Communications in Mathematical Physics

, Volume 144, Issue 2, pp 303–323 | Cite as

The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition

  • Daniel W. Stroock
  • Boguslaw Zegarlinski
Article

Abstract

Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Time Distribution 

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References

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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Daniel W. Stroock
    • 1
  • Boguslaw Zegarlinski
    • 1
    • 2
  1. 1.2-272, Mathematics DepartmentM.I.T.CambridgeUSA
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochum 1Germany

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