A central limit theorem for the overlap in the Hopfield model. (English) Zbl 0872.60015

Summary: We consider the Hopfield model with \(n\) neurons and an increasing number \(p=p(n)\) of randomly chosen patterns. Under the condition \((p^3 \log p)/n \to 0\), we prove for every fixed choice of overlap parameters a central limit theorem as \(n\to\infty\), which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition \((p^2 \log p)/n \to 0\) suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.


60F05 Central limit and other weak theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82C32 Neural nets applied to problems in time-dependent statistical mechanics
Full Text: DOI


[1] Amit, D. J., Gutfreund, H. and Sompolinsky, H. (1985). Spin-glass models of neural networks. Phy s. Rev. A 32 1007-1018.
[2] Bovier, A., Gay rard, V. and Picco, P. (1994). Gibbs states of the Hopfield model in the regime of perfect memory. Probab. Theory Related Fields 100, 329-363. · Zbl 0810.60094
[3] Bovier, A. and Gay rard, V. (1996). An almost sure large deviation principle for the Hopfield model. Ann. Probab. 24 1444-1475. · Zbl 0871.60022
[4] Bovier, A. and Gay rard, V. (1996). The retrieval phase of the Hopfield model: a rigorous analysis of the overlap distribution. Probab. Theory Related Fields. · Zbl 0866.60085
[5] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York. · Zbl 0566.60097
[6] Figotin, A. L. and Pastur, L. A. (1977). Exactly soluble model of a spin glass. Sov. J. Low Temp. Phy s. 3 378-383.
[7] Figotin, A. L. and Pastur, L. A. (1978). Theory of disordered spin sy stems. Theoret. and Math. Phy s. 35 403-414.
[8] Gentz, B. (1996). An almost sure central limit theorem for the overlap parameters in the Hopfield model. Stochast. Proc. Appl. 62 243-262. · Zbl 0863.60019
[9] Hertz, J., Krogh, A. and Palmer, R. G. (1991). Introduction to the Theory of Neural Computation. Addison-Wesley, Reading, MA.
[10] Hopfield, J. J. (1982). Neural networks and physical sy stems with emergent collective computational abilities. Proc. Nat. Acad. Sci. U.S.A. 79 2554-2558. JSTOR: · Zbl 1369.92007
[11] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin. · Zbl 0748.60004
[12] Martin-L öf, A. (1973). Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Comm. Math. Phy s. 32 75-92.
[13] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. IHES 81 73-205. · Zbl 0864.60013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.