Weak Poincar\'e Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model

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 $\beta < \frac{1}{2}$. 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 $\beta < \frac{1}{2}$. 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].