Critical overlap fluctuations in SK models
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At β=1, what is the critical scaling law of the two-replica overlap in the Ising and spherical Sherrington–Kirkpatrick models? The target concerns the quenched law of N^(1/3) times overlap and the limit of its disorder-averaged squared overlap after scaling by N^(2/3).
Why it matters and what counts as progress
Why it matters: The critical overlap law and second moment refine the finite-size transition in two standard spin-glass models.
What counts as progress: Identify the limiting random law and prove moment convergence, including Talagrand’s proposed second-moment limit.
How the problem developed
- 2011 · Open question [1]
Michel Talagrand Asked for the critical scaling limit of the squared two-replica overlap at inverse temperature one in the SK model.
- 9 Aug 2026 · Reported theorem [2]
Hang Du and Brice Huang Reported an N^(−1/3) overlap scale and an explicit limiting quenched distribution for both Ising and spherical critical SK models, determining the limit of N^(2/3) times the mean squared overlap.
AI contribution
Overlap distribution of the critical Sherrington-Kirkpatrick model
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.
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