A modified Poincare inequality and its application to First Passage Percolation

TL;DR

This paper extends a Gaussian inequality to product measures and applies it to show that First Passage Percolation exhibits sublinear variance for a broad class of continuous edge time distributions, including exponential.

Contribution

It introduces a modified Poincare inequality for Gaussian measures and applies it to prove sublinear variance in First Passage Percolation for various continuous distributions.

Findings

First Passage Percolation has sublinear variance for broad continuous distributions.

The inequality improves upon classical Gaussian Poincare inequalities.

Extension includes exponential and other continuous distributions.

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.