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
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.