Talagrand’s convolution conjecture

On the Boolean hypercube $G = \{-1,1\}^n$ with uniform measure $\lambda$, let $T_\mu f(x) = \int_G f(x \odot y)\,d\mu(y)$ be convolution by a finite positive measure $\mu$, and set $$\psi_\mu(u) = \sup\{u\,\lambda(\{T_\mu f \ge u\}) : f \ge 0,\ \|f\|_1 = 1\},$$ which measures how much better than Markov's inequality convolution makes the tail. In 1989 Talagrand conjectured that for the biased-coin product measure $\mu_a = (\tfrac{1+a}{2}\delta_1 + \tfrac{1-a}{2}\delta_{-1})^{\otimes n}$ with $0 < a < 1$, $$\psi_{\mu_a}(u) \le \frac{C_a}{\sqrt{\log u}} \qquad (u > 1),$$ with $C_a$ depending on $a$ alone and not on the dimension $n$. He offered a \$1000 prize for a proof. The Gaussian analogue was settled by Eldan and Lee; the hypercube case, the original, stayed open. This paper claims the conjectured bound.

The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$. What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and Xiang-Zhang's localized terminal-discrepancy method are taken as given; the addition is a power coupling that splits each reverse edge ratio into two geometric powers, producing a switched exponential weight that restores the exact reverse jump rate of the perturbed coordinate. Because the frozen exponent then has a fixed numerator, no growing stopping buffer is needed and the $\log\log u$ factor disappears. That last $\log\log$ is what stood between the previous work and Talagrand's statement.