# Functional inequalities for discrete gradients and application to the geometric distribution

Aldéric Joulin; Nicolas Privault

ESAIM: Probability and Statistics (2004)

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

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topJoulin, Aldéric, and Privault, Nicolas. "Functional inequalities for discrete gradients and application to the geometric distribution." ESAIM: Probability and Statistics 8 (2004): 87-101. <http://eudml.org/doc/245603>.

@article{Joulin2004,

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; Geometric distribution},

language = {eng},

pages = {87-101},

publisher = {EDP-Sciences},

title = {Functional inequalities for discrete gradients and application to the geometric distribution},

url = {http://eudml.org/doc/245603},

volume = {8},

year = {2004},

}

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

PY - 2004

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; Geometric distribution

UR - http://eudml.org/doc/245603

ER -

## References

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