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

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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 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.

DOI : 10.1051/ps:2004004
Classification : 60E07, 60E15, 60K35
Keywords: geometric distribution, isoperimetry, logarithmic Sobolev inequalities, spectral gap, Herbst method, deviation inequalities, Gibbs measures 
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
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     author = {Joulin, Ald\'eric and Privault, Nicolas},
     title = {Functional inequalities for discrete gradients and application to the geometric distribution},
     journal = {ESAIM: Probability and Statistics},
     pages = {87--101},
     year = {2004},
     publisher = {EDP-Sciences},
     volume = {8},
     doi = {10.1051/ps:2004004},
     mrnumber = {2085608},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2004004/}
}
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