Rigorous results on the thermodynamics of the dilute Hopfield model.
- Bovier, Anton
- Gayrard, Veronique
2010 Mathematics Subject Classification
- Neural networks, Hopfield model, random graphs, mean-field theory
We study the Hopfield model of an autoassociative memory on a random graph on N vertices where the probability of two vertices being joined by a link is p(N). Assuming that p(N) goes to zero more slowly than O(1/N), we prove the following results: 1) If the number of stored patterns, m(N), is small enough such that m(N)/(Np(N)) ↓ 0, as N ↑ ∞, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. 2) If in addition m(N) > ln N / ln 2, we prove that there exists, for T > 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.
- J. Stat. Phys. 72 (1993), pp. 79-112