Weak Poincaré Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model

Abstract: We develop a new method for proving a weak functional inequality by first proving it for a sufficiently regular sequence of distributions approximating the stochastic localization (SL) process, and then transferring it to the desired distribution via regularity of the SL process and conductance arguments. We use this strategy to prove a weak Poincaré inequality (WPI) holds for the Gibbs measure of the Sherrington-Kirkpatrick model when $β< \frac{1}{2}$. A prior result of the authors [arXiv:2605.03718, 2026] proves the ASL process for the Sherrington-Kirkpatrick model satisfies the required regularity conditions. A consequence of the WPI is that a much simpler algorithm -- Glauber dynamics with a warm-start -- efficiently samples the Gibbs measure of the SK model at $β< \frac{1}{2}$. This is a significant structural step towards resolution of the conjecture that Glauber dynamics mixes fast in the replica-symmetric regime for the Sherrington-Kirkpatrick model [arXiv:2504.20539, Open-Problem 15, 2025].

• The paper develops an approximate stochastic localization framework to establish weak Poincaré inequalities for complex high-dimensional measures.
• It introduces a three-step methodology—constructing the ASL process, transferring functional inequalities, and reducing algorithms—to derive explicit constants for the SK model.
• The results enable practical MCMC methods using Glauber dynamics with rapid mixing guarantees in the high-temperature phase.

Weak Poincaré Inequalities via Approximate Stochastic Localization with Application to Sampling the Sherrington-Kirkpatrick Model

Overview

This paper develops a framework for establishing weak Poincaré inequalities (WPI) for high-dimensional probability measures using approximate stochastic localization (ASL). The method is designed to relax the regularity requirements typically demanded by stochastic localization (SL), widening the class of distributions for which functional inequalities can be established. The approach is then instantiated on the Sherrington-Kirkpatrick (SK) spin glass model, yielding both new functional inequalities and efficient sampling algorithms via Glauber dynamics in the high-temperature/replica-symmetric phase.

Approximate Stochastic Localization and Weak Poincaré Inequalities

The key idea is to prove a WPI for a complicated high-dimensional measure $\mu$ (such as a Gibbs measure for a spin glass) via a three-step approximation pipeline:

1. Construct an ASL Process: Develop an SDE-driven process that approximates the true SL process up to controlled, bounded error, with relaxed regularity assumptions relative to classical localization.
2. Functional Inequality Transfer: Prove that if the localized (tilted) distributions produced by the ASL process satisfy WPIs with high probability, then (via conductance and coupling arguments) the original measure $\mu$ also satisfies a WPI, with controlled degradation in parameters.
3. Algorithmic Reduction: Demonstrate that for measures admitting such a WPI, efficient sampling from $\mu$ can be achieved by natural Markov chains (e.g., Glauber dynamics) started from a suitable warm start, obviating the need for more elaborate or non-standard algorithms.

The approach is technical: it relies on precise quantitative regularity assumptions on the approximate drift (magnetization) of the SDE, explicit $L^2$ error bounds between true and approximate processes (via Girsanov), and "decomposition theorems" that bridge conditional WPIs (on the locally tilted measures) with a global inequality for $\mu$.

The main theorem provides explicit WPI constants for $\mu$ in terms of the regularity and error control of the ASL process, and the WPI constants for the (random) localized measures. This enables application to models where only integrated (e.g., second-moment) control is available, a significant reduction in required regularity compared to classic SL arguments.

Application to the Sherrington-Kirkpatrick Model

The SK model is the canonical mean-field spin glass: configurations $\sigma \in \{\pm 1\}^n$ are sampled from the Gibbs measure

$$\mu_{\beta A}(\sigma) \propto \exp\left( \frac{\beta}{2} \sigma^\top A \sigma \right)$$

where $A$ is a GOE random matrix and $\beta$ is the inverse temperature.

Prior work had two main limitations:

• Functional inequalities (PI/LSI) were only known up to $\mu$0.
• Sampling algorithms (polynomial time, negligible TV distance) either only worked up to $\mu$1, or required complex algorithms leveraging the full machinery of ASL and reweightings via Jarzynski's equality.

The paper shows that (i) the required regularity assumptions for the ASL process (Lipschitzness and magnetization error bounds) hold for the SK model with high probability up to $\mu$2, and (ii) the localized SK measure satisfies a WPI with strength sufficient for Glauber dynamics to mix rapidly from a warm start.

The main corollary for the SK model is:

For any fixed $\mu$3, with high probability over $\mu$4, the Gibbs measure $\mu$5 satisfies a $\mu$6-WPI for all $\mu$7, with explicit dependence of constants on $\mu$8 and $\mu$9.

Thus, efficient sampling of SK states in the high-temperature phase can be achieved by:

1. Obtaining a warm start (using, e.g., Langevin dynamics for the annealed measure or a short run of ASL).
2. Running standard Glauber dynamics for $\mu$0 steps.

Technical Foundations

Decomposition Theorem for WPI

A novel technical contribution is the development of a decomposition theorem for WPIs. Given a mixture (or Gibbs-type) measure, if the conditional distributions satisfy WPIs, and the mixing dynamics between mixture components are sufficiently regular, then one can deduce a WPI for the overall measure, with explicit parameter control. This is achieved via careful analysis of the associated Dirichlet forms and a comparison argument reliant on a joint chain that alternates between conditional and marginal dynamics.

Conductance and Gaussian Regularization

The authors show that low-conductance "gaps" (weak connections between high-density regions) are overcome under additional Gaussian noising, as induced by the localization process. This is formalized via coupling arguments and the Gaussian isoperimetric inequality, which provides robust control over the boundary measure in high dimensions.

From Path Space to Finite Dimensions: Annealed Distributions

To obtain an explicit, efficiently samplable warm start, the authors introduce and analyze the annealed measure---a Gaussian reweighting by an explicit "free energy" function. They prove that this measure satisfies log-Sobolev inequalities with dimension-free constants, via delicate analysis on Wiener space (using the Cameron-Martin structure and perturbation theory), and show that sampling from this measure via discretized Langevin yields warm starts in $\mu$1 for the true SL measure.

Composition with Markov Chains

The warm-start reduction is closed via a two-phase algorithm:

1. Use annealed Langevin (or discretized ASL) to reach a warm start in $\mu$2 steps.
2. Run Glauber dynamics for $\mu$3 steps to converge in total variation, with the WPI ensuring rapid mixing from the warm start.

Compared to prior algorithms involving rejection sampling from complex ASL-based distributions, this approach is substantially simpler both in implementation and theoretical analysis.

Implications and Future Directions

This work broadens the practical and theoretical landscape for proving functional inequalities in high-dimensional spin glasses and other complex multimodal distributions. By lowering the barrier to the required regularity, it brings functional inequality theory into contact with algorithmic regimes that are less accessible using classic tools.

Practical implications:

• The results provide a concrete, analyzable, and implementable Markov chain Monte Carlo (MCMC) method for the SK model in temperature regimes where previously only intricate or fragile algorithms (or none at all) were available.
• The modularity of the decomposition approach suggests applicability to other mean-field disordered systems, including spherical models or models on more general state spaces.

Theoretical directions:

• Exploring the quantitative sharpness of the resulting WPI (e.g., improving constants to unlock simulated annealing reductions).
• Tightening the link between algorithmic stochastic localization and simulated annealing, both for SK and more general settings.
• Extending the path-space perturbation tools to more general non-Gaussian or non-symmetric models.
• Investigating the dependence of the method on the precise form of the error control between true and approximate stochastic localization.

Conclusion

The paper establishes a framework for proving weak Poincaré inequalities for high-dimensional, non-log-concave distributions using only approximate stochastic localization, and illustrates its efficacy via new results for the SK spin glass model. This leads to structurally simpler, rigorous sampling algorithms whose performance is underpinned by a transparent functional-analytic argument, rather than the more fragile mechanics of exact stochastic localization or operator norm bounds on the covariance. The methods introduced are likely to be influential for both theoretical and applied work in high-dimensional probability and sampling for complex random systems.