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arXiv · 2506.09852

Functional inequalities and random walks on increasing subsets of the hypercube

Fan Chang·Guowei Sun·Lei Yu
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Abstract

Motivated by random walks on subsets of the hypercube, we prove two discrete functional inequalities on the hypercube by the technique of induction-by-restrictions. 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 functions supported on increasing sets, 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 known as Glauber dynamics.

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Fan Chang, Guowei Sun, Lei Yu. 2025-06-11. Functional inequalities and random walks on increasing subsets of the hypercube. https://arxiv.org/abs/2506.09852

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