Probability Theory and Related Fields

, Volume 109, Issue 4, pp 507–538

Logarithmic Sobolev inequality for generalized simple exclusion processes

  • Horng-Tzer Yau

DOI: 10.1007/s004400050140

Cite this article as:
Yau, HT. Probab Theory Relat Fields (1997) 109: 507. doi:10.1007/s004400050140


Let \(\) be a probability measure on the set {0,1, . . .,R} for some R∈ℕ and ΛL a cube of width L in d. Denote by μgcΛL the (grand canonical) product measure on the configuration space on ΛL with \(\) as the marginal measure; here the superscript indicates the grand canonical ensemble. The canonical ensemble, denoted by μcΛL,n, is defined by conditioning μgcΛL given the total number of particles to be n. Consider the exclusion dynamics where each particle performs random walk with rates depending only on the number of particles at the same site. The rates are chosen such that, for every n and L fixed, the measure μcΛL,n is reversible. We prove the logarithmic Sobolev inequality in the sense that ∫flogfdμcΛL,n\(\) for any probability density f with respect to μcΛL,n; here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal.

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Horng-Tzer Yau
    • 1
  1. 1.Courant Institute, New York University New York, NY 10012, USA e-mail: US

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