Functional inequalities for discrete gradients and application to the geometric distribution

Abstract

We present several functional inequalities for finite difference gradients, such as a Cheeger inequality, Poincare 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 Poincare 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.