Chernoff-type concentration for spectrally independent and TD-threshold complexes

Determine whether spectrally independent simplicial complexes and complexes at the Trickling-Down threshold satisfy true Chernoff-type concentration (i.e., subgaussian tails) beyond exponential concentration at the top level. Concretely, establish whether these classes admit optimal inclusion sampling bounds rather than only exponential concentration on the top dimension; a positive resolution would, for example, yield a 2^{Omega(d^2)} degree lower bound for the top level of hyper-regular lambda-Trickling-Down complexes.

Background

The paper proves optimal inclusion sampling (Chernoff-type bounds) for broad classes of high dimensional expanders but only establishes exponential concentration at the very top level in weaker regimes such as spectral independence and at the Trickling-Down (TD) threshold.

The authors point out that this may be a technical limitation of their methods or a genuine barrier for these regimes. Establishing subgaussian concentration in these settings would have significant consequences, including stronger degree lower bounds for certain HDX families.