## The Logarithmic Sobolev Inequality For Discrete Spin Systems On A Lattice (1992)

Venue: | Commun. Math. Phys |

Citations: | 39 - 3 self |

### BibTeX

@ARTICLE{Stroock92thelogarithmic,

author = {Daniel W. Stroock and Boguslaw Zegarlinski},

title = {The Logarithmic Sobolev Inequality For Discrete Spin Systems On A Lattice},

journal = {Commun. Math. Phys},

year = {1992},

volume = {149},

pages = {175--194}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper, our local specification will come from a shift invariant, finite range Gibbs potential \Phi j f\Phi X gX2F . That is, (1) for each X 2 F, \Phi X 2 CX(\Omega\Gamma2

### Citations

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Logarithmic sobolev inequalities
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Citation Context ...) 2 satisfying psqs1 + (p \Gamma 1)e 2t c : Whensis a probability measure which is \Phi T t : t 2 (0; 1) \Psi -reversing and E is the associated Dirichlet form, then L. Gross's integration lemma (cf. =-=[G]-=- or, for the general case, Corollary 6.1.17 in [DS]) says that H is equivalent to the logarithmic Sobolev inequality LSs\Gamma f 2 log f \DeltascE(f; f) + kfk 2 L 2 () log kfk L 2 () for all positive ... |

43 | Constructive Criterion for the Uniqueness of Gibbs Fields - Dobrushin, Shlosman - 1985 |

30 | R.: Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin-Shlosman regime - Aizenman, Holley - 1987 |

17 | The logarithmic Sobolev inequality for continuous spin systems on a lattice - Stroock, Zegarliński - 1992 |

10 | uniqueness theorem and logarithmic Sobolev inequalities - Zegarliński, Dobrushin - 1992 |

4 |
An example in the theory of hypercontractive semigroups
- KORZENIOWSKI, STROOCK
- 1985
(Show Context)
Citation Context ...though (as we pointed out in our discussion of hypercontractivity) LS always implies SG, it is not at all clear under what circumstances one can go the other direction. Indeed, the only examples (cf. =-=[KS]-=-) when we know for sure that the logarithmic Sobolev constant fails to be equal to the reciprocal of the spectral gap are, in some sense, degenerate. Thus, there is a possibility that ii) () iii) is a... |

3 |
A unified existence and ergodic theorem for Markov evolution of random fields
- SULLIVAN
- 1974
(Show Context)
Citation Context ...rresponding to the operators in (1.18) so long as the b k 's in (1.18) are uniformly positive. x2: The Proof of Parts a) and b) of Theorem 1.8 We begin with an argument which goes back to W. Sullivan =-=[Sul]-=- and has since then been adapted to various situations in [HS] and [A & H]. However, the proof which we give below has some new features which we believe clarify what is happening. 2.1 Theorem. The co... |

2 |
Applications of the stochastic Ising model to the Gibbs states
- HOLLEY, STROOCK
- 1976
(Show Context)
Citation Context ... (1.18) are uniformly positive. x2: The Proof of Parts a) and b) of Theorem 1.8 We begin with an argument which goes back to W. Sullivan [Sul] and has since then been adapted to various situations in =-=[HS]-=- and [A & H]. However, the proof which we give below has some new features which we believe clarify what is happening. 2.1 Theorem. The condition DSU(Y) and DSM(Y) imply (1.9) and (1.12), respectively... |

1 |
Infinite Particle Systems, Grundlehren Series #276
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Citation Context ...Gamma! A 0(\Omega\Gamma by L Y ' = X k2\Gamma L Y k ' where \Theta L Y k ' (j) j \Theta Ek+Y ' (j) \Gamma '(j): Although L Y is no longer bounded, it is nonetheless well-known (eg. see Theorem 3.9 in =-=[L]-=-) that our assumptions about \Phi are more than enough to guarantee that there exists precisely one Markov semigroup \Phi P Y t : t 2 (0; 1) \Psi on A(\Omega\Gamma with the property that (1.4) P Y t '... |

1 |
Ergodicity of probabilistic automata
- Maes, Shlosman
- 1991
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Citation Context ...te spin systems, and so we have been forced to adopt here an argument which derives from the one used in [Z, 1] and [Z, 2] and bears a close relation to ones employed recently by Maes and Shlosman in =-=[MS]-=-; and it turns out that this new argument is somewhat simpler than the one given in [SZ, 1]. In particular, although we were unable to transfer the argument given in [SZ, 1] to the discrete spin conte... |

1 | log-Sobolev inequalities for infinite lattice systems - On - 1990 |

1 | inequalities for infinite one-dimensional lattice systems - Log-Sobolev - 1990 |