972
Zentralblatt MATH: 0232.42007
47 STROOCK, D. 1993. Logarithmic Sobolev inequalities for Gibbs states. Lecture Notes in Math. 1563. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR96c:60120
Zentralblatt MATH: 0801.60056
Digital Object Identifier: doi:10.1007/BFb0074094
48 STROOCK, D. and ZEGARLINSKI, B. 1992. The logarithmic Sobolev inequality for discrete spin sy stems on a lattice. Comm. Math. Phy s. 149 175 193.
Mathematical Reviews (MathSciNet): MR93j:82013
Zentralblatt MATH: 0758.60070
Digital Object Identifier: doi:10.1007/BF02096629
Project Euclid: euclid.cmp/1104251144
49 SU, F. 1995. Ph.D. dissertation, Harvard Univ.
50 YAU, T. H. 1995. Logarithmic Sobolev inequality for zero range process. Report, Dept. Mathematics, New York Univ.
CNRS, UNIVERSITE PAUL SABATIER ´ HARVARD UNIVERSITY STATISTIQUE ET PROBABILITES ´ DEPARTMENT OF MATHEMATICS 31062 TOULOUSE CEDEX
CAMBRIDGE, MASSACHUSETTS 02138 FRANCE EMAIL: lsc.cict.fr
previous :: next
The Annals of Applied Probability
