Holden Lee

ALT 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< 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< 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].

ALT We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington–Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature β< 1/2. Prior work obtained TVD error guarantees only up to β≈ 0.295, while results covering the entire replica-symmetric regime β< 1 gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an O(1) Wasserstein error guarantee for finite-time localization, improving on the previous o(n). A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling t