Prelude: what stochastic localization does, and what it costs

This chapter is a short primer on stochastic localization, written for the two mechanisms that follow one object through it: the fixed eigenfunction (Chapter The fixed eigenfunction: following one eigenfunction through localization) and the fixed cut kept in the archive (Chapter The fixed cut: approach and lessons). It states no identity precisely and proves nothing. Each exact statement is linked where it is first needed, and the full apparatus is in the technical foundations, Chapters Analytic conventions and the two-color localization setup to Model geometries: the Gaussian and product brackets. A reader willing to take the mechanism on trust can go directly to either chapter.

The process, and why one would want it¶

The difficulty in KLS is that a log-concave measure can be badly conditioned in a way no single observable sees. Stochastic localization is a way of improving the measure continuously while keeping track of what the improvement costs.

One runs a measure-valued process $p_t$, started at $\mu$, in which the density is multiplied by a Gaussian factor driven by a Brownian motion: informally, at time $t$ the measure has been tilted by $e^{\inner{\theta_t}{x}-t|x|^2/2}$ with $\theta_t$ a martingale. Three things happen at once, and all three matter.

(1) The measure becomes strongly log-concave. The quadratic factor $e^{-t|x|^2/2}$ makes $p_t$ at least $t$-strongly log-concave, so by Brascamp–Lieb its Poincaré constant is at most $1/t$. Waiting is therefore free progress: at any time $t$, the localized measure is as good as a Gaussian of variance $1/t$. Formally this is (23.10).

(2) The measure is preserved on average. $p_t$ is a martingale in the measure: $\E p_t=\mu$ for every $t$. Nothing is lost by running the process; the information is redistributed, not destroyed. This is what makes step (3) legitimate.

(3) The covariance moves, and can move badly. The conditional covariance $A_t=\Cov(p_t)$ obeys its own SDE, (23.7), whose drift is not sign-definite. This is the only place a cost is incurred, and the whole of both approaches is about paying it.

Transferring an isoperimetric statement back¶

The reason (2) is worth having is that isoperimetry transfers. Fix a would-be bottleneck set $E$ and follow its mass $m_t=p_t(E)$. Because $p_t$ is a measure martingale, $m_t$ is a bounded martingale, so it converges; and its quadratic variation is exactly an integral of the correlation between $\one_E$ and the localization direction. Two readings of the same fact drive the two approaches.

If the mass stays balanced for a while, then at that time the localized measure is both $t$-strongly log-concave and still genuinely cut by $E$, and a strongly log-concave measure with a balanced cut has boundary. Integrating that back through the martingale gives a lower bound on $\mu^+(E)$, which is a Cheeger statement about $\mu$ itself. This is Lemma 30.1, and its stopped form, Theorem 30.1, is the exact bridge the fixed-cut approach uses.

If the mass is identified too quickly — $m_t$ rushing to 0 or 1 — then the quadratic variation was large, which means the cut was strongly correlated with the localization direction, which is information about the geometry rather than a failure. The fixed-cut approach is the attempt to show that the second case cannot happen for every balanced cut at once.

The fixed-eigenfunction approach changes one object and nothing else: it follows $M_t(f)=\E_{p_t}f$ for a fixed near-first eigenfunction $f$ instead of $m_t=\E_{p_t}\one_E$. The SDE is the same shape, with $\Cov_{p_t}(X,f)$ in place of the set correlation, and the payoff is that the tensor orientation of $f$ survives, where a set indicator has already discarded it.

Source against damping: how to read the Riccati equation¶

Everything technical in the two approaches is a contest between two terms, and it is worth naming them before meeting them. Along the process, the scalar quantity the argument actually tracks obeys an equation of the form

a martingale increment plus a drift split into exactly one positive source $S_t$ and one coercive damping $D_t$. This is Theorem 24.1; the matrix identity behind it is Lemma 24.1, and both are derived in Chapter The two-color Riccati identities.

The source is where the argument can lose. It measures how strongly the localization direction is correlated with the object being followed, and Section The Stein dictionary gives it a Stein representation — Lemma 32.1 converts between a Stein norm and the source while the mass stays near balance — which is what makes it estimable at all. The damping is coercive, $D_t\gtrsim r_t^2$, so a bounded source is absorbed: the process cannot run away. Every argument in either approach is, in the end, an attempt to absorb the source into the damping for long enough.

The gap between what can be absorbed and what can be estimated has a name, and it is the same gap in both approaches: control is available at trace scale and needed at operator scale. Section The Stein dictionary states that operator-to-trace gap explicitly, and Conjecture 29.1 in Chapter The fixed cut: remaining problems is the fixed cut’s version of it.

The warning that constrains both approaches¶

One thing must be carried into both approaches from Section Obstacles for alternative arguments, because it rules out the argument a reader is most likely to try to construct.

There is no uniform bound on $\norm{A_t}_\op$ along the path, even for measures that satisfy KLS. Proposition 0.1 exhibits the counterexample, and it is a product of centered exponentials — a measure that is dimension-free by tensorization.

So an argument may not simply bound the covariance better. It must either follow an object that notices when a spike is harmless — the fixed cut follows one cut, the fixed eigenfunction one eigenfunction, and both retain structure that $\norm{A_t}_\op$ has thrown away — or leave the method entirely, which is what the moment map and conditional fibers do. The covariance control of Chapter Small-time operator-norm control of the covariance is correspondingly a small-time theory: it controls the covariance for a short while, not forever, and the two localization arguments are built to need only that.

Product measures are the standing stress test for exactly this reason, and Chapter Model geometries: the Gaussian and product brackets collects them alongside the Gaussian model, where the covariance is deterministic, damping is active and the excess vanishes identically. The two models bracket the problem: whatever a proposal does, it must be right on both.

Where to go from here¶

Chapter The static quadratic-chaos input and the two-tail obstruction states the static input the fixed-cut approach uses and the two-tail obstruction that limits what it can supply; it sits among the shared foundations but is worth reading before the fixed-cut chapters, because it is an obstruction, not a tool. Chapter The fixed eigenfunction: following one eigenfunction through localization then opens the fixed eigenfunction, and Chapter The fixed cut: approach and lessons the fixed-cut archive.