The complexity of spherical pp-spin models—A second moment approach

• Eliran Subag ^[1]
  1. [1] Weizmann Institute of Science (Israel)
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 3385-3450
• Idioma: inglés
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• Resumen

  • Recently, Auffinger, Ben Arous and Černý initiated the study of critical points of the Hamiltonian in the spherical pure pp-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than NuNu by CrtN(u)CrtN⁡(u), they computed the asymptotics of 1Nlog(ECrtN(u))1Nlog⁡(ECrtN(u)), as NN, the dimension of the sphere, goes to ∞∞. We compute the asymptotics of the corresponding second moment and show that, for p≥3p≥3 and sufficiently negative uu, it matches the first moment:

    E{(CrtN(u))2}/(E{CrtN(u)})2→1.

    As an immediate consequence we obtain that CrtN(u)/E{CrtN(u)}→1CrtN⁡(u)/E{CrtN⁡(u)}→1, in L2L2, and thus in probability. For any uu for which ECrtN(u)ECrtN⁡(u) does not tend to 00 we prove that the moments match on an exponential scale.