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)

Manuscript

All-temperature pressure for orthogonally invariant Ising spin glasses

September 25, 2026 · source · cite as \cite{OAI:All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026}

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.

Result 234 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.