The spin-glass phase-transition in the Hopfield model with p-spin interactions
We study the Hopfield model with pure $p$-spin interactions with even $p\geq 4$, and a number of patterns, M(N) growing with the system size, $N$, as $M(N) = \a N^{p-1}$. We prove the existence of a critical temperature $\b_p$ characterized as the first time quenched and annealed free energy differ. We prove that as $p\uparrow\infty$, $\b_p\to\sqrt {\a 2\ln 2}$. Moreover, we show that for any $\a>0$ and for all inverse temperatures $\b$, the free energy converges to that of the REM at inverse temperature $\b/\sqrt\a$. Moreover, above the critical temperature the distribution of the replica overlap is concentrated at zero. We show that for large enough $\a$, there exists a non-empty interval of in the low temperature regime where the distribution has mass both near zero and near $\pm 1$. 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 $\a_p\sim 1/p!$ such that for $\a>\a_p$ 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 fromulas have to be derived by expansion techniques.