Self-averaging in a class of generalized Hopfield models
- Bovier, Anton
2010 Mathematics Subject Classification
- 60K35 82C32
- Hopfield model, neural networks, self-averaging, law of large numbers
We prove the almost sure convergence to zero of the fluctuations of the free energy, resp. local free energies, in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: 1) Multi-spin interactions are permitted and 2) the random variables ξμi i describing the "patterns" can have arbitrary distributions with mean zero and finite 4+∈-th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the systemsize. This generalizes a previous result of Bovier, Gayrard, and Picco [BGP3] for the standard Hopfield model, and improves a result of Feng and Tirozzi [FT] that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven.
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