Family 6: parallel coupling of exponential tilts

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared; as presented in this manuscript, none of them uses the parallel coupling, the tool behind the thin-shell theorem. This chapter describes what the coupling controls on its own and what it does not give.

Object followed. The family of log-affine perturbations of $\mu$, its exponential tilts

followed jointly along stochastic localization: the localization processes started from $\mu$ and from its tilts are driven together, so that what one learns about $X$ can be compared across the whole family.

What it buys. Control of how the law reacts to a linear perturbation, uniformly in the direction of the perturbation. The tilts of (6.1) are the log-affine factor of the measures stochastic localization produces, before its Gaussian factor $e^{-t\abs x^2/2}$ (Chapter Family 2: stochastic localization), and they are the priors of the nonlinear-filtering reading of localization (Remark 2.1): observing $X$ through Gaussian noise turns the prior $\mu$ into a random exponential tilt of its Gaussian-weighted version $e^{-t\abs x^2/2}\mu$. Coupling the processes started from neighboring tilts measures how sensitive the posterior is to the prior along this $n$-parameter family, and that sensitivity is what the radial question asks about. Differentiating at $\theta=0$,

so, to first order, the tilts probe the correlation of a test function with linear functions, and nothing else. By (3.6), the thin-shell problem reduces to the low spectral mass of exactly those linear functions.

Sharpest result. The thin-shell theorem: for every isotropic log-concave $X$ in $\R^n$, $\E(\abs X-\sqrt n)^2\le C$ with $C$ universal Klartag & Lehec, 2025, a preprint (version 2). The coupling is combined there with Guan’s covariance technique Guan, 2024, already used in the proof of the slicing conjecture (Section Solved neighbors that are not KLS). Along the way the argument controls how many covariance eigenvalues of the localized measure are large, and for how long; these two consequences are recorded in this manuscript as Theorem 26.3 and Theorem 26.4 (Chapter Small-time operator-norm control of the covariance). The sharp constant, $\Var(\abs X^2)\le8n$, is the later Theorem 4.4.

What it does not reach alone. A coupling for functional perturbations $(1+\eps f)\mu$, with a cost controlled by $\int\abs{\nabla f}^2\dd\mu$ rather than by $\abs\theta^2$. Applied to a first eigenfunction $f$, such a coupling would test the slowest mode directly instead of its correlation with the coordinates. The family (6.1) is fixed before the test function is: it has $n$ parameters, and to first order, by (6.2), it sees a test function only through its covariance with linear functions. That is exactly enough for $\abs x^2$, whose derivatives are linear (Remark 3.1), and it says nothing about a first eigenfunction whose gradient is not. The parallel coupling is thus one more instance of the difficulty of Section Fixed and averaged, against adaptive and uniform: an estimate uniform over a fixed family, where a spectral argument needs one adapted to the extremal function. The rank estimates it produces count large covariance eigenvalues but do not orient them against a cut or an eigenfunction (the paragraph after Theorem 26.4).

Where this family meets the alternative mechanisms. Through its rank estimates only (Theorem 26.3, Theorem 26.4), which bear on the early-time covariance control of the fixed eigenfunction and of the fixed-cut archive but discharge none of their open estimates. Extending the coupling itself beyond linear tilts is the further direction of Section A further direction: parallel coupling beyond linear tilts; none of the alternative mechanisms carries it, and this manuscript formulates no statement for it.