Potential Hessian Ascent IV: Sampling the Sherrington-Kirkpatrick model at $β< 1$

2026-09-24

AI Summary

This study addresses the challenge of polynomial-time efficient sampling from the Gibbs measure of the Sherrington-Kirkpatrick model at low-temperature parameters $\beta<1$. The proposed method employs the TAP free energy as a surrogate, integrating algorithmic stochastic localization with rejection sampling based on the Jarzynski equality. By overcoming global regularity limitations through local strong convexity analysis, this work extends the applicable regime from $\beta<1/2$ to the full range $\beta<1$ and introduces a general local regularity framework. Furthermore, it synthesizes cavity interpolation theory with tools from free probability. Ultimately, this approach achieves high-precision approximate sampling in polynomial time with vanishing total variation distance error, significantly enhancing the simulation efficiency of complex spin glass models.

Abstract

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick (SK) model with $o_n(1)$ error in total-variation distance (TVD) at any inverse-temperature $β< 1$. The algorithm combines algorithmic stochastic localization (ASL) with rejection sampling over path-space via Jarzynski's equality (JE). The analysis extends the authors' prior $β< 1/2$ result [arXiv:2605.03718] by replacing all global regularity requirements in the stochastic differential equation (SDE) error analysis with local regularity around likely trajectories. The relaxed regularity is established using Celentano's proof of the local strong convexity of the TAP free energy [arXiv:2208.09550]. The analysis utilizes the cavity interpolation theory and free probability toolkit developed in the authors' previous result, where the former applies nearly verbatim and the latter applies supplemented with Lipschitz and $C^2$ extensions of various functions. The ASL and JE analysis arises from using the TAP free energy as an efficiently computable proxy for the actual free energy of the stochastically localized Gibbs measure [$ §$ 3, arXiv:2605.03718]. We give a list of \vocab{desiderata} encapsulating the approximation and regularity properties required of the TAP free energy, relaxing those of [$ §$ 2.5, arXiv:2605.03718] to only require local regularity. These generic desiderata are potentially applicable in other settings where a free energy surrogate exists, giving algorithmic sampling guarantees while bypassing the usual functional inequalities based approach.

Problem

• Sherrington-Kirkpatrick model
• Gibbs measure sampling
• inverse temperature
• total-variation distance
• polynomial-time algorithm

Innovation

• Sherrington-Kirkpatrick model
• Algorithmic Stochastic Localization
• Jarzynski's equality
• TAP free energy
• Local regularity