math.PR — Probability

The L¹ Poincaré constant of the Hamming cube

Contributed by Tejasvi Singh Tomar

Let $C$ be the smallest constant such that $\mathbb{E}|f-\mathbb{E} f|\le C\,\mathbb{E}|\nabla f|$ for every real-valued function $f$ on every Hamming cube $\{-1,1\}^n$, where $|\nabla f|$ is the Euclidean length of $(D_1f,\dots,D_nf)$ and $D_jf(x)=(f(x)-f(x^{(j)}))/2$. Ben Efraim and Lust-Piquard proved $C\le\pi/2$, the Gaussian case gives $C\ge\sqrt{\pi/2}$, and Ivanisvili, Li, van Handel and Volberg asked whether $C=\sqrt{\pi/2}$. We prove that it is, with an explicit constant $c_n<\sqrt{\pi/2}$ in dimension $n$. The proof turns on an integral of first absolute moments of sums of independent biased random signs: we compute it exactly when the coefficients are nonnegative, and show, with a sampled Fejér kernel, that it can only grow when the coefficients are replaced by their absolute values. The equality $C=\sqrt{\pi/2}$ is also proved in Lean 4 with Mathlib.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715900