Potential Hessian Ascent IV: Sampling the Sherrington-Kirkpatrick model at β< 1
Abstract
We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington-Kirkpatrick (SK) model with on(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 C2 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.
Related
- Sampling from the Sherrington-Kirkpatrick Model up to Criticality
- Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models
- Free energy fluctuation of soft spherical Sherrington-Kirkpatrick model
- Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model
- Parisi measure of the Sherrington-Kirkpatrick model with external field
BibTeX
@misc{lee2026potential,
title = {{Potential Hessian Ascent IV: Sampling the Sherrington-Kirkpatrick model at $β< 1$}},
author = {Holden Lee and Juspreet Singh Sandhu and Jonathan Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30590},
eprint = {2609.30590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30590},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Potential Hessian Ascent IV: Sampling the Sherrington-Kirkpatrick model at β< 1
AU - Holden Lee
AU - Juspreet Singh Sandhu
AU - Jonathan Shi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30590
UR - https://arxiv.org/abs/2609.30590
AB - 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.
ER -