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.
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.