Applying HDX global hypercontractivity and booster theorem to sharp thresholds

Determine whether the global hypercontractivity inequality for partite high dimensional expanders (Theorem 1.2, Global Hypercontractivity on HDX in Bonami form) or the booster theorem for partite high dimensional expanders (Theorem 1.3, A Booster Theorem for HDX) can be leveraged to establish sharp threshold results for graph properties analogous to those derived from Bourgain’s booster theorem on product spaces.

Background

The paper proves an optimal global hypercontractivity inequality and a booster theorem for partite high dimensional expanders (γ-products), extending classical techniques such as symmetrization beyond product spaces. In product settings, Bourgain’s booster theorem was pivotal in the development of the theory of sharp thresholds for graph properties (Friedgut–Bourgain).

Motivated by these classical connections, the authors explicitly raise the question of whether their HDX analogues—global hypercontractivity and the booster theorem—can similarly yield sharp threshold results in sparse, high-dimensional settings modeled by HDX.

References

Classically, Bourgain's booster theorem lead to the famous theory of sharp thresholds for graph properties . It is an interesting open problem whether \Cref{thm:Bonami-intro} or \Cref{thm:intro-booster} could be used in this context.

— Hypercontractivity on HDX II: Symmetrization and q-Norms  (2408.16687 - Hopkins, 2024) in Section 1, Results (following Theorem 1.3)