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.

Keywords: geometric distribution, isoperimetry, logarithmic Sobolev inequalities, spectral gap, Herbst method, deviation inequalities, Gibbs measures

@article{PS_2004__8__87_0, 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}, publisher = {EDP-Sciences}, volume = {8}, year = {2004}, doi = {10.1051/ps:2004004}, mrnumber = {2085608}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps:2004004/} }

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 SP - 87 EP - 101 VL - 8 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2004004/ DO - 10.1051/ps:2004004 LA - en ID - PS_2004__8__87_0 ER -

%0 Journal Article %A Joulin, Aldéric %A Privault, Nicolas %T Functional inequalities for discrete gradients and application to the geometric distribution %J ESAIM: Probability and Statistics %D 2004 %P 87-101 %V 8 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2004004/ %R 10.1051/ps:2004004 %G en %F PS_2004__8__87_0

Joulin, Aldéric; Privault, Nicolas. Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM: Probability and Statistics, Volume 8 (2004), pp. 87-101. doi : 10.1051/ps:2004004. http://archive.numdam.org/articles/10.1051/ps:2004004/

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