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Stretched-exponential barriers for typical SK initial states
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Abstract
At every fixed inverse temperature β > 1, we prove a stretched-exponential obstruction to mixing for the zero-field Sherrington–Kirkpatrick model from typical equilibrium configurations held fixed as initial states. The Gibbs mass of states whose total-variation distance from equilibrium at time $\exp(n^{1/10000})$ exceeds 1/4 tends to one in probability over the disorder. This holds for rate-one-per-site heat-bath dynamics and after the same stated number of discrete update attempts.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- P. M. Aronow, Patrick Lopatto, Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model (hexagon:2610.00402v1, 20 Oct 2026). Follows this paper's proof structure and improves the typical-start lower bound from time exp(n^{1/10000}) to exp(A_β n^{7/30}) for every fixed β > 1, using a three-replica comparison with a cubic pressure deficit. [AI-generated per source]
Formalization
- OpenAI, Family 227 Lean scope (lean/docs/227.md, 2026-10-06). States that the release's Lean development proves the main theorem for fixed $\beta>1$: at time $\exp(n^{1/10000})$, the expected Gibbs mass of starts still farther than 1/4 from equilibrium tends to one, for both per-site continuous time and discrete update attempts (comparator
SKBarriers.lean). [family-level] - P. M. Aronow, Patrick Lopatto, Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model (hexagon:2610.00402v1, 20 Oct 2026). Reports a Lean 4 formalization of the strengthened n^{7/30} barrier; the source says it uses a weaker form of the local expansion than the written proof. [AI-generated per source]
Digests and expositions
- Latent Seminar, SK dynamics across the temperature transition (2026-10-06). Explains the eight companions and selected proof mechanisms without independently verifying their proofs. [family-level]
Elsewhere
- Hexagon: Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model (AI-generated; human understanding: some)
- Emergent Mind: family 227
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026,
author = {{OpenAI}},
title = {{Stretched-exponential barriers for typical SK initial states}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf}{OAI:Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/227.md. The Lean statements have not been checked by people against the paper's claims.
Family 227: Critical SK autocorrelation processes and dynamics across the temperature transition
For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $\log n$ scale for fixed $0\le\beta\lt 1$, mixing time $n^{2/3+o(1)}$ at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.
Other papers in this family:
- Universality of critical quench autocorrelations in the Sherrington–Kirkpatrick model · October 5, 2026
- Functional universality of critical SK autocorrelations · October 5, 2026
- A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase · September 24, 2026
- Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase · September 24, 2026
- Critical slowing down in the Sherrington–Kirkpatrick model · September 24, 2026
- A typical-start upper bound for low-temperature SK Glauber dynamics · September 24, 2026
- Critical mixing in the Sherrington–Kirkpatrick model · September 25, 2026