The Annals of Applied Probability

Logarithmic Sobolev inequalities for finite Markov chains

P. Diaconis and L. Saloff-Coste

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab. Volume 6, Number 3 (1996), 695-750.

Dates
First available in Project Euclid: 18 October 2002

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

Subjects
Primary: 60J15 60J27: Continuous-time Markov processes on discrete state spaces 60F05: Central limit and other weak theorems

Keywords
Markov chains logarithmic Sobolev inequalities

Citation

Diaconis, P.; Saloff-Coste, L. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996), no. 3, 695--750. doi:10.1214/aoap/1034968224. http://projecteuclid.org/euclid.aoap/1034968224.


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References

  • 1 ALDOUS, D. and DIACONIS, P. 1987. Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69 97.
  • 2 ALON, N. 1987. Eigenvalues and expanders. Combinatorica 5 83 96.
  • 3 ALON, N. and MILMAN, V. 1985., isoperimetric inequalities for graphs and superconcen1 trators. J. Combin. Theory Ser. B 38 78 88.
  • 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.
  • 7 DAVIES, E. B. 1989. Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • 8 DEUSCHEL, J-D. and STROOCK, D. 1989. Large Deviations. Academic Press, New York.
  • 9 DIACONIS, P. 1988. Group Representations in Probability and Statistics. IMS, Hay ward, CA.
  • 10 DIACONIS, P. and SALOFF-COSTE, L. 1996. Nash's inequalities for finite Markov chains. Journal of Theoretical Probability 9 459 510.
  • 11 DIACONIS, P. and SALOFF-COSTE, L. 1993. Comparison techniques for random walk on finite groups. Ann. Probab. 21 2131 2156.
  • 12 DIACONIS, P. and SALOFF-COSTE, L. 1993. Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696 730.
  • 13 DIACONIS, P. and SALOFF-COSTE, L. 1994. Moderate growth and random walk on finite groups. Geometry and Functional Analy sis 4 1 34.
  • 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.
  • 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.
  • 17 DIACONIS, P. and SALOFF-COSTE, L. 1996. Random walks on generating sets of finite Abelian groups. Probab. Theory Related Fields. To appear.
  • 18 DIACONIS, P. and SALOFF-COSTE, L. 1996. What do we know about the Metropolis algorithm J. Comput. Sy stem Sci. To appear.
  • 19 DIACONIS, P. and SHAHSHAHANI, M. 1981. Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159 179.
  • 20 DIACONIS, P. and SHAHSHAHANI, M. 1987. Time to reach stationarity in the Bernoulli Laplace diffusion model. SIAM J. Math. Anal. 18 208 218.
  • 21 DIACONIS, P. and STROOCK, D. 1991. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36 61.
  • 22 FELLER, W. 1968. An Introduction to Probability Theory and Its Applications, 3rd ed., 1. Wiley, New York.
  • 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.
  • 24 GROSS, L. 1976. Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061 1083.
  • 25 GROSS, L. 1993. Logarithmic Sobolev inequalities and contractivity properties of semigroups. Lecture Notes in Math. 1563. Springer, Berlin.
  • 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.
  • 28 HORN, P. and JOHNSON, C. 1985. Matrix Analy sis. Cambridge Univ. Press.
  • 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.
  • 31 LIGGETT, T. 1985. Interacting Particle Sy stems. Springer, New York.
  • 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.
  • 33 LUBOTZKY, A. 1994. Discrete Groups, Expanding Graphs and Invariant Measures. Birkhauser, Boston. ¨
  • 34 MARGULIS, G. 1973. Explicit construction of concentrators. Problemy Peredachi Informatsii
  • 35 MICLO, L. 1995a. Sur les problemes de sortie discrets inhomogenes. Preprint.
  • 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.
  • 38 ROTHAUS, O. 1980. Logarithmic Sobolev inequalities and the spectrum of Sturm Liouville operators. J. Funct. Anal. 39 42 56.
  • 39 ROTHAUS, O. 1981. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42 102 109.
  • 40 ROTHAUS, O. 1981. Logarithmic Sobolev inequalities and the spectrum of Schrodinger ¨ operators. J. Funct. Anal. 42 110 120.
  • 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.
  • 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.
  • 46 STEIN, E. and WEISS, G. 1971. Introduction to Fourier Analy sis on Euclidean Spaces. Princeton Univ. Press.
  • 47 STROOCK, D. 1993. Logarithmic Sobolev inequalities for Gibbs states. Lecture Notes in Math. 1563. Springer, Berlin.
  • 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.
  • 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.
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