THE COMPLEXITY OF SPHERICAL p-SPIN MODELS-A SECOND MOMENT APPROACH

• 成果类型: Article
• 署名作者: Subag, Eliran
• 署名单位: Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1139
• 发表日期: 2017
• 页码: 3385-3450
• 关键词: random smooth functions; critical-points; supersymmetric vacua; glass model; number; asymptotics

摘要

Recently, Auffinger, Ben Arous and Cerny initiated the study of critical points of the Hamiltonian in the spherical pure p-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than Nu by Crt(N) (u), they computed the asymptotics of 1/N log(ECrt(N) (u)), as N, the dimension of the sphere, goes to infinity. We compute the asymptotics of the corresponding second moment and show that, for p >= 3 and sufficiently negative u, it matches the first moment: E{(Crt(N)(u))(2)}/(E{Crt(N)(u)})(2) -> 1. As an immediate consequence we obtain that Crt(N) (u)/E{Crt(N)(u)} -> 1, in L-2, and thus in probability. For any u for which E Crt(N) (u) does not tend to 0 we prove that the moments match on an exponential scale.