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

Yuansi Chen

TL;DR

Talagrand's convolution conjecture asks for a dimension-free tail bound stronger than Markov's inequality for the heat semigroup acting on $L^1$ functions on the Boolean cube. The authors develop a reverse-heat/coupling framework by perturbing the reverse jump dynamics and coupling two Markov processes through a joint generator, employing a multi-stage Duhamel approach to obtain an anti-concentration bound. The main result yields a dimension-free tail bound of order $\frac{\log\log \eta}{\eta\sqrt{\log \eta}}$ (up to a $\tau$-dependent constant) for $\eta>e^3$, thereby resolving Talagrand's conjecture up to a $\log\log \eta$ factor. This work connects discrete heat-regularization on the Boolean cube with Gaussian-semigroup ideas via a carefully engineered perturbation, advancing understanding of regularization phenomena in high-dimensional discrete spaces and offering a blueprint for similar coupling-based strategies.

Abstract

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.

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

TL;DR

Talagrand's convolution conjecture asks for a dimension-free tail bound stronger than Markov's inequality for the heat semigroup acting on functions on the Boolean cube. The authors develop a reverse-heat/coupling framework by perturbing the reverse jump dynamics and coupling two Markov processes through a joint generator, employing a multi-stage Duhamel approach to obtain an anti-concentration bound. The main result yields a dimension-free tail bound of order (up to a -dependent constant) for , thereby resolving Talagrand's conjecture up to a factor. This work connects discrete heat-regularization on the Boolean cube with Gaussian-semigroup ideas via a carefully engineered perturbation, advancing understanding of regularization phenomena in high-dimensional discrete spaces and offering a blueprint for similar coupling-based strategies.

Abstract

We prove that under the heat semigroup on the Boolean hypercube, any nonnegative function exhibits a uniform tail bound that is better than that by Markov's inequality. Specifically, for any and , \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 is the uniform measure on the Boolean hypercube and is a constant that only depends on . This resolves Talagrand's convolution conjecture up to a dimension-free 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.

Paper Structure

This paper contains 33 sections, 15 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. A few comments on the conjecture
  3. Remarks on the Gaussian counterpart
  4. Notation
  5. Preliminaries
  6. Boolean function analysis basics
  7. Inner product for functions on $\left\{ -1,1 \right \}^n$.
  8. Multilinear expansion of a function on $\left\{ -1,1 \right \}^n$
  9. Explicit form of the heat semigroup
  10. Continuous-time Markov processes
  11. Homogeneous case
  12. In-homogeneous case
  13. Heat semigroup and its associated Markov process
  14. Forward jump process
  15. Time reversal of a Markov process
  16. ...and 18 more sections

Key Result

Theorem 1

For every $\tau > 0$, there exists a universal constant $c > 0$, such that for every nonnegative function $f: \left\{ -1, 1 \right \}^n \to \mathbb{R}_+$ with $\left\| f\right\|_{1} \neq 0$, and any $\eta > e^3$, we have

Theorems & Definitions (37)

  • Conjecture 1: talagrand1989conjecture
  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1: Anti-concentration
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 2: Total variation control
  • Lemma 3: Approximate monotone coupling at tail
  • proof : Proof of Lemma \ref{['lem:anti_concentration']}
  • Remark 3
  • ...and 27 more