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All-temperature pressure for orthogonally invariant Ising spin glasses
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Abstract
We determine the limiting pressure of an Ising spin glass with Haar orthogonal eigenvectors, a compact limiting spectral law, and no asymptotic outliers, at every fixed temperature. The pressure converges in expectation and almost surely to a variational formula. We also give the limiting pressure with deterministic external fields whose empirical laws converge in first-moment transport distance, and derive a formula for the zero-field ground-state energy by taking temperature to zero.
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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.
Formalization
- OpenAI, All-temperature pressure of orthogonally invariant Ising spin glasses: Lean scope note (lean/docs/234.md, OpenAI Math Release, 2026-10-06). The release's scope note covers zero-field and Wasserstein-field pressure limits at all positive temperatures, the ground-state limit, and random-spectrum and Gaussian-pattern specializations, and links the comparator statement
InvariantIsing.lean. [family-level] - OpenAI, InvariantIsing.lean (comparator statements) (openai/math, commit adc7f12). Lean 4 statements of 11 results matching the paper's theorems and corollaries, among them the Gaussian-pattern operator-norm limits; in this comparator file each one ends in
sorry, as challenge statements do. - OpenAI, Unconditional.lean (openai/math, commit adc7f12). The solution module: it derives the unconditional theorems from conditional versions and from inputs proved in other files of the release, such as
haarConcentration_provedandpanchenkoTalagrandFieldPair_proved.
Possibly relevant literature (found by the agent, not checked)
- G. Parisi and M. Potters, Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices (arXiv:cond-mat/9503009, J. Phys. A 28 (1995) 5267). Gives TAP equations and a replica count of metastable states for the random orthogonal Ising model, alongside the replica predictions the paper cites (Marinari–Parisi–Ritort; Cherrier–Dean–Lefèvre).
- Zhou Fan, Yufan Li and Subhabrata Sen, TAP equations for orthogonally invariant spin glasses at high temperature (arXiv:2202.09325, 2022-02-18; revised 2022-12-20). Characterizes the magnetization at high temperature through TAP equations and AMP convergence for the same orthogonally invariant Ising model, a possible comparison for the paper's external-field pressure formula.
- A. Maillard, L. Foini, A. Lage Castellanos, F. Krzakala, M. Mézard and L. Zdeborová, High-temperature Expansions and Message Passing Algorithms (arXiv:1906.08479, J. Stat. Mech. (2019) 113301). Derives free entropies for rotationally invariant couplings through Plefka-type expansions, which bears on how the paper's formula relates to replica-symmetric predictions expressed through the $R$-transform.
Digests and expositions
- Latent Seminar, All-temperature pressure for invariant Ising spin glasses (2026-10-06). Explains the spectral transform, entropy functional, normalization conventions and selected upper-bound and cavity arguments, while explicitly stating that it does not independently verify the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 234
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026,
author = {{OpenAI}},
title = {{All-temperature pressure for orthogonally invariant Ising spin glasses}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026/paper.pdf}{OAI:All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/234.md. The Lean statements have not been checked by people against the paper's claims.
Family 234: All-temperature pressure of orthogonally invariant Ising spin glasses
Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.
This is the only paper in this family.