Boguslaw Zegarlinski

The logarithmic Sobolev inequality for discrete spin systems on a lattice

For finite range lattice gases with a finite spin space, it is shown that the Dobrushin-Shlosman mixing condition is equivalent to the existence of a logarithmic Sobolev inequality for the associated (unique) Gibbs state. In addition, implications of these considerations for the ergodic properties of the corresponding Glauber dynamics are examined.

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

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.

Entropy bounds and isoperimetry

Introduction and notations Poincare-type inequalities Entropy and Orlicz spaces

The logarithmic Sobolev inequality for continuous spin systems on a lattice

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The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice

and Hardy-type inequalities on the line Probability measures satisfying

Dobrushin uniqueness theorem and logarithmic Sobolev inequalities

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Log-Sobolev inequalities for infinite one dimensional lattice systems

-inequalities on the real line Exponential integrability and perturbation of measures

On log-Sobolev inequalities for infinite lattice systems

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On the Ergodic Properties of Glauber Dynamics

-inequalities for Gibbs measures with super Gaussian tails

QUANTUM STOCHASTIC DYNAMICS II

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