Formulating the ‘It Ain’t Over Till It’s Over’ analog for HDX

Characterize the correct formulation of the ‘It Ain’t Over Till It’s Over’ conjecture (random restriction tail bounds) for high dimensional expanders or product spaces, and establish corresponding tail bounds.

Background

The paper generalizes a noise-operator tail bound to HDX, paralleling tools used to resolve the classical conjecture on the hypercube. However, the appropriate version for HDX or even product spaces remains unclear.

A precise HDX analog would likely require tailoring random restriction models to simplicial complexes and developing corresponding analytic or probabilistic tools.

References

It is not clear what the correct form of this conjecture should be for HDX, or even for product spaces. We leave this as an open question.

— Chernoff Bounds and Reverse Hypercontractivity on HDX (2404.10961 - Dikstein et al., 17 Apr 2024) in Section: Analytic, Geometric, and Combinatorial Applications, subsection "It Ain't Over Till It's Over"