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

Abstract

We prove that under the heat semigroup $(P_{\ddot{I}}„)$ on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than Markov's inequality. Specifically, for any $\ddot{I}„>0$, $n\ge 1$, $\hat{I}\cdot >e^3$, and $f:\{-1,1{\}}^n\to {\mathbb{R}}_{+}$ with $\int fd\hat{I}¼>0$, 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 $\hat{I}¼$ is the uniform measure on the Boolean hypercube $\{-1,1{\}}^n$ and $c_{\ddot{I}}„$ is a constant that depends only on $\ddot{I}„$. This result resolves Talagrand's convolution conjecture up to a dimension-free $(\log \log \hat{I}\cdot {)}^{\frac{3}{2}}$ 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.

43 pages, fixed a mistake in the previous draft which was kindly pointed out by Joseph Lehec