Chernoff bounds at the TD-threshold and under spectral independence

Determine whether d-uniform complexes that are either at the Trickling-Down (TD) threshold or satisfy η-spectral independence admit true Chernoff-type (subgaussian) concentration inequalities for lifted functions and inclusion sampling beyond the exponential top-level bounds established in this work. Concretely, establish subgaussian tails (with exponent proportional to ε^2 times the relevant set size) for these complexes across levels, not only at the top dimension.

Background

The paper proves optimal inclusion sampling and reverse hypercontractivity for several classes of high dimensional expanders, but under weaker assumptions—namely spectral independence and at the TD-threshold—it only shows exponential concentration at the top level. Subgaussian (Chernoff) tails are stronger and would directly improve associated degree lower bounds and related results.

A positive resolution would, for example, yield a 2{Ω(d2)} degree lower bound at the top level for hyper-regular λ-TD complexes, strengthening the threshold phenomena discussed in the paper.

References

While we are able to show optimal concentration for a fairly broad class of high dimensional expanders, in the weakest settings (namely under spectral independence and at the TD-Threshold), we are only able to prove exponential concentration at the top level. It is unclear whether this is a fundamental or purely technical barrier: do such complexes satisfy a true Chernoff bound? As discussed above, a resolution of this question in the positive leads to better degree lower bounds for HDX, and in particular a $2{\Omega(d2)}$ lower bound for the top level of hyper-regular $\lambda$-TD complexes.

— Chernoff Bounds and Reverse Hypercontractivity on HDX (2404.10961 - Dikstein et al., 17 Apr 2024) in Open questions section