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September 2017 The complexity of spherical $p$-spin models—A second moment approach
Eliran Subag
Ann. Probab. 45(5): 3385-3450 (September 2017). DOI: 10.1214/16-AOP1139


Recently, Auffinger, Ben Arous and Černý 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 $\operatorname{Crt}_{N}(u)$, they computed the asymptotics of $\frac{1}{N}\log (\mathbb{E}\mbox{Crt}_{N}(u))$, as $N$, the dimension of the sphere, goes to $\infty$. We compute the asymptotics of the corresponding second moment and show that, for $p\geq3$ and sufficiently negative $u$, it matches the first moment:

\[\mathbb{E}\{(\operatorname{Crt}_{N}(u))^{2}\}/(\mathbb{E} \{\operatorname{Crt}_{N}(u)\})^{2}\to1.\] As an immediate consequence we obtain that $\operatorname{Crt}_{N}(u)/\mathbb{E}\{\operatorname{Crt}_{N}(u)\}\to1$, in $L^{2}$, and thus in probability. For any $u$ for which $\mathbb{E}\operatorname{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.


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Eliran Subag. "The complexity of spherical $p$-spin models—A second moment approach." Ann. Probab. 45 (5) 3385 - 3450, September 2017.


Received: 1 May 2015; Revised: 1 June 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812208
MathSciNet: MR3706746
Digital Object Identifier: 10.1214/16-AOP1139

Primary: 60G15 , 60G60 , 82D30
Secondary: 60B20

Keywords: critical points , second moment , Spin glasses

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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