Functional inequalities for discrete gradients and application to the geometric distribution

Aldéric Joulin; Nicolas Privault

ESAIM: Probability and Statistics (2010)

  • Volume: 8, page 87-101
  • ISSN: 1292-8100

Abstract

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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 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.

How to cite

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Joulin, Aldéric, and Privault, Nicolas. "Functional inequalities for discrete gradients and application to the geometric distribution." ESAIM: Probability and Statistics 8 (2010): 87-101. <http://eudml.org/doc/104325>.

@article{Joulin2010,
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. },
author = {Joulin, Aldéric, Privault, Nicolas},
journal = {ESAIM: Probability and Statistics},
keywords = {Geometric distribution; isoperimetry; logarithmic Sobolev inequalities; spectral gap; Herbst method; deviation inequalities; Gibbs measures. ; logarithmic Sobolev inequalities; deviation inequalities; Gibbs measures},
language = {eng},
month = {3},
pages = {87-101},
publisher = {EDP Sciences},
title = {Functional inequalities for discrete gradients and application to the geometric distribution},
url = {http://eudml.org/doc/104325},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Joulin, Aldéric
AU - Privault, Nicolas
TI - Functional inequalities for discrete gradients and application to the geometric distribution
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 87
EP - 101
AB - 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.
LA - eng
KW - Geometric distribution; isoperimetry; logarithmic Sobolev inequalities; spectral gap; Herbst method; deviation inequalities; Gibbs measures. ; logarithmic Sobolev inequalities; deviation inequalities; Gibbs measures
UR - http://eudml.org/doc/104325
ER -

References

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