Functional inequalities for discrete gradients and application to the geometric distribution

Laboratoire de Mathématiques, Université de La Rochelle, avenue Michel Crépeau, 17042 La Rochelle Cedex, France; nprivaul@univ-lr.fr.; ajoulin@univ-lr.fr.

Received: 29 July 2003
Revised: 3 February 2004

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 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.