Letwin: third-moment bound and covariance control - The KLS theorem and its methods

Part of the results of the literature written out here, Chapter Small-time operator-norm control of the covariance; the reading order is on the full proofs page.

Overview. A quadratic-variance bound controls the directional third-moment tensor by duality. Substitution into the published Klartag--Lehec fixed-time moment theorem gives a covariance window of order $1/\log n$ . Taking its first and second moments yields the claimed interface consequences. The independent published sup-over-time estimate supplies the shorter fallback window without the quadratic input.

Source matching. We use Klartag & Lehec, 2022, Corollary 5.4, checked in arXiv:2203.15551v2, and Klartag & Lehec, 2025, Theorem 61, checked in arXiv:2406.01324v2. In the former source $\|\cdot\|_2$ is the Schatten two-norm, hence exactly our Hilbert--Schmidt norm. Its parameter $\kappa_n$ is the one below. Corollary 5.4 has a time constant independent of the moment exponent; its moment constant depends on that exponent. The two citations concern, respectively, fixed-time moments and a supremum-over-time tail. We do not upgrade one into the other.

Proof 107.1

Fix the law and direction, and put $M=\mathbb E[\langle X,\theta\rangle X\otimes X]$ . It is a finite symmetric matrix. Centering and isotropy imply

\mathbb E\langle X,\theta\rangle=0,\quad \mathbb E\langle X,\theta\rangle^2=1,\quad \|M\|_{\mathrm{HS}}^2 =\mathbb E[\langle X,\theta\rangle(X^TMX-\operatorname{Tr}M)].

(107.2)

Cauchy--Schwarz and the quadratic-chaos antecedent give $\|M\|_{\mathrm{HS}}^4\le\operatorname{Var}(X^TMX) \le8\|M\|_{\mathrm{HS}}^2$ . If $M=0$ there is nothing to prove; otherwise divide by its squared norm. Taking both suprema gives the result, with no dimensional factor.

Proof 107.2

The published Corollary 5.4 states that, for an absolute $C$ ,

\mathbb E\|A_t\|_{\mathrm{op}}^p\le C_p \quad\hbox{if }0<t\le(C\kappa_n^2\log n)^{-1}.

(107.4)

Its proof combines Lemma 5.2’s fixed-time tail with $\|A_t\|_{\mathrm{op}}\le t^{-1}$ , giving $2^p+t^{-p}e^{-1/(Ct)}$ . For every real $p\ge1$ the second term has finite supremum: setting $y=1/t$ , it is $y^pe^{-y/C}$ , maximized at $y=Cp$ . Thus no integer restriction on $p$ or loss in the time window is needed. The smooth positive initial-density convention of that paper can be removed as follows.

Let $X\sim\mu$ and let $G,Z$ be independent standard Gaussians, independent of $X$ . For $\varepsilon>0$ put $X_\varepsilon=(X+\varepsilon G)/\sqrt{1+\varepsilon^2}$ . Its law $\mu_\varepsilon$ is smooth, positive, isotropic and log-concave, and converges weakly to $\mu$ . For fixed $t>0$ , write the tilted posterior covariance as $A^\mu(t,b)$ , with likelihood $e^{b\cdot x-t|x|^2/2}$ . The observation representation of localization says that $A_t$ has the law of $A^\mu(t,tX+\sqrt t Z)$ : this is the Gaussian Bayes likelihood, equivalently the law of the tilt parameter in the proof of Lemma 2.2 of the cited paper. Use the same representation for $\mu_\varepsilon$ and $b_\varepsilon=tX_\varepsilon+\sqrt t Z$ .

Almost surely $b_\varepsilon\to b=tX+\sqrt t Z$ . For $b$ in a compact set, each polynomial of degree at most two multiplied by $e^{b\cdot x-t|x|^2/2}$ is bounded and continuous in $x$ , with tails tending uniformly to zero as $|x|\to\infty$ . Weak convergence of the initial laws and $b_\varepsilon\to b$ therefore give convergence of their normalizing integrals and of their first two unnormalized moments. The limiting normalizer is positive. Dividing and subtracting the squared mean gives

A^{\mu_\varepsilon}(t,b_\varepsilon) \longrightarrow A^\mu(t,b) \quad\hbox{almost surely}.

(107.5)

Fatou transfers the uniform $p$ -moment estimate to the arbitrary initial law. At $t=0$ , $A_0=I_n$ , so the bound holds after enlarging $C_p$ to at least one. This argument needs no convergence of entire stochastic paths.

Finally Proposition 26.1, with its antecedent discharged by the input Theorem 25.1, gives $\kappa_n^2\le8$ . Taking $c=1/(8C)$ therefore places the asserted window inside the published window and proves the claim. The proof of Lemma 5.2 uses $\beta=2\log n>0$ and is applicable for every $n\ge2$ ; the stated dimensional restriction avoids interpreting $1/\log1$ .

Proof 107.3

Take $p=1$ in Corollary 26.1. Choose its universal $c_0$ smaller if necessary so that $c_0\le\log3$ , making $t_1\le1$ for $n\ge3$ . This supplies exactly the constants in Assumption 26.1, with $C_2=1$ . For $0\le t\le t_1$ , $X_t\le\|A_t\|_{\mathrm{op}}$ gives $\mathbb EX_t\le C_1$ . For $t>0$ , the posterior is $t$ -uniformly log-concave; the Brascamp--Lieb covariance cap is $A_t\preceq t^{-1}I$ . Thus on $[t_1,T]$ , $X_t\le t^{-1}$ , and integration proves the first displayed inequality. Since $T\le1$ and $n\ge3$ , its right side is at most $C_1+\log\log n+|\log c_0|$ , which proves the second after adjusting a universal constant. This repeats the interface integration used in Corollary 33.1; it claims no removal of the logarithmic loss and does not change any near-worstness or geometric premises downstream.

Proof 107.4

For the first bound, take $p=2$ in Corollary 26.1 and use $X_t^2\le\|A_t\|_{\mathrm{op}}^2$ . Integrating over any deterministic subinterval $I$ gives $\int_I\mathbb EX_t^2\,dt\le K_0|I|$ . Apply Corollary 31.2 to obtain the stated all-cut inequality. Its product implication is unconditional, but this verification on the longer window still uses the quadratic input.

For the independent fallback, use Theorem 26.1, the published Theorem 61 in the notes cited above. At any $0<t\le1/(C(\log n)^2)$ , the event $\{\|A_t\|_{\mathrm{op}}\ge2\}$ is contained in its sup-over-time event and has probability at most $e^{-1/(Ct)}$ . On the complement $X_t^2\le1$ , and on the event the covariance cap gives $X_t^2\le t^{-2}$ . Therefore

\mathbb EX_t^2\le1+t^{-2}e^{-1/(Ct)} \le1+4C^2e^{-2}.

(107.8)

The last bound follows by maximizing $y^2e^{-y/C}$ at $y=2C$ . At zero $X_0=0$ . Choose $c_0$ no larger than both available window constants and $K_0$ at least both available bounds. Integration again gives (V2). The product branch of Corollary 31.2 then yields its unconditional all-cut consequence; the general branch retains its Letwin antecedent. Both windows tend to zero with dimension, so neither statement is Assumption 28.1, which requires a universal positive horizon.

Dependencies, hypotheses and fences. The third-moment proposition is a conditional implication with Theorem 25.1 in assumes. The unconditional moment-window corollary uses that theorem and the third-moment proposition as proof dependencies. The two interface statements retain the manuscript’s Letwin antecedent. The second-moment statement also uses Theorem 26.1 and Corollary 31.2. No target has an explicit bounded_by edge. The statements avoid dimension one wherever $\log n$ is a denominator; the third-moment bound itself includes dimension one. No posterior singularity arises from a full-dimensional isotropic initial law at finite time, since its likelihood is positive. The bounds are fixed-time moments, with $C_p$ allowed to depend on $p$ ; there is no asserted dimension-free time window or moment of a time supremum.

References¶

Klartag, B., & Lehec, J. (2022). Bourgain’s Slicing Problem and KLS Isoperimetry up to Polylog. Geometric and Functional Analysis, 32(5), 1134–1159. 10.1007/s00039-022-00612-9
Klartag, B., & Lehec, J. (2025). Isoperimetric Inequalities in High-Dimensional Convex Sets. Bulletin of the American Mathematical Society, 62(4), 575–642. 10.1090/bull/1869