## The Annals of Statistics

### Estimation in spin glasses: A first step

Sourav Chatterjee

#### Abstract

The Sherrington–Kirkpatrick model of spin glasses, the Hopfield model of neural networks and the Ising spin glass are all models of binary data belonging to the one-parameter exponential family with quadratic sufficient statistic. Under bare minimal conditions, we establish the $\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural parameter in this family, even at critical temperatures. Since very little is known about the low and critical temperature regimes of these extremely difficult models, the proof requires several new ideas. The author’s version of Stein’s method is a particularly useful tool. We aim to introduce these techniques into the realm of mathematical statistics through an example and present some open questions.

#### Article information

Source
Ann. Statist., Volume 35, Number 5 (2007), 1931-1946.

Dates
First available in Project Euclid: 7 November 2007

https://projecteuclid.org/euclid.aos/1194461717

Digital Object Identifier
doi:10.1214/009053607000000109

Mathematical Reviews number (MathSciNet)
MR2363958

Zentralblatt MATH identifier
1126.62128

#### Citation

Chatterjee, Sourav. Estimation in spin glasses: A first step. Ann. Statist. 35 (2007), no. 5, 1931--1946. doi:10.1214/009053607000000109. https://projecteuclid.org/euclid.aos/1194461717

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