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Functional Inequalities and Random Walks on Increasing Subsets of the Hypercube

Fan Chang, Guowei Sun, Lei Yu

TL;DR

This work analyzes functional inequalities on non-product spaces arising from increasing subsets of the hypercube, focusing on censored random walks and biased measures. It yields a simple induction-on-dimension proof of a Poincaré inequality on increasing sets, giving an $O(n^2)$ mixing bound for censored walks and a near-optimal constant, and it extends Samorodnitsky's edge-isoperimetric inequality to the $p$-biased setting for real-valued increasing functions, with equality exactly for increasing subcubes. The authors provide a probabilistic interpretation in terms of mean first exit time on weighted hypercubes and compute exact exit times for increasing subcubes, linking geometric isoperimetry to Markov-chain exit behavior. The results illuminate constrained sampling and mixing under non-product measures and point toward further development of log-Sobolev-type tools on increasing sets to approach the full Ding--Mossel conjecture. Open questions include establishing suitable log-Sobolev inequalities in these non-product spaces and extending the induction framework to nonlinear functionals.

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 Poincaré 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.

Functional Inequalities and Random Walks on Increasing Subsets of the Hypercube

TL;DR

This work analyzes functional inequalities on non-product spaces arising from increasing subsets of the hypercube, focusing on censored random walks and biased measures. It yields a simple induction-on-dimension proof of a Poincaré inequality on increasing sets, giving an mixing bound for censored walks and a near-optimal constant, and it extends Samorodnitsky's edge-isoperimetric inequality to the -biased setting for real-valued increasing functions, with equality exactly for increasing subcubes. The authors provide a probabilistic interpretation in terms of mean first exit time on weighted hypercubes and compute exact exit times for increasing subcubes, linking geometric isoperimetry to Markov-chain exit behavior. The results illuminate constrained sampling and mixing under non-product measures and point toward further development of log-Sobolev-type tools on increasing sets to approach the full Ding--Mossel conjecture. Open questions include establishing suitable log-Sobolev inequalities in these non-product spaces and extending the induction framework to nonlinear functionals.

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 Poincaré inequality on increasing subsets of the cube recently established by Fei and Ferreira Pinto Jr, which yields an upper bound on the mixing time of censored random walks, improving upon previous bounds. Second, adapting Samorodnitsky's induction method to the -biased setting, we establish a sharp -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.

Paper Structure

This paper contains 15 sections, 16 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Poincaré inequality and random walks on increasing sets
  3. Biased Samorodnitsky's inequality and exit time of random walks from increasing sets
  4. Organization
  5. Proof of Theorem \ref{['thm:main thm2']}
  6. Preliminaries
  7. Induction part
  8. Five-point Inequality Part
  9. Proof of Theorem \ref{['thm:main thm1']}
  10. Preliminaries
  11. Induction Part
  12. Two-point Inequality Part
  13. Proof of Theorem \ref{['thm:mean first exit time']}
  14. Proof of mean first exit time for increasing subcubes
  15. Conclusion and open problems

Key Result

Theorem 1.1

Let $A\subseteq \{0,1\}^n$ be a non-empty increasing set. It holds that for all $f:A\to \mathbb{R}$, Here ${\rm Var}_A[f]$ stands for the variance of $f(x)$ where $x$ is a uniformly random element of $A$, and $\mathcal{E}_A(f)$ is called "restricted" Dirichlet form of a function on $A$ defined as follows where the expectation is taken with respect to the uniform probability measure on the increa

Theorems & Definitions (29)

  • Theorem 1.1: fei2025spectral
  • Theorem 1.2
  • Theorem 1.3: fei2025spectral
  • Theorem 1.4: Samorodnitsky Samorodnitsky17
  • Definition 1
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 1: Mean first exit time for the $p$-biased walk on increasing subcubes
  • Lemma 2.1
  • ...and 19 more