Chernoff Bounds and Reverse Hypercontractivity on HDX

We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> be a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex>-dimensional HDX. We show for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$i \leq k$</tex> and function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f: X(i)\rightarrow [0, 1]$</tex>: \begin{equation*}\underset{s \in X(k)}{\mathbb{P}}[\vert \underset{t \subseteq s}{\mathbb{E}}[f(t)]-\mu\vert \geqslant \varepsilon] \leqslant \exp \left(-\varepsilon^2 \frac{k}{i}\right). \end{equation*} Using this fact, we prove that high dimensional expanders are reverse hypercontractive, a powerful functional inequality from discrete analysis implying that for any sets <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A, B \subset X(k)$</tex>, the probability a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\rho$</tex>-correlated pair passes between them is at least \begin{equation*}\underset{s, s^{\prime} \sim T_\rho}{\mathbb{P}}\left[s \in A, s^{\prime} \in B\right] \geqslant \mathbb{P}[A]^{O(1)} \mathbb{P}[B]^{O(1)}.\end{equation*} Our results hold under weak spectral assumptions on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex>. Namely we prove exponential concentration of measure for any complex below the ‘Trickling-Down Threshold’ (beyond which concentration may be arbitrarily poor), and optimal concentration for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\sqrt{k}$</tex>. skeletons of such complexes. We also show optimal bounds for the top dimension of stronger HDX among other settings. We leverage our inequalities to prove several new agreement testing theorems on high dimensional expanders, including a new 99%-regime test for subsets, and a variant of the ‘Z-test’ achieving inverse exponential soundness under the stronger assumption of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{\infty}$</tex> -expansion. The latter gives rise to the first optimal testers beyond the complete complex and products, a stepping stone toward the use of HDX in strong soundness PCPs. We also give applications within expansion, analysis, combinatorics, and coding theory, including a proof that two-sided HDX have optimal geometric overlap (giving the first explicit bounded-degree construction), near-optimal double samplers, new super-exponential degree lower bounds for certain HDX, distance-amplified list-decodable and locally testable codes, a Frankl+Rödl Theorem, and more.