We determine the logarithmic Sobolev constant for the Bernoulli- Laplace model and the time to stationarity for the symmetric simple exclusion model up to the leading order. Our method for proving the logarithmic Sobolev inequality is based on a martingale approach and is applied to the random transposition model as well. The proof for the time to stationarity is based on a general observation relating the time to stationarity to the hydrodynamical limit.
References
1 DAVIES, E. B. 1989. Heat Kernels and Spectral Theory. Cambridge Univ. Press. MR990239 0699.350061 DAVIES, E. B. 1989. Heat Kernels and Spectral Theory. Cambridge Univ. Press. MR990239 0699.35006
2 DAVIES, E. B., GROSS, L. and SIMON, B. 1992. Hypercontractivity: a bibliographical review. In Ideas and Methods of Mathematics and Physics, in Memoriam of Raphael HoeghKrohn S. Albeverio, J. E. Fenstand, H. Holden and T. Lindstrom, eds. Cambridge Univ. Press. MR1190534 0790.460552 DAVIES, E. B., GROSS, L. and SIMON, B. 1992. Hypercontractivity: a bibliographical review. In Ideas and Methods of Mathematics and Physics, in Memoriam of Raphael HoeghKrohn S. Albeverio, J. E. Fenstand, H. Holden and T. Lindstrom, eds. Cambridge Univ. Press. MR1190534 0790.46055
3 DEUSCHEL, J. and STROOCK, D. W. 1989. Large Deviations. Academic Press, San Diego. MR90h:60026 0675.600863 DEUSCHEL, J. and STROOCK, D. W. 1989. Large Deviations. Academic Press, San Diego. MR90h:60026 0675.60086
4 DIACONIS, P. and SALOFF-COSTE, L. 1995. Random walks on finite groups. In ProbabilityMeasures on Groups and Related Structures XI H. Heyer, ed. World Scientific, River Edge, NJ. MR1414925 0918.600594 DIACONIS, P. and SALOFF-COSTE, L. 1995. Random walks on finite groups. In ProbabilityMeasures on Groups and Related Structures XI H. Heyer, ed. World Scientific, River Edge, NJ. MR1414925 0918.60059
5 DIACONIS, P. and SALOFF-COSTE, L. 1996. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695 750. MR97k:60176 0867.60043 10.1214/aoap/1034968224 euclid.aoap/1034968224
5 DIACONIS, P. and SALOFF-COSTE, L. 1996. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695 750. MR97k:60176 0867.60043 10.1214/aoap/1034968224 euclid.aoap/1034968224
6 DIACONIS, P. and SHAHSHAHANI, M. 1981. Generating a random permutation with random transpositions.Wahrsch. Verw. Gebiete 57 159 179. MR82h:60024 0485.60006 10.1007/BF005354876 DIACONIS, P. and SHAHSHAHANI, M. 1981. Generating a random permutation with random transpositions.Wahrsch. Verw. Gebiete 57 159 179. MR82h:60024 0485.60006 10.1007/BF00535487
7 DIACONIS, P. and SHAHSHAHANI, M. 1987. Time to reach stationarity in the Bernoulli Laplace diffusion model. SIAM J. Math. Anal. 18 208 218. MR88e:60014 0617.60009 10.1137/05180167 DIACONIS, P. and SHAHSHAHANI, M. 1987. Time to reach stationarity in the Bernoulli Laplace diffusion model. SIAM J. Math. Anal. 18 208 218. MR88e:60014 0617.60009 10.1137/0518016
9 GROSS, L. 1976. Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061 1083. 0318.46049 MR420249 10.2307/23736889 GROSS, L. 1976. Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061 1083. 0318.46049 MR420249 10.2307/2373688
10 HOLLEY, R. and STROOCK, D. 1987. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 1159 1194. MR89e:82013 0682.60109 10.1007/BF0101116110 HOLLEY, R. and STROOCK, D. 1987. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 1159 1194. MR89e:82013 0682.60109 10.1007/BF01011161
11 KIPNIS, C., OLLA, S. and VARADHAN, S. R. S. 1989. Hydrodynamics and large deviations for simple exclusion process. Comm. Pure Appl. Math. 42 115 137. MR91h:60115 0644.76001 10.1002/cpa.316042020211 KIPNIS, C., OLLA, S. and VARADHAN, S. R. S. 1989. Hydrodynamics and large deviations for simple exclusion process. Comm. Pure Appl. Math. 42 115 137. MR91h:60115 0644.76001 10.1002/cpa.3160420202
12 MARTINELLI, F. and OLIVIERI, E. 1994. Approach to equilibrium of Glauber dynamics in the one phase region, I and II. Comm. Math. Phys. 161 447 514. MR96c:82041 0793.60111 10.1007/BF02101930 euclid.cmp/1104270007
12 MARTINELLI, F. and OLIVIERI, E. 1994. Approach to equilibrium of Glauber dynamics in the one phase region, I and II. Comm. Math. Phys. 161 447 514. MR96c:82041 0793.60111 10.1007/BF02101930 euclid.cmp/1104270007
13 QUASTEL, J. 1992. Diffusion of colour in the simple exclusion process. Comm. Pure Appl. Math. 45 321 379. MR1162368 0769.60097 10.1002/cpa.316045060213 QUASTEL, J. 1992. Diffusion of colour in the simple exclusion process. Comm. Pure Appl. Math. 45 321 379. MR1162368 0769.60097 10.1002/cpa.3160450602
14 STROOCK, D. W. 1992. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet Forms. Lecture Notes in Math. 1563. Springer, Berlin. MR1292280 0801.60056 10.1007/BFb007409414 STROOCK, D. W. 1992. Logarithmic Sobolev inequalities for Gibbs states. Dirichlet Forms. Lecture Notes in Math. 1563. Springer, Berlin. MR1292280 0801.60056 10.1007/BFb0074094
15 STROOCK, D. and ZEGARLINSKI, B. 1992. The equivalence of the logarithmic Sobolev inequality and the Dobrushin Shlosman mixing condition. Comm. Math. Phys. 144 303 323; see also, The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 175 193. MR93b:82005 0745.60104 10.1007/BF02101094 euclid.cmp/1104249318
15 STROOCK, D. and ZEGARLINSKI, B. 1992. The equivalence of the logarithmic Sobolev inequality and the Dobrushin Shlosman mixing condition. Comm. Math. Phys. 144 303 323; see also, The logarithmic Sobolev inequality for discrete spin systems on a lattice. Comm. Math. Phys. 149 175 193. MR93b:82005 0745.60104 10.1007/BF02101094 euclid.cmp/1104249318
16 YAU, H.-T. 1991. Relative entropy and hydrodynamics of a Ginzburg Landau model. Lett. Math. Phys. 22 63 80. MR93e:82035 0725.60120 10.1007/BF0040037916 YAU, H.-T. 1991. Relative entropy and hydrodynamics of a Ginzburg Landau model. Lett. Math. Phys. 22 63 80. MR93e:82035 0725.60120 10.1007/BF00400379
17 YAU, H.-T. 1996. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 367 408. Also, Logarithmic Sobolev inequality for generalized simple exclusion processes. In Probab. Theory Related Fields. To appear. MR98e:82021 10.1007/BF02101009 euclid.cmp/1104287767
17 YAU, H.-T. 1996. Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phys. 181 367 408. Also, Logarithmic Sobolev inequality for generalized simple exclusion processes. In Probab. Theory Related Fields. To appear. MR98e:82021 10.1007/BF02101009 euclid.cmp/1104287767