Eliminating the extra log log factor in the dimension-free tail bound

Improve the current dimension-free tail bound for \(P_\tau f\) on the Boolean hypercube by eliminating the extra \(\log\log \eta\) factor, potentially through a refined exponential martingale analysis within the reverse heat coupling framework.

Background

The main theorem resolves Talagrand’s conjecture up to a log⁡log⁡η\log\log \eta factor in the numerator, using a coupling constructed from perturbations of the reverse heat process.

The authors explicitly conjecture that this extra factor might be removed via a similar strategy, hinting that stronger process-level bounds—such as improved control of score differences—could achieve the full conjectured rate.

References

In view of the exponential martingale bound in Lemma~\ref{lem:log_diff_bound}, it is reasonable to conjecture that one might be able to obtain a better bound to save the extra $\log\log\eta$ factor via a similar strategy.

— Talagrand's convolution conjecture up to loglog via perturbed reverse heat (2511.19374 - Chen, 24 Nov 2025) in Remark after Lemma 3.3, Section 3.3 (Proof of approximate monotone coupling)