 |
|
|
| | | | | |
|
|
|
|
|
Diffusive Long-time Behavior of Kawasaki Dynamics
|
Nicoletta Cancrini, Università: dell'Aquila, Italy
Filippo Cesi, Università di Roma ``La Sapienza", Italy
Cyril Roberto, Università di Roma 3, Italy
|
Abstract
If $P_t$ is the semigroup associated with
the Kawasaki dynamics on $Z^d$ and $f$ is a local function
on the configuration space, then the variance with
respect to the invariant measure $mu$ of $P_t f$ goes to zero
as $ttooo$ faster than $t^{-d/2+e}$, with $e$ arbitrarily
small. The fundamental assumption is a mixing condition on the
interaction of Dobrushin and Schlosman type.
|
Full text: PDF
Pages: 216-249
Published on: March 4, 2005
|
Bibliography
| Ané, Cécile;
Blachère, Sébastien; Chafa"i, Djalil; Fougères, Pierre; Gentil, Ivan; Malrieu, Florent; Roberto, Cyril; Scheffer, Grégory. Sur les inégalités de Sobolev logarithmiques.
(French) [Logarithmic Sobolev inequalities] With a preface by Dominique Bakry and Michel Ledoux.
Panoramas et Synthèses [Panoramas and Syntheses], 10. Société Mathématique de France, Paris, 2000. xvi+217 pp. ISBN: 2-85629-105-8 MR1845806 (2002g:46132) |
| Bertini, Lorenzo; Cancrini, Nicoletta; Cesi, Filippo. The spectral gap for a Glauber-type dynamics in a continuous gas.
Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 1, 91--108. MR1899231 (2003d:82073) |
| Bertini, Lorenzo; Cirillo, Emilio N. M.; Olivieri, Enzo. Renormalization-group transformations under strong mixing conditions: Gibbsianness and convergence of renormalized interactions.
J. Statist. Phys. 97 (1999), no. 5-6, 831--915. MR1734386 (2001a:82029) |
| Bertini, Lorenzo; Zegarlinski, Bogusl aw. Coercive inequalities for Gibbs measures.
J. Funct. Anal. 162 (1999), no. 2, 257--286. MR1682059 (2000c:60158) |
| Bertini, L.; Zegarlinski, B. Coercive inequalities for Kawasaki dynamics. The product case.
Markov Process. Related Fields 5 (1999), no. 2, 125--162. MR1762171 (2001h:60174) |
| Cancrini, N.; Martinelli, F. Comparison of finite volume canonical and grand canonical Gibbs measures under a mixing condition.
Markov Process. Related Fields 6 (2000), no. 1, 23--72. MR1758982 (2002f:60186) |
| Cancrini, N.; Martinelli, F. On the spectral gap of Kawasaki dynamics under a mixing condition revisited.
Probabilistic techniques in equilibrium and nonequilibrium statistical physics.
J. Math. Phys. 41 (2000), no. 3, 1391--1423. MR1757965 (2002j:82075) |
| Cancrini, N.; Martinelli, F.; Roberto, C. The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited.
Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 4, 385--436. MR1914934 (2003j:82057) |
| Davies, E. B. Heat kernels and spectral theory.
Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. x+197 pp. ISBN: 0-521-36136-2 MR0990239 (90e:35123) |
| Dobrushin, R. L.; Shlosman, S. B. Completely analytical interactions: constructive description.
J. Statist. Phys. 46 (1987), no. 5-6, 983--1014. MR0893129 (88h:82006) |
| Janvresse, E.; Landim, C.; Quastel, J.; Yau, H. T. Relaxation to equilibrium of conservative dynamics. I. Zero-range processes.
Ann. Probab. 27 (1999), no. 1, 325--360. MR1681098 (2000e:60183) |
| Ledoux, Michel. Concentration of measure and logarithmic Sobolev inequalities.
Séminaire de Probabilités, XXXIII,
120--216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995 (2002j:60002) |
| Lu, Sheng Lin; Yau, Horng-Tzer. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics.
Comm. Math. Phys. 156 (1993), no. 2, 399--433. MR1233852 (95f:60122) |
| Landim, C.; Yau, H. T. Convergence to equilibrium of conservative particle systems on $Bbb Zsp d$.
Ann. Probab. 31 (2003), no. 1, 115--147. MR1959788 (2003m:60282) |
| Martinelli, Fabio. Lectures on Glauber dynamics for discrete spin models.
Lectures on probability theory and statistics (Saint-Flour, 1997),
93--191, Lecture Notes in Math., 1717, Springer, Berlin, 1999. MR1746301 (2002a:60163) |
| Spohn, H..
Large scale dynamics of interacting particles,
Springer, Berlin, 1991.
| | Varadhan, S. R. S.; Yau, Horng-Tzer. Diffusive limit of lattice gas with mixing conditions.
Asian J. Math. 1 (1997), no. 4, 623--678. MR1621569 (99k:60256) |
| Yau, Horng-Tzer. Logarithmic Sobolev inequality for lattice gases with mixing conditions.
Comm. Math. Phys. 181 (1996), no. 2, 367--408. MR1414837 (98e:82021) |
| |