Dimension-free L2 bound on the terminal discrepancy of the coupled processes

Derive a dimension-free upper bound on the Euclidean distance \(\|W_{T_o} - V_{T_o}\|_2\) between the perturbed and unperturbed reverse heat processes at time \(T_o\), sufficient to make the semi-log-convexity-based second-order Taylor expansion effective for total variation distance control.

Background

The authors also explored adapting the Gaussian proofs that use semi-log-convexity and second-order Taylor expansions. On the Boolean hypercube, this requires bounding the L2 distance between the terminal states of the coupled processes.

They explicitly note being unable to obtain a good dimension-free L2 bound, which prevents this approach from succeeding in the discrete setting.

References

However, again, the difficulty to proceed from here is that the author was unable to obtain a good dimension-free L2 bound on #2{W_{T_o} - V_{T_o}{2}, which is intuitively quite large as the distance between two points on the Boolean hypercube.

— Talagrand's convolution conjecture up to loglog via perturbed reverse heat (2511.19374 - Chen, 24 Nov 2025) in Remark after Lemma 3.2, Section 3.2 (Proof of Lemma on total variation control)