Log-Sobolev, isoperimetry and transport inequalities on graphs

TL;DR

This paper investigates various functional inequalities for reversible Markov processes on finite graphs, providing estimates of constants using the path method, which enhances understanding of their geometric and probabilistic properties.

Contribution

It introduces new estimates for constants in functional inequalities on graphs using the path method, advancing the analysis of Markov processes.

Findings

Derived bounds for constants in Poincaré and log-Sobolev inequalities

Established relationships between isoperimetric and transportation inequalities

Applied path method to finite graph Markov processes

Abstract

In this paper, we study some functional inequalities (such as Poincar\'e inequalities, logarithmic Sobolev inequalities, generalized Cheeger isoperimetric inequalities, transportation-information inequalities and transportation-entropy inequalities) for reversible nearest-neighbor Markov processes on a connected finite graph by means of (random) path method. We provide estimates of the involved constants.