The Annals of Probability

Convergence to equilibrium of conservative particle systems on $\mathbb{Z}^{\bm{d}}$

C. Landim and H. T. Yau

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We consider the Ginzburg--Landau process on the lattice $\mathbb{Z}^d$ whose potential is a bounded perturbation of the Gaussian potential. We prove that the decay rate to equilibrium in the variance sense is $t^{-d/2}$ up to a~logarithmic correction, for any function $u$ with finite triple norm; that is, $|\!|\!| u |\!|\!| \;=\; \sum_{x\in \mathbb{Z}^d} \Vert \partial_{\eta_x} u \Vert_\infty < \infty$.

Article information

Ann. Probab., Volume 31, Number 1 (2003), 115-147.

First available in Project Euclid: 26 February 2003

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82A05

Interacting particle systems polynomial convergence to equilibrium Nash inequality


Landim, C.; Yau, H. T. Convergence to equilibrium of conservative particle systems on $\mathbb{Z}^{\bm{d}}$. Ann. Probab. 31 (2003), no. 1, 115--147. doi:10.1214/aop/1046294306.

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  • [1] BERTINI, L. and ZEGARLINSKI, B. (1999). Coercive inequalities for Kawasaki dy namics: The product case. Markov Process. Related Fields 5 125-162.
  • [2] BERTINI, L. and ZEGARLINSKI, B. (1999). Coercive inequalities for Gibbs measures. J. Funct. Anal. 162 257-289.
  • [3] CANCRINI, N. and MARTINELLI, F. (2000). On the spectral gap of Kawasaki dy namics under a mixing condition revisited. J. Math. Phy s. 41 1391-1423.
  • [4] DAVIES, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • [5] FERRARI, P. A., GALVES, A. and LANDIM, C. (2000). Rate of convergence to equilibrium of sy mmetric simple exclusion processes. Markov Process. Related Fields 6 73-88.
  • [6] JANVRESSE, E., LANDIM, C., QUASTEL, J. and YAU, H. T. (1999). Relaxation to equilibrium of conservative dy namics I: Zero range processes. Ann. Probab. 27 325-360.
  • [7] LANDIM, C. (1998). Decay to equilibrium in L of finite interacting particle sy stems in infinite volume. Markov Process. Related Fields 4 517-534.
  • [8] LANDIM, C., PANIZO GARCIA, G. and YAU, H. T. (2000). Spectral gap and logarithmic Sobolev inequalities for Ginzburg-Landau processes. Ann. Inst. H. Poincaré Ser. B. To appear.
  • [9] LANDIM C., SETHURAMAN S. and VARADHAN S. R. S. (1996). Spectral gap for zero range dy namics. Ann. Probab. 24 1871-1902.
  • [10] LIGGETT, T. M. (1991). L2 rates of convergence for attractive reversible nearest neighbor particle sy stems: The critical case. Ann. Probab. 19 935-959.
  • [11] LU, S. L. (1995). Hy drody namic scaling limits with deterministic initial configurations. Ann. Probab. 23 1831-1852.
  • [12] LU, S. L. and YAU, H. T. (1993). Spectral Gap and Logarithmic Sobolev Inequality for Kawasaki and Glauber Dy namics. Comm. Math. Phy s. 156 399-433.
  • [13] YAU, H. T. (1996). Logarithmic Sobolev inequality for lattice gases with mixing conditions. Comm. Math. Phy s. 181 367-408.
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