We prove that under the heat semigroup $(P_τ)$ on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than Markov's inequality. Specifically, for any $τ> 0$, $n \geq 1$, $η> e^3$, and $f: \{-1,1\}^n \to \mathbb{R}_+$ with $\int f dμ> 0$, 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 $μ$ is the uniform measure on the Boolean hypercube $\{-1,1\}^n$ and $c_τ$ is a constant that depends only on $τ$. This result resolves Talagrand's convolution conjecture up to a dimension-free $(\log \log η)^{\frac32}$ 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.

翻译：我们证明，在布尔超立方体上的热半群$(P_τ)$作用下，任意非负函数展现出优于马尔可夫不等式的均匀尾部界。具体而言，对于任意$τ> 0$，$n \geq 1$，$η> e^3$，以及满足$\int f dμ> 0$的函数$f: \{-1,1\}^n \to \mathbb{R}_+$，有 \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*} 其中$μ$为布尔超立方体$\{-1,1\}^n$上的均匀测度，$c_τ$是仅依赖于$τ$的常数。该结果将Talagrand卷积猜想推进至无维数因子$(\log \log η)^{\frac32}$。我们的证明采用了布尔超立方体上的反向热过程、精心设计的跳跃率扰动耦合构造以及时间平滑反集中估计。