# 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

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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 (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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