Shattering in the Ising p-spin glass model
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440_2025_Article_1414.pdf
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Author(s)
Gamarnik, David
Jagannath, Aukosh
Kızıldağ, Eren C.
Date Issued
September 11, 2025
Journal
Probability Theory and Related Fields
Publisher
Springer Berlin Heidelberg
Citation
Gamarnik, D., Jagannath, A. & Kızıldağ, E.C. Shattering in the Ising p-spin glass model. Probab. Theory Relat. Fields 193, 89–141 (2025).
Version
Final published version
Abstract
We study the Ising p-spin glass model for large p. We show that for any inverse temperature ln 2 < β < 2 ln 2 and any large p, the model exhibits shattering: w.h.p. as n → ∞ , there exists exponentially many well-separated clusters such that (a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist of configurations with energy near β . Range of temperatures for which shattering occurs is within the replica symmetric region. To the best of our knowledge, this is the first shattering result regarding the Ising p-spin glass models. Furthermore, we show that for any γ > 0 and any large enough p, the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value γ 2 ln 2 . Our proofs are elementary, and in particular based on simple applications of the first and the second moment methods.
MIT Department
Sloan School of Management
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DOI of Published Version
https://doi.org/10.1007/s00440-025-01414-4
