Eldan's Stochastic Localization

• Eldan's stochastic localization is a measure-valued martingale process that deforms a logconcave density to include a strong Gaussian component, yielding sharper isoperimetric inequalities.
• The approach improves Cheeger, thin-shell, and log-Sobolev constant bounds, leading to enhanced concentration measures and faster mixing rates for high-dimensional random walks.
• It informs algorithmic design by providing tighter spectral gap estimates and reduced mixing times, directly benefiting sampling techniques in convex geometry.

Eldan's stochastic localization is a measure-valued continuous-time martingale process designed to deform an initial logconcave probability density into a form with a strong Gaussian component. This approach provides a systematic methodology for deriving dimension-dependent isoperimetric inequalities, concentration bounds, and mixing rates for high-dimensional random walks. The technique has led to significant improvements on central open problems in convex geometry, including bounds related to the KLS (Kannan–Lovász–Simonovits) conjecture.

1. Cheeger Constant and Isoperimetry Improvements

For any $n$-dimensional isotropic logconcave measure $p$ in $\mathbb{R}^n$, the Cheeger (KLS) constant $\psi_p$ is bounded as

$$\psi_p = O\bigl(\operatorname{Tr}(A^2)\bigr)^{1/4},$$

where $A$ is the covariance matrix of $p$. In the isotropic case, $\operatorname{Tr}(A^2) \lesssim n$, yielding

$\psi_p = O(n^{1/4}).$

This improves upon the previous best $O(n^{1/3}\sqrt{\log n})$ bound. The Cheeger constant governs fundamental inequalities such as Poincaré, thin-shell, and log-Sobolev constants. A smaller $\psi_p$ directly implies sharper concentration of measure, improved spectral gap estimates, and faster mixing for MCMC algorithms (e.g., ball walk).

2. Thin-Shell, Poincaré, and Slicing Constants

Consequent to the improved Cheeger constant, the thin-shell constant $\sigma_p$ and the isotropic (slicing) constant $L_p$ also satisfy $O(n^{1/4})$ bounds. Explicitly, for any Lipschitz function $g$: $$\operatorname{Var}_p(g) \leq \psi_p^2 \cdot \mathbb{E}_p(\|\nabla g\|^2).$$ Thin-shell estimates ensure that the norm $\|x\|$ for $x \sim p$ is concentrated within an annulus of width $O(1)$ around $\sqrt{n}$, refining the understanding of high-dimensional geometric distributions beyond previous results.

3. Ball Walk Mixing and Sampling Algorithms

The ball walk Markov chain with step size $\delta = \Theta(1/\sqrt{n})$ achieves mixing in $O(n^2 D)$ proper steps from any starting point, where $D$ is the support diameter, improving the prior $O(n^2 D^2)$ bound. For densities supported in $\ell_2$-balls of radius $O(\sqrt{n})$, the overall mixing time is $O^*(n^{2.5})$ from a warm start. This result is asymptotically tight and directly enhances the efficiency of algorithms for sampling from isotropic logconcave distributions.

The improved mixing bounds arise from the stochastic localization technique's ability to transfer isoperimetric information along the localization path, resulting in tighter control over the spectral gap and conductance of the random walk.

4. Stochastic Localization Martingale and Gaussian Factorization

At the core of Eldan's approach is the gradual transformation of the target density $p$ via a continuous-time martingale. The process generates a family $\{p_t\}_{t\geq0}$ such that

$$p_t(x) \propto e^{c_t^\top x - \frac{t}{2}\|x\|^2} p(x),$$

where $c_t$ evolves stochastically. This transformation introduces a significant Gaussian factor, particularly as $t$ increases—the terminal measure resembles a Gaussian times a logconcave remainder. Spectral control is maintained throughout by monitoring $\operatorname{Tr}(A_t^2)$, ensuring the measure does not become too anisotropic.

The martingale property guarantees that for any set $A$, the expected measure $\mathbb{E}[p_t(A)]$ remains constant. This probabilistic "tilting" is crucial for propagating concentration and isoperimetric properties from the Gaussian regime back to the original logconcave measure.

5. Log-Sobolev Inequalities

Via a refined localization argument employing a Stieltjes-type potential, the log-Sobolev constant $\rho_p$ of any isotropic logconcave density with support of diameter $D$ satisfies

$\rho_p = \Omega(1/D),$

which is sharp and improves upon the previous $\Omega(1/D^2)$ bound. This resolves a question posed by Frieze and Kannan (1997) and aligns the behavior of logconcave measures with Gaussian analogues. The enhanced log-Sobolev constant immediately translates into improved mixing upper bounds for reversible Markov chains.

6. Concentration and Small Ball Probabilities

For an $L$-Lipschitz function $g$ over an isotropic logconcave density $p$,

$$\Pr_{x \sim p}(|g(x) - \bar{g}| \geq L t) \leq \exp\left(-\frac{c t^2}{t + \sqrt{n}}\right),$$

where $\bar{g}$ is the mean or median of $g$. This generalizes prior estimates by Paouris and Guedon–Milman, providing sharper tail behavior even when the density is not compactly supported.

Additionally, for the small ball probability, if $\psi_p = O((\operatorname{Tr}(A^k))^{1/(2k)})$, then for $0 \leq \varepsilon \leq c_1$,

$$\Pr_{x \sim p}(\|x\|^2 \leq \varepsilon n) \leq \varepsilon^{c_2 / k \cdot n^{1 - 1/k}},$$

with universal constants $c_1,c_2$. For $k = 2$, this matches the best-known bounds $\varepsilon^{c\sqrt{n}}$ due to Paouris, quantifying the rarity of extreme deviations—a central aspect in convex geometry.

7. Methodological and Algorithmic Implications

Eldan's stochastic localization framework introduces a technical paradigm shift: it leverages measure-valued martingales to propagate geometric properties through a sequence of Gaussian-tilted densities. This enables one to establish dimension-dependent bounds for isoperimetric, spectral, and concentration phenomena in convex geometry. The approach has direct consequences for the design and analysis of high-dimensional algorithms, particularly those relying on efficient sampling, spectral gap estimates, and rapid mixing properties.

The following formulas encapsulate the principal results:

• Cheeger constant: $\psi_p = O(\operatorname{Tr}(A^2))^{1/4}$
• Poincaré inequality: $$\operatorname{Var}_p(g) \leq \psi_p^2 \cdot \mathbb{E}_p(\|\nabla g\|^2)$$
• Large deviation for $g$: $$\Pr\left(|g(x) - \bar{g}| \geq L t\right) \leq \exp\left(-\frac{c t^2}{t + \sqrt{n}}\right)$$
• Log-Sobolev constant: $\rho_p = \Omega(1/D)$

These contributions clarify and improve sharp bounds in convex geometry and have immediate ramifications for high-dimensional sampling and optimization, matching or surpassing previous best-known results across multiple structural inequalities for logconcave measures.