Talagrand's convolution conjecture up to loglog via perturbed reverse heat

We prove that under the heat semigroup $(P_{\tau })$ on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than Markov’s inequality. Specifically, for any $\tau >0$, $n\ge 1$, $\eta >e^3$, and $f:\{-1,1{\}}^n\to {\mathbb{R}}_{+}$ with $\int fd\mu >0$, we have \begin{align*} $\mathbb{P}$_{X \sim μ}\left( P_τf(X) > η\int f dμ\right) \leq c_τ\frac{ (\log \log η)^{\frac32} }{η\sqrt{\log η}}, \end{align*} where $\mu$ is the uniform measure on the Boolean hypercube $\{-1,1{\}}^n$ and $c_{\tau }$ is a constant that depends only on $\tau$. This result resolves Talagrand’s convolution conjecture up to a dimension-free $(\log \log \eta {)}^{\frac{3}{2}}$ factor. Our proof uses the reverse heat process on the Boolean hypercube, a coupling construction with carefully engineered perturbations of jump rates and a time-smoothed anti-concentration estimate.