Talagrand’s convolution conjecture on the Boolean hypercube

Prove, for the heat semigroup (P_τ) on the Boolean hypercube with uniform measure μ, that for every τ > 0 there exists a constant c_τ depending only on τ such that for all nonnegative functions f: {-1,1}^n → R_+ with ∥f∥_1 ≠ 0 and all η > 1, the dimension-free tail bound P_{X∼μ}(P_τ f(X) > η ∥f∥_1) ≤ c_τ/(η √(log η)) holds.

Background

The paper studies the regularization effect of the heat semigroup on L1 functions over the Boolean hypercube. Markov’s inequality yields a baseline tail bound of 1/η, while Talagrand conjectured a universal improvement to 1/(η√(log η)) with a constant depending only on the time parameter τ but not on the dimension.

The authors prove a bound up to an extra log log η factor, with a dimension-free constant. The exact conjectured form without the log log η factor remains the central open problem, historically known as Talagrand’s convolution conjecture.

References

Talagrand's convolution conjecture, also restated on his website with the title "Regularization from $L1$ by convolution", aims to precisely quantify the regularization effect of the heat semigroup when applied to $L1(\mu)$ functions, as follows. For every $\tau > 0$, there exists a constant $c_\tau > 0$ only depends on $\tau$, such that for every nonnegative function $f: #1 {-1, 1}n \to +$ with $#2{f}{1} \neq 0$, and any $\eta > 1$, we have \begin{align*} {{X \sim \mu}(P_\tau f(X) > \eta #2{f}{1} ) \leq \frac{c_\tau}{\eta \sqrt{\log \eta}. \end{align*}

— Talagrand's convolution conjecture up to loglog via perturbed reverse heat (2511.19374 - Chen, 24 Nov 2025) in Conjecture (Section 1, Introduction)