2025/11/24 by Chen, Yuansi
Mathematics · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2511.19374
We prove that under the heat semigroup (Pτ) 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 ≥ 1, η> e3, and f: \-1,1\n → ℝ+ with ∫ f dμ> 0, we have ℙX ∼ μ( Pτf(X) gt; η∫ f dμ) ≤ cτ\frac (log log η)\frac32 η√(log η), where μ is the uniform measure on the Boolean hypercube \-1,1\n and cτ 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.