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Functional inequalities for discrete gradients and application to the geometric distribution

Published online by Cambridge University Press:  15 September 2004

Aldéric Joulin  and
Nicolas Privault
Aldéric Joulin
Affiliation:
Laboratoire de Mathématiques, Université de La Rochelle, avenue Michel Crépeau, 17042 La Rochelle Cedex, France; [email protected].; [email protected].
Nicolas Privault
Affiliation:
Laboratoire de Mathématiques, Université de La Rochelle, avenue Michel Crépeau, 17042 La Rochelle Cedex, France; [email protected].; [email protected].
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Abstract

We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities, associated deviation estimates, and an exponential integrability property. In the particular case of the geometric distribution on ${\mathbb{N}}$ we use an integration by parts formula to compute the optimal isoperimetric and Poincaré constants, and to obtain an improvement of our general logarithmic Sobolev inequality. By a limiting procedure we recover the corresponding inequalities for the exponential distribution. These results have applications to interacting spin systems under a geometric reference measure.

Keywords

Geometric distributionisoperimetrylogarithmic Sobolev inequalitiesspectral gapHerbst methoddeviation inequalitiesGibbs measures.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 87 - 101
Copyright
© EDP Sciences, SMAI, 2004

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