The spin-glass phase-transition in the Hopfield model with $p$-spin interactions

Abstract

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) = αN p−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