We prove that under the heat semigroup $(P_τ)$ on the Boolean hypercube, any nonnegative function $f: \{-1,1\}^n \to \mathbb{R}_+$ exhibits a uniform tail bound that is better than that by Markov's inequality. Specifically, for any $η> e^3$ and $τ> 0$, \begin{align*} \mathbb{P}_{X \sim μ}\left( P_τf(X) > η\int f dμ\right) \leq c_τ\frac{ \log \log η}{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 only depends on $τ$. This resolves Talagrand's convolution conjecture up to a dimension-free $\log\log η$ factor. Its proof relies on properties of the reverse heat process on the Boolean hypercube and a coupling construction based on carefully engineered perturbations of this reverse heat process.

翻译：我们证明了在布尔超立方体上的热半群$(P_τ)$作用下，任意非负函数$f: \\{-1,1\\}^n \\to \\mathbb{R}_+$均展现出优于马尔可夫不等式给出的均匀尾部界。具体而言，对于任意$η> e^3$和$τ> 0$，\\begin{align*} \\mathbb{P}_{X \\sim μ}\\left( P_τf(X) > η\\int f dμ\\right) \\leq c_τ\\frac{ \\log \\log η}{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，$c_τ$为仅依赖于$τ$的常数。该结果将Talagrand卷积猜想在维度无关的$\\log\\log η$因子内予以解决。证明依赖于布尔超立方体上反向热过程的性质，以及基于对该反向热过程进行精细构造的扰动耦合方法。