Boolean Heat Semigroup Insights

• Boolean heat semigroup is a family of linear operators acting on functions on the hypercube, facilitating noise smoothing via diffusion processes.
• It employs both spectral (Fourier–Walsh) and Markovian constructions to exponentially damp higher order Fourier coefficients, enhancing signal regularity.
• Sharp, dimension-free tail bounds are established through reverse chains and perturbed coupling, nearly resolving Talagrand’s convolution conjecture.

The Boolean heat semigroup is a family of linear operators $(P_\tau)_{\tau\ge0}$ acting on functions on the discrete hypercube $\{-1,1\}^n$ and fundamental to discrete analysis, probabilistic combinatorics, and the modern theory of influences and noise sensitivity. It provides the canonical means of diffusion or noise-smoothing on the Boolean cube, forming the discrete counterpart to the classical heat semigroup on continuous spaces. Rigorous dimension-free tail estimates for its action under convolution have yielded resolutions of longstanding conjectures, notably Talagrand’s convolution conjecture up to tight $\log\log\eta$ factors (Chen, 24 Nov 2025).

1. Formal Definition and Characterizations

The Boolean heat semigroup $(P_\tau)_{\tau\ge0}$ acts on real-valued functions $f:\{-1,1\}^n\to\mathbb{R}$, preserving positivity and linearity. Two principal and equivalent constructions are standard:

• Generator (Markovian) Form: Consider the pure-jump Markov process $U_t$ on $\{-1,1\}^n$ with infinitesimal generator

$$L^U h(x) = \frac12 \sum_{i=1}^n [h(\sigma_i x) - h(x)],$$

where $\sigma_i x$ denotes flipping the $i$th coordinate of $x$. The induced semigroup is

$P_t h(x) = \mathbb{E}[h(U_t) \mid U_0 = x],$

satisfying $P_t = e^{t L^U}$. The law of $U_t$ is reversible with respect to the uniform measure $\mu$.

• Spectral (Fourier–Walsh) Form: Every $g:\{-1,1\}^n \to \mathbb{R}$ decomposes into a Walsh–Fourier expansion:

$$g(x) = \sum_{S\subseteq[n]} \hat{g}(S) x^S,\qquad x^S = \prod_{i\in S} x_i.$$

The semigroup acts diagonally:

$$P_t g(x) = \sum_{S\subseteq[n]} e^{-t|S|} \hat{g}(S) x^S.$$

Thus, $P_\tau f(x) = \sum_S \hat{f}(S) e^{-\tau|S|} x^S$. Spectrally, $P_\tau$ exponentially damps higher order Fourier–Walsh coefficients, reflecting its smoothing action.

2. Uniform Tail Bounds and Talagrand’s Convolution Conjecture

A central probabilistic problem is bounding the upper tail of $P_\tau f$ for nonnegative $f:\{-1,1\}^n\to\mathbb{R}_+$ under $\mu$. In (Chen, 24 Nov 2025), a dimension-free upper bound with sharp dependence on $\eta$ resolves Talagrand's conjecture up to a $\log\log\eta$ factor. For any $\tau>0$ and $\eta>e^3$, with $c_\tau>0$ depending only on $\tau$ and not on $n$ or $f$: $$\Pr_{X\sim\mu}\left(P_\tau f(X) > \eta\int f\,d\mu\right)\leq c_\tau\,\frac{\log\log\eta}{\eta\sqrt{\log\eta}}.$$ This rate is asymptotically optimal except for the $\log\log\eta$ factor, and dramatically outperforms Markov’s inequality in the high-$\eta$ regime. The result is dimension-free: all constants are independent of $n$, and $f$ need only be nonnegative with $\|f\|_{L^1(\mu)}\neq 0$.

3. Proof Architecture: Reverse Chains and Perturbed Coupling

The tail estimate is obtained via a novel argument involving the reverse heat process and a coupling construction leveraging controlled perturbations. The core stages are:

1. Anti-concentration Reduction: The problem reduces to bounding the $\nu_{P_\tau f}$-measure of shells of the form $\{y: P_\tau f(y) \in (\eta, e\eta]\}$, then summing over dyadic scales.
2. Reverse Process Construction:
   • The forward process interprets $P_t f$ as law at time $t$ of $U_t$ started from $\nu_f = f\cdot\mu$.
   • The reverse chain, $V_t = U_{T-t}$, equipped with a time-inhomogeneous generator,

   $$L^V_t h(x) = \frac12\sum_{i=1}^n \frac{f(\sigma_i(e^{-(T-t)}x))}{f(e^{-(T-t)}x)}[h(\sigma_i x) - h(x)],$$

   is scaled onto $[-1,1]^n$ for analytic tractability.

3. Perturbed Coupling: Introduce a second chain $W_t$ on $[-1,1]^n$ sharing the Poisson noise of $V_t$ but with jump rates slightly perturbed by state-dependent $\delta_i(x)$ up to a stopping time $\theta$. Two key properties:

   • Total Variation Closeness: At time $T-\tau$, $\mathrm{TV}(V_{T-\tau}, W_{T-\tau})$ is $O((e^{-\tau}/(1-e^{-\tau}))/\sqrt{\log\eta})$.
   • Monotonicity at the Tail: On $\{P_\tau f(W_{T-\tau}) \geq \eta\}$, $P_\tau f(V_{T-\tau})$ typically exceeds $P_\tau f(W_{T-\tau})$ by $+1$.

The coupling argument yields the anti-concentration needed for the main tail bound.

4. Central Analytical Estimates and Lemmas

Several critical lemmas drive the proof, each uniform in $n$ and depending on $\tau$ through $\alpha=(1-e^{-\tau})/(1+e^{-\tau})$:

• Level-1 Inequality (Lemma 4.1): For Boolean $h$ on $(-1,1)^n$,

$$\sum_{i=1}^n (1-x_i^2)[\partial_i h(x)]^2 \leq h(x) - h(x)^2.$$

• Expected Squared Score Bound (Lemma 4.2):

$$\int_0^{T_0}\sum_{i=1}^n S_i(V_s)^2\,ds \leq \frac{4}{\alpha}\left(\log\eta+\frac12\log\log\eta+O(1)\right).$$

• Reverse-Time Martingale Identity (Lemma 4.3): For blocked perturbations $W^{(k)}$, increments in $0$–$1$ observables can be represented as differences in smoothed expectations, reflecting gain in regularity from $P_\tau$.
• Exponential-Martingale Tail Controls (Lemmas 4.6–4.8): The log-martingale processes of $f(V_t)$ and $f(W_t)$ satisfy sharp concentration and drift inequalities, crucially supporting the +1 monotonicity in the coupled tail event.

5. Interpretation, Optimality, and Significance

The established tail bound for $P_\tau f$ is the first dimension-free, sharp (up to an unavoidable $\log\log\eta$) theorem for the Boolean heat semigroup convolution. For Talagrand’s convolution conjecture, this closes the problem—the $\log\log\eta$ factor is proved in (Chen, 24 Nov 2025) to be the last remaining gap. The methodology—reverse Markov process analysis, carefully engineered coupling, and multi-stage Duhamel interpolation—supersedes earlier approaches based on hypercontractivity, measure concentration, or isoperimetry, none of which yielded optimal $\eta$-dependence for large $\eta$ in the discrete setting.

A plausible implication is that these analytic and probabilistic tools, especially the reverse-process/coupling techniques, can be adapted to other high-dimensional, discrete diffusions where classical hypercontractivity becomes suboptimal.

6. Related Contexts and Connections

The Boolean heat semigroup is fundamental in several contexts:

• Noise Sensitivity and Influence: $P_\tau$ is the operator underlying discrete noise stability, crucial in the theory of influences.
• Discrete Isoperimetry: Many isoperimetric and concentration results exploit semigroup techniques, with $P_\tau$ acting as an averaging mechanism.
• Martingale Methods: The reverse-process and exponential-martingale constructions parallel tools in stochastic calculus and statistical mechanics.

The result achieves uniform control in $n$ and, by exploiting the Walsh–Fourier structure, leverages the hypercube’s rich algebraic and spectral symmetries, suggesting direct connections to discrete functional inequalities and the analysis of Boolean functions.

7. Tabulation of Principal Definitions and Estimates

Concept	Formula or Bound	Comments
Generator $L^U$	$\frac12\sum_{i=1}^n [h(\sigma_ix)-h(x)]$	Markov process on $\{-1,1\}^n$
Spectral form	$P_\tau f(x) = \sum_S \hat{f}(S) e^{-\tau|S|} x^S$	Walsh–Fourier diagonalization
Main tail bound	$$\Pr(P_\tau f > \eta\int f d\mu) \le c_\tau \frac{\log\log\eta}{\eta\sqrt{\log\eta}}$$	$c_\tau$ depends only on $\tau$
Level-1 inequality (Lemma 4.1)	$$\sum_i (1-x_i^2)(\partial_i h(x))^2 \le h(x) - h(x)^2$$	Boolean $h$ on $(-1,1)^n$
Expected squared-score (Lemma 4.2)	$$\int_0^{T_0}\sum_{i=1}^n S_i(V_s)^2\,ds \le \frac4\alpha(\log\eta+\frac12\log\log\eta+O(1))$$	$\alpha=(1-e^{-\tau})/(1+e^{-\tau})$

The Boolean heat semigroup constitutes the canonical analytical tool for smoothing and probabilistic estimates on the Boolean cube, and the tail bound of (Chen, 24 Nov 2025) sets a dimension-free optimal benchmark for nonlinear convolution inequalities.