Functional Inequalities and Random Walks on Increasing Subsets of the Hypercube

TL;DR

This paper proves new functional inequalities on the hypercube related to random walks, improving mixing time bounds and characterizing extremizers for biased edge-isoperimetric inequalities, with implications for probabilistic exit times.

Contribution

It provides a simplified proof of the Poincaré inequality on increasing subsets and establishes a sharp $p$-biased edge-isoperimetric inequality for real-valued functions.

Findings

O(n^2) upper bound on mixing time of censored random walks

Sharp $p$-biased edge-isoperimetric inequality for increasing functions

Increasing subcubes are extremizers for the inequality

Abstract

Motivated by random walks on subsets of the hypercube, we prove two discrete functional inequalities on the hypercube. First, we give a short, elementary proof of the Poincar\'e inequality on increasing subsets of the cube recently established by Fei and Ferreira Pinto Jr, which yields an $O(n^2)$ upper bound on the mixing time of censored random walks, improving upon previous bounds. Second, adapting Samorodnitsky's induction method to the $p$-biased setting, we establish a sharp $p$-biased edge-isoperimetric inequality for real-valued increasing functions, which recovers the classic biased edge-isoperimetric inequality for increasing sets and identifies increasing subcubes as the extremizers. This result also admits a probabilistic interpretation in terms of maximizing the mean first exit time of biased random walks.