The spin-glass phase-transition in the Hopfield model with p-spin interactions
- Bovier, Anton
- Niederhauser, Beat
2010 Mathematics Subject Classification
- 82A87 60K35
- spin glasses, Hopfield models, phase transition, overlap distribution
We study the Hopfield model with pure 𝑝-spin interactions with even 𝑝 ≥ 4, and a number of patterns, M(N) growing with the system size, N, as M(N) = αN𝑝-1. We prove the existence of a critical temperature β𝑝 characterized as the first time quenched and annealed free energy differ. We prove that as 𝑝 ↑ ∞, β𝑝 → √α2ln2. Moreover, we show that for any α > 0 and for all inverse temperatures β, the free energy converges to that of the REM at inverse temperature β / √α. Moreover, above the critical temperature the distribution of the replica overlap is concentrated at zero. We show that for large enough α, there exists a non-empty interval of in the low temperature regime where the distribution has mass both near zero and near ±1. As was first shown by M. Talagrand in the case of the 𝑝-spin SK model, this implies the the Gibbs measure at low temperatures is concentrated, asymptotically for large N, on a countable union of disjoint sets, no finite subset of which has full mass. Finally, we show that there is α𝑝 ∼ 1 / 𝑝! such that for α > α𝑝 the set carrying almost all mass does not contain the original patterns. In this sense we describe a genuine spin glass transition. Our approach follows that of Talagrand's analysis of the 𝑝-spin SK-model. The more complex structure of the random interactions necessitates, however, considerable technical modifications. In particular, various results that follow easily in the Gaussian case from integration by parts formulas have to be derived by expansion techniques.
- Adv. Theor. Math. Phys. 5 (2001), pp. 1001-1046