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Overlap distribution of the critical Sherrington-Kirkpatrick model

Hang Du, Brice Huang

At inverse temperature β=1, the quenched law of N^(1/3) times the two-replica overlap converges to an explicit random Airy₁-based measure for both Ising and spherical Sherrington–Kirkpatrick models. Its second moment gives the limit of N^(2/3) E⟨overlap²⟩, answering Talagrand’s Conjecture 11.7.5.

Sources for this record

What was already known

Talagrand posed the critical second-moment question. Du and Huang’s earlier work established its N^(−2/3) order, but not the limit constant or quenched law.

AI involvement AI-centered

Du and Huang credit GPT-5.6 Pro with most formal arguments and a Hilbert-space kernel idea. They supplied the limiting objects, comparison theorem formulation and overall spherical-to-Ising strategy.

The credited model-generated arguments and kernel device are central to the retained critical-overlap proof, while the authors supplied the main formulation and strategy.

Importance Notable

Identifies the critical overlap law and the conjectured second-moment limit.

Limitations

The result concerns β=1 and the stated two-replica overlap. The paper does not document a completed independent check of the model-generated argument.

Technical verification Reported

The v1 manuscript contains a proof and an author responsibility statement, but no completed human or machine check of this argument is documented; Reported is an evidence default.

Human understanding Full

The human authors explicitly supplied the limiting objects and proof architecture and explain the sphere-to-cube and Hilbert-space steps, including the model's kernel idea, in their proof-ideas section.

Correction and publication history