A modified Poincare inequality and its application to First Passage Percolation

Abstract

We extend a Gaussian functional inequality to a countable product of Gaussian measures. This inequality improves on the classical Poincare inequality for Gaussian measures. As an application, we prove that First Passage Percolation has sublinear variance when the edge times distribution belongs to a wide class of continuous distributions, including the exponential one. This extends a result by Benjamini, Kalai and Schramm, valid for positive Bernoulli edge times.

After having completed a first version of this paper, we were informed that our main inequality, which we claimed to be new, was implied by a work of Bobkov and Houdre. See details in the revised paper