Functional inequalities for discrete gradients and application to the geometric distribution
ESAIM: Probability and Statistics, Tome 8 (2004), pp. 87-101.

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.

DOI : 10.1051/ps:2004004
Classification : 60E07, 60E15, 60K35
Mots clés : geometric distribution, isoperimetry, logarithmic Sobolev inequalities, spectral gap, Herbst method, deviation inequalities, Gibbs measures
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Joulin, Aldéric; Privault, Nicolas. Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 87-101. doi : 10.1051/ps:2004004. http://www.numdam.org/articles/10.1051/ps:2004004/

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