The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature

Gentz B, Lowe M (1999)
PROBABILITY THEORY AND RELATED FIELDS 115(3): 357-381.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
We investigate the limiting fluctuations of the order parameter in the Hopfield model of spin glasses and neural networks with finitely many patterns at the critical temperature 1/beta(c), = 1, At the critical temperature, the measure-valued random variables given by the distribution of the appropriately scaled order parameter under the Gibbs measure converge weakly towards a random measure which is non-Gaussian in the sense that it is not given by a Dirac measure concentrated in a Gaussian distribution. This remains true in the case of beta = beta(N) --> beta(c) = 1 as N --> infinity provided beta(N) converges to beta(c) = 1 fast enough, i.e., at speed partial derivative(1/root N). The Limiting distribution is explicitly given by its (random) density. Mathematics Subject Classification (1991): 60F05, 60K35 (primary), 82C32 (secondary).
Stichworte
critical temperature; theorems; limit; random disorder; Hopfield model; spin glasses; neural networks; non-Gaussian fluctuations
Erscheinungsjahr
1999
Zeitschriftentitel
PROBABILITY THEORY AND RELATED FIELDS
Band
115
Ausgabe
3
Seite(n)
357-381
ISSN
0178-8051
Page URI
https://pub.uni-bielefeld.de/record/1621090

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Gentz B, Lowe M. The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. PROBABILITY THEORY AND RELATED FIELDS. 1999;115(3):357-381.
Gentz, B., & Lowe, M. (1999). The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. PROBABILITY THEORY AND RELATED FIELDS, 115(3), 357-381.
Gentz, B., and Lowe, M. (1999). The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. PROBABILITY THEORY AND RELATED FIELDS 115, 357-381.
Gentz, B., & Lowe, M., 1999. The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. PROBABILITY THEORY AND RELATED FIELDS, 115(3), p 357-381.
B. Gentz and M. Lowe, “The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature”, PROBABILITY THEORY AND RELATED FIELDS, vol. 115, 1999, pp. 357-381.
Gentz, B., Lowe, M.: The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature. PROBABILITY THEORY AND RELATED FIELDS. 115, 357-381 (1999).
Gentz, Barbara, and Lowe, M. “The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature”. PROBABILITY THEORY AND RELATED FIELDS 115.3 (1999): 357-381.