We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i\leq k$ and $f:X(i)\to [0,1]$: \[\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq s}}{\mathbb{E}}[f(t)]-\mu\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).\] Using this fact, we prove that high dimensional expanders are reverse hypercontractive, a powerful functional inequality from discrete analysis implying that for any sets $A,B \subset X(k)$, the probability a $\rho$-correlated pair passes between them is at least \[\Pr_{s,s' \sim T_\rho}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.\] Our results hold under weak spectral assumptions on $X$. 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 $\sqrt{k}$-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 $\ell_\infty$-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-R\"odl Theorem and more.

翻译：我们证明了高维扩展子（HDX）上提升函数的最优测度集中性。设 $X$ 为一个 $k$ 维 HDX。我们证明对于任意 $i\leq k$ 和函数 $f:X(i)\to [0,1]$：\[\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq s}}{\mathbb{E}}[f(t)]-\mu\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).\] 利用这一事实，我们证明高维扩展子具有反向超压缩性，这是离散分析中的一个强大函数不等式，意味着对于任意集合 $A,B \subset X(k)$，一对 $\rho$ 相关的样本点穿过它们之间的概率至少为 \[\Pr_{s,s' \sim T_\rho}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.\] 我们的结果在 $X$ 满足弱谱假设的条件下成立。具体而言，我们证明了对于任何低于“滴流阈值”（超出该阈值集中性可能任意差）的复形，测度具有指数集中性；并且对于此类复形的 $\sqrt{k}$-骨架，我们证明了最优集中性。我们还展示了更强 HDX 在顶维等其他设置下的最优界。我们利用这些不等式证明了高维扩展子上若干新的协议测试定理，包括一个新的针对子集的 99% 机制测试，以及在更强的 $\ell_\infty$-扩展假设下实现指数级逆声明的“Z-测试”变体。后者产生了超越完全复形与乘积结构的最优测试器，为在强声明概率可检查证明（PCP）中使用 HDX 奠定了基础。我们还给出了在扩展性、分析学、组合数学和编码理论中的应用，包括证明双侧 HDX 具有最优几何重叠性（首次给出显式的有界度构造）、近最优的双重采样器、针对特定 HDX 的新超指数级度数下界、距离放大的列表可解码与局部可测试编码、一个弗兰克尔-勒德尔定理及其他结果。