All-temperature pressure for orthogonally invariant Ising spin glasses at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

A
D
V
E
R
T

I
S
E
M
E
N
T

All-temperature pressure for orthogonally invariant Ising spin glasses

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

thinking... about 3 hours remaining

If this game doesn't work on your computer, go here for help. (Lean version available!)

expertly designed by an internal OpenAI model

Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	luck, magnets	Levels:	1
Category:	Probability and statistical mechanics	Lean version:	YES! ✔
Rate this game! 4.0 out of 5 (1,072 votes)

>>> How to Play <<<

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.

>>> Level Select <<<

LEVEL 1 All-temperature pressure for orthogonally invariant Ising spin glasses

released 2026-09-25 | 3 theorems · 18 lemmas · 35 proofs · 25,461 words | PLAY LEVEL 1 » (pdf)

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.

More Probability and statistical mechanics Games!
Critical logarithmic corrections and BKT scaling for the planar XY model	The low-temperature Sherrington–Kirkpatrick fluctuation law	Conformal universality for weakly interacting and disordered Ising models	GOE universality for random regular graphs with weak disorder
Directional zero–one laws and ballisticity in random environments	The Mézard–Parisi formula for diluted spin glasses	Perceptron free energies and microscopic jamming	Conformal limits of square-lattice random-cluster interfaces

COOLMAITH GAMES SPONSORS: (not real)
Big Al's USED GPUs	Epsilon & Delta LLP attorneys at limit	QED Bail Bonds	Hilbert's Hotel always a vacancy!

Cool Links: openai/math Lean Mathlib arXiv the real Coolmath Games

You are visitor number 000719

Thanks for visiting CoolmAIth Games
© 1997-2026 CoolmAIth Games. A parody. Not affiliated with Coolmath.com LLC or OpenAI.
Every proof here was written by an internal OpenAI model and lives at github.com/openai/math.
Best viewed in Netscape Navigator 4.0 at 800x600