Estimation in spin glasses: a first step. (English) Zbl 1126.62128

Chatterjee, Sourav

Ann. Stat. 35, No. 5, 1931-1946 (2007).

Summary: The Sherrington-Kirkpatrick [see D. Sherrington, Neural networks: The spin glass approach. J. G. Taylor (ed.), Math. Approaches Neural Networks. North-Holland Math. Libr. 51, 261–291 (1993; Zbl 0785.92005)] 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.

MSC:

62P35	Applications of statistics to physics
62F12	Asymptotic properties of parametric estimators
60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
62F10	Point estimation
82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

Keywords:

neural networks; estimation; consistency; exponential families

Citations:

Zbl 0785.92005

References:

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