Sparse 1% agreement testers without \ell_\infty-expansion

Determine whether the Z-test analysis in the 1% regime can be completed without assuming \ell_\infty-expansion, using only reverse hypercontractivity and spectral gap of the down-up walk, thereby yielding sparse low-acceptance agreement testers under weaker conditions (e.g., recent topological criteria).

Background

The paper proves an optimal 1% agreement tester under a strong \ell_\infty-expansion assumption, which holds for dense complexes but typically fails for sparse HDX.

The authors note that the core argument relies only on reverse hypercontractivity and spectral gap and suggest that topological conditions from recent works might suffice, leaving open whether the \ell_\infty-expansion assumption can be removed to obtain sparse testers.

References

It is possible \pref{thm:intro-local-agreement} could be propogated to a true Z-test under much weaker conditions than \lambda-globality, e.g.\ under the recent topological notions of . We leave this as an open question for the 1\%-regime.

— Chernoff Bounds and Reverse Hypercontractivity on HDX  (2404.10961 - Dikstein et al., 2024) in Section 4 (Agreement Testing), following Theorem (The Local Agreement Theorem)