Result 234, Probability and statistical mechanics

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.

Lean formalization Classification or exact value

The bigger picture

Why it matters

In an Ising spin glass, binary spins interact through competing couplings. The manuscript reports an exact large-system formula that summarizes this competition at every fixed temperature for a class of interactions built from uniformly random rotations.

What changes?

Pressure is the logarithm of the total thermal weight of all spin configurations, divided by the number of spins. The manuscript gives an exact variational formula, expressed through an optimization, for its infinite-size limit at every fixed temperature. The coupling matrix has uniformly random orthogonal eigenvectors and deterministic eigenvalues whose empirical distribution approaches a compactly supported law. The extreme eigenvalues must approach that law's support edges. Pressure converges both almost surely and in expectation.

What does that help mathematicians do?

Taking temperature to zero yields a formula for the zero-field ground-state energy, connecting equilibrium pressure to the lowest energy attainable by spin configurations. The manuscript also treats deterministic external fields whose empirical distributions converge in first-moment transport distance: the average absolute displacement needed to match the distributions tends to zero. This extends the equilibrium description to qualifying nonuniform fields, rather than only systems without an applied field.

Are there practical applications?

The immediate value is foundational for statistical mechanics: it supplies an exact large-system target for studying disordered binary-spin models with prescribed limiting spectra. The zero-temperature result also characterizes an optimization problem, but a variational formula alone does not establish an efficient solver or practical predictions for particular magnetic materials.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

All-temperature pressure for orthogonally invariant Ising spin glasses

September 25, 2026 59 pages

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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Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/234.md.

All-temperature pressure of orthogonally invariant Ising spin glasses

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives variational limits for orthogonally invariant Ising spin glasses whose eigenvectors have Haar law and whose empirical eigenvalue distributions converge to a compactly supported law with the stated control of extreme eigenvalues. The pressure converges both in expectation and almost surely to an explicit functional of the limiting spectral law. A version with external fields assumes convergence of their empirical laws in Wasserstein distance and gives the corresponding magnetic-field functional.

The formalization also covers every positive temperature, a ground-state limit, random spectral data with the stated conditional Haar law, and Gaussian-pattern specializations. These hypotheses identify the invariant models to which the limits apply.

Comparator links

Result Comparator statement
Pressure and ground-state limits for invariant Ising models InvariantIsing.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.