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Zbl 1055.82004
Helffer, Bernard
Remarks on decay of correlations and Witten Laplacians. III: Application to logarithmic Sobolev inequalities. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 35, No. 4, 483-508 (1999). ISSN 0246-0203

Summary: This is the continuation of our two previous articles devoted to the use of Witten Laplacians for analyzing Laplace integrals in statistical mechanics. The main application treated in Part I [J. Funct. Anal. 155, No. 2, 571--586 (1998; Zbl 0921.35141)] was a semi-classical one. The second application [Rev. Math. Phys. 11, No. 3, 321--336 (1999; Zbl 1054.82002)] was more perturbative in spirit and gave very explicit estimates for the lower bound of the Witten Laplacian in the case of a quartic model.\par We shall relate in this third part our studies of the Witten Laplacian with the existence of uniform logarithmic Sobolev inequalities through a criterion of {\it B. Zegarliński} [Commun. Math. Phys. 175, No. 2, 401--432 (1996; Zbl 0844.46050)]. More precisely, our main contribution is to show how to control the decay of correlations uniformly with respect to various parameters, under a natural condition of strict convexity at infinity of the single-spin phase and when the nearest neighbor interaction is small enough.

MSC 2000:: *82B20 Lattice systems
35J99 Elliptic equations and systems
82B26 Phase transitions (general)

Citations: Zbl 1054.82002; Zbl 0921.35141; Zbl 0844.46050

Cited in: Zbl 1054.82002

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