Chernoff Bounds and Reverse Hypercontractivity on HDX
Yotam Dikstein, Max Hopkins
TL;DR
This work proves optimal concentration of lifted functions on high dimensional expanders (HDX), establishing Chernoff-type bounds for inclusion sampling and, crucially, reverse hypercontractivity for HDX under weak spectral assumptions. The authors introduce a combinatorial method to derive reverse hypercontractivity from local sampling properties, enabling strong bounds for both 99% and 1% agreement testers on HDX. They develop a comprehensive framework connecting HDX concentration, sampler graphs, and high-order random walks, and apply it to agreement testing, geometric overlap, double samplers, and robust coding-theoretic constructions. The results close a long-standing gap between known Chebyshev-type bounds and Chernoff-type tails in sparse settings, with implications for PCPs, sampling, and combinatorial geometry. They also showcase an explicit separation between MLSI and reverse hypercontractivity and provide near-optimal degree lower bounds, underscoring the depth and impact of the HDX toolkit in discrete analysis and theoretical CS.
Abstract
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 $f:X(i)\to [0,1]$: \[\Pr_{s\in X(k)}\left[\left|\underset{t\subseteq s}{\mathbb{E}}[f(t)]-μ\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).\] 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 $ρ$-correlated pair passes between them is at least \[\Pr_{s,s' \sim T_ρ}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.\] 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-Rödl Theorem and more.