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

Abstract

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 $\tau > 0$, $n \geq 1$, $\eta > e^3$, and $f: \{-1,1\}^n \to \mathbb{R}_+$ with $\int f d\mu > 0$, we have \begin{align*} \mathbb{P}_{X \sim \mu}\left( P_\tau f(X) > \eta \int f d\mu \right) \leq c_\tau \frac{ (\log \log \eta)^{\frac32} }{\eta \sqrt{\log \eta}}, \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 $(\log \log \eta)^{\frac32}$ 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.