Talagrand's convolution conjecture up to loglog via perturbed reverse heat
For researchers in probability and analysis on the hypercube, this provides a near-optimal resolution of a long-standing conjecture, though the result is not fully optimal due to the extra loglog factor.
The authors prove a uniform tail bound for the heat semigroup on the Boolean hypercube, improving over Markov's inequality by a factor of √log η, up to a loglog factor. This resolves Talagrand's convolution conjecture up to a (log log η)^{3/2} factor.
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.