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The Mézard–Parisi formula for diluted spin glasses
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Abstract
We prove the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. This resolves the variational equality conjecture for that class: the limiting pressure equals the infimum of the trial functional over all finite hierarchy depths and trial laws. Only first moments of the interaction and external field are required.
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Notes
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work in progress
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 math release, Lean scope notes for family 221 (release documentation, 2026-10-06). States that the formalization proves convergence of the finite-system pressure to the infimum of the trial functional over all finite hierarchy depths and trial laws, for even arity at least two and positive density, and links the comparator statement
DilutedSpin.lean. - OpenAI math release, DilutedSpin.lean (comparator statement) (release Lean code, 2026-10-06). The formal statement of the Mézard–Parisi variational equality against which the release's Lean proof is checked.
- OpenAI math release, OAI/Probability/DilutedSpin/Main.lean (openai/math at commit adc7f12). The solution module for the comparator challenge.
Related prior literature mentioned by others
- Dmitry Panchenko and Michel Talagrand, Bounds for diluted mean-fields spin glass models (arXiv:math/0405357; Probability Theory and Related Fields 130, 2004). Identified as the source of the equality conjecture, stated after Theorem 4, that the release manuscript addresses.
Possibly relevant literature (found by the agent, not checked)
- D. Panchenko, Hierarchical exchangeability of pure states in mean field spin glass models (arXiv:1307.2207; Probab. Theory Related Fields 161 (2015)). Identifies as the missing step toward the Mézard–Parisi ansatz that multi-overlaps of pure states depend only on their overlaps, which bears on the paper's stated main obstacle of controlling all finite multioverlaps.
- D. Panchenko, Structure of 1-RSB asymptotic Gibbs measures in the diluted p-spin models (arXiv:1308.1944; J. Stat. Phys. 155 (2014)). Earlier verification of the Mézard–Parisi ansatz for diluted p-spin models in the 1-RSB case, a precursor to the finite-RSB result [Pan16] that the paper cites.
- S. Franz, M. Leone, F. L. Toninelli, Replica bounds for diluted non-Poissonian spin systems (arXiv:cond-mat/0307367, 2003). Extends the Franz–Leone replica bounds beyond Poisson degree distributions and derives multi-overlap Ghirlanda–Guerra-type identities, relevant to the paper's restriction to Poisson dilution and to its multioverlap concentration step.
Digests and expositions
- OpenAI, The Mézard–Parisi formula for diluted spin glasses — reasoning summary (OpenAI Math Release, 2026). The published summary describes the model's proposed route through marked Poisson perturbations, shifted branching depths, averaged multioverlap concentration, and truncation to first-moment disorder.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 221
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026,
author = {{OpenAI}},
title = {{The M{\'e}zard--Parisi formula for diluted spin glasses}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf}{OAI:The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/221.md. The Lean statements have not been checked by people against the paper's claims.
Family 221: The Mézard–Parisi formula for diluted spin glasses
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.
This is the only paper in this family.