Family 2: stochastic localization - The KLS theorem and its methods

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared, and all three run a stochastic localization at some step. This chapter describes what the process controls on its own and what it does not give without the polynomial estimates of those proofs.

Object followed. A measure-valued martingale $t\mapsto\mu_t$ , together with its centroid $a_t$ and covariance $A_t$ , along a Brownian filtration.

What it buys. Every general improvement of the KLS bound since 2012 has used this construction. It replaces the needle dichotomy of Chapter Family 1: classical needle localization — “one dimension, no covariance” — by a process that keeps the ambient dimension and tracks covariance explicitly, at the price of controlling it only in law.

This chapter fixes the process and the three identities used by the two stochastic-localization approaches of this manuscript, the fixed cut and the fixed eigenfunction. This manuscript’s own two-color refinement of them is Chapter Analytic conventions and the two-color localization setup onward; what follows is the scalar/matrix backbone as it appears in the literature.

The process¶

In the Lee–Vempala form of Eldan’s process Lee & Vempala, 2018Lee & Vempala, 2024, start from $p_0=p$ and define

p_t(x)=\frac{\exp\bigl(\inner{c_t}{x}-\frac t2|x|^2\bigr)\,p(x)} {\int\exp\bigl(\inner{c_t}{y}-\frac t2|y|^2\bigr)\,p(y)\dd y},

(2.1)

where

\dd c_t=\dd W_t+a_t\dd t, \qquad a_t=\E_{p_t}X, \qquad A_t=\Cov(p_t).

(2.2)

Itô calculus gives the three identities that carry the whole method:

\begin{aligned} \dd p_t(x)&=p_t(x)\inner{x-a_t}{\dd W_t}, \end{aligned}

(2.3)

\begin{aligned} \dd a_t&=A_t\dd W_t, \end{aligned}

(2.4)

\begin{aligned} \dd A_t&=\calT_t(\dd W_t)-A_t^2\dd t, \end{aligned}

(2.5)

where the matrix-valued third-moment tensor is

\calT_t(u)=\E_{p_t}\bigl[(X-a_t)^{\otimes2}\inner{X-a_t}{u}\bigr].

(2.6)

Two features are decisive, and they pull in opposite directions:

(i) By (2.3), $p_t(x)$ is a measure-valued martingale: $\E p_t(x)=p_0(x)$ for every $x$ . Whatever is proved about $p_t$ can therefore be averaged back to time zero.

(ii) The factor $e^{-t|x|^2/2}$ in (2.1) makes $p_t$ $t$ -strongly log-concave. Curvature is manufactured, at a known rate, out of nothing.

Feature (ii) creates the good event; feature (i) transports it back. The entire difficulty is that the drift $-A_t^2\dd t$ in (2.5) that would keep $A_t$ small competes with the noise $\calT_t(\dd W_t)$ , which is a third-moment quantity.

How isoperimetry is transferred back¶

Fix a Borel set $E$ and let $g_t=p_t(E)$ . By (2.3), $g_t$ is a martingale with

\dd g_t=\Bigl\langle\int_E(x-a_t)\,p_t(x)\dd x,\ \dd W_t\Bigr\rangle,

(2.7)

whose quadratic variation is controlled by $A_t$ . Lee–Vempala’s criterion is the clean statement of the transfer Lee & Vempala, 2018:

\Prob\Bigl\{\int_0^T\norm{A_s}_\op\dd s\le\tfrac1{64}\Bigr\}\ge\tfrac34 \qquad\Longrightarrow\qquad \PsiKLS_{p_0}\lesssim T^{-1/2}.

(2.8)

The mechanism has three steps, and it is worth separating them because the fixed-cut and fixed-eigenfunction approaches each modify exactly one of them:

(1) the covariance bound keeps $g_T$ away from 0 and 1 with positive probability — the cut is not identified too fast;

(2) $p_T$ is $T$ -strongly log-concave, hence $h(p_T)\gtrsim\sqrt T$ by Bakry–Émery;

(3) averaging the boundary measure of $E$ under $p_T$ back to time zero, using the martingale property, transfers that expansion to $p_0$ .

So KLS becomes a question about the covariance process — and specifically about its largest eigenvalue, which is where (2.8) charges its cost. The fixed-cut approach replaces step (1) by a two-color estimate for one fixed $E$ (Section The mass martingale); the fixed-eigenfunction approach replaces the set $E$ by a first eigenfunction (Chapter The fixed eigenfunction: following one eigenfunction through localization).

Why third moments appear¶

Take the basic Schatten-2 potential $\Phi_t=\Tr(A_t^2)$ . Itô’s formula applied to (2.5) gives

\dd\Phi_t =\dd M_t -2\Tr(A_t^3)\dd t +\E_{X,Y\sim p_t}\inner{X-a_t}{Y-a_t}^3\dd t ,

(2.9)

with $M_t$ a martingale. The term $-2\Tr(A_t^3)$ is the stabilizing drift inherited from $-A_t^2$ . The positive Itô correction is exactly the squared Hilbert–Schmidt norm of the third-moment tensor: writing $a=a_t$ ,

\E_{X,Y\sim p_t}\inner{X-a}{Y-a}^3 =\sum_{i,j,k}\Bigl(\E\bigl[(X-a)_i(X-a)_j(X-a)_k\bigr]\Bigr)^2 .

(2.10)

Controlling this noise term while retaining information about $\lmax(A_t)$ is the central stochastic-localization calculation, and it is why a sharp bound on the third-moment tensor — the parameter $\kappa_n$ of (0.13) — is the crucial input rather than an incidental one.

Evolution of the method¶

Paper	Main technical device	Result for $\PsiKLS_n$
Eldan Eldan, 2013	Covariance-normalized localization $C_t=A_t^{-1}$ , high Schatten powers, relation to thin shell	$n^{1/3}\sqrt{\log n}$ with the thin-shell bounds of 2012
Lee–Vempala Lee & Vempala, 2018Lee & Vempala, 2024	Fixed Gaussian tilt and the $\Tr(A_t^2)$ stopping argument	$n^{1/4}$
Chen Chen, 2021	Iterated high-Schatten potentials, affine preconditioning, induction on dimension	$\exp(C\sqrt{\log n\log\log n})$
Klartag–Lehec Klartag & Lehec, 2022	Heat-flow duality, spectral projections, $H^{-1}$ , covariance growth	$(\log n)^5$
Jambulapati–Lee–Vempala Jambulapati et al., 2022	Two localization representations and spiked-spectrum estimates	$(\log n)^{3.2226}$
Klartag Klartag, 2023	Improved Lichnerowicz plus short-time covariance control	$\sqrt{\log n}$
Letwin, v1 Letwin, 2026	Sharp quadratic Poincaré $\Rightarrow\kappa_n=O(1)$ , inserted into Klartag’s bridge	$(\log n)^{1/4}$ (July 2026 preprint)
Song–Zhang, v1 Song & Zhang, 2026	Appell coefficient bounds fed into curvature profiles and back through localization	$4^{\log^*(n+2)}$ (1 October 2026 preprint)
Bizeul–Klartag–Lehec (BKL) Bizeul et al., 2026	Cumulants of every order along a covariance-normalized localization, and suspension	$O(1)$ (4 October 2026 preprint)
Song–Zhang, v2 Song & Zhang, 2026	Repeated refinement of one coefficient radius with summable losses	$O(1)$ (4 October 2026 preprint)
Balasubramanian–Kasiviswanathan (BK) Balasubramanian & Kasiviswanathan, 2026	Compatible integration and a direct Appell induction	$O(1)$ , explicit: $\PsiKLS_n\le\sqrt{\pi(1+2\cdot10^{16})}$ (6 October 2026 preprint)

Where the surviving logarithm lives¶

This subsection makes precise the claim of Section Letwin: localization and improved Lichnerowicz, because it is what an effective-rank potential would have to remove (Section An effective rank in place of the soft maximum).

To use (2.8) one needs a potential that both dominates $\lmax(A_t)$ and admits an Itô calculus. Klartag–Lehec use log-trace-exp,

\Phi_t=\frac1\beta\log\Tr\bigl(e^{\beta A_t}\bigr), \qquad \norm{A_t}_\op\le\Phi_t\le\norm{A_t}_\op+\frac{\log n}\beta .

(2.11)

The two-sided bound in (2.11) is the whole story. Approximating $\lmax$ to within a constant requires $\beta\asymp\log n$ . The Itô drift generated by the noise (2.6) then has size $O(\kappa_n^2\log n)$ , so covariance control survives only up to

t_*\asymp\frac1{\kappa_n^2\log n} .

(2.12)

Improved Lichnerowicz (Theorem 3.1) converts a curvature time into a Poincaré constant,

\CP(\mu)\lesssim t_*^{-1/2}\lesssim\kappa_n\sqrt{\log n},

(2.13)

which is the bridge (0.14).

Remark 2.2 (The logarithm is an entropy cost, not a moment cost)

The $\log n$ in (2.11) is the entropy of the uniform distribution on $n$ directions: it is what one pays to replace a maximum over $n$ eigenvalues by a smooth surrogate, and it would be present even if $\kappa_n$ were known exactly and equal to 1. Since $\kappa_n=O(1)$ is now available (Proposition 26.1), this term accounts for the remaining dimension dependence in Letwin’s use of this bridge. The Song–Zhang, BKL and BK arguments use different conversions, described in Chapters Song–Zhang, first version: polynomial estimates and curvature, Bizeul–Klartag–Lehec: cumulants and suspension, Song–Zhang, second version: repeated refinement with summable losses and Balasubramanian–Kasiviswanathan: compatible integration. A potential that depends only on the directions actually relevant to a near-extremizer, or on an effective rank rather than the ambient dimension $n$ , would convert $\kappa_n=O(1)$ into $\CP=O(1)$ .

What it does not reach alone. Control of $\lmax(A_t)$ along the whole path at a universal time, without paying the $\log n$ of (2.11). Sharpening the covariance bound cannot supply it: a uniform pathwise bound on $\norm{A_t}_\op$ is false even for measures that satisfy KLS (Section The covariance spike, and why the direct repair fails). A potential that recognizes when a covariance spike is harmless is needed instead, and products of centered exponentials, the canonical harmless spike, are the stress test for any such potential (Chapters Model geometries: the Gaussian and product brackets and The fixed cut: product stress test). The three proofs of KLS do not supply this control either: they replace the conversion through $\lmax(A_t)$ by spectral criteria on polynomials of every degree (Chapter The proofs of KLS compared).

Where this family meets the alternative mechanisms. It is the engine of the two localization arguments, which differ only in what they refuse to average away. The fixed eigenfunction (Chapter The fixed eigenfunction: following one eigenfunction through localization) keeps one fixed eigenfunction and its covariance tensor; the fixed cut, kept as an archive (Chapter The fixed cut: approach and lessons), keeps one fixed cut and its two-color covariance. The conceptual summary they share is Chapter Prelude: what stochastic localization does, and what it costs, and the apparatus is in the shared technical foundations, Chapters Analytic conventions and the two-color localization setup–Model geometries: the Gaussian and product brackets.

References¶

Lee, Y. T., & Vempala, S. S. (2018). The Kannan–Lovász–Simonovits Conjecture. https://arxiv.org/abs/1807.03465
Lee, Y. T., & Vempala, S. S. (2024). Eldan’s Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing. Annals of Mathematics, 199(3), 1043–1092. 10.4007/annals.2024.199.3.2
Eldan, R. (2013). Thin Shell Implies Spectral Gap up to Polylog via a Stochastic Localization Scheme. Geometric and Functional Analysis, 23(2), 532–569. 10.1007/s00039-013-0214-y
Chen, Y. (2021). An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geometric and Functional Analysis, 31(1), 34–61. 10.1007/s00039-021-00558-4
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
Jambulapati, A., Lee, Y. T., & Vempala, S. S. (2022). A Slightly Improved Bound for the KLS Constant.
Klartag, B. (2023). Logarithmic Bounds for Isoperimetry and Slices of Convex Sets. Ars Inveniendi Analytica, 2023(4), 1–17. 10.15781/jsjy-0b06
Letwin, B. (2026). The KLS Constant is O(\log1/4 n).
Song, Z., & Zhang, X. (2026). An O(4\log^* n) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v1
Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a Proof of the Kannan–Lovasz–Simonovits Conjecture. https://arxiv.org/abs/2610.05474v1
Song, Z., & Zhang, X. (2026). An O(1) Bound for the KLS Constant. https://arxiv.org/abs/2610.01447v2
Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728v1