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

We study the Hopfield model with pure $p$-spin interactions with even $p\\geq\n4$, and a number of patterns, M(N) growing with the system size, $N$, as $M(N)\n= \\a N^{p-1}$. We prove the existence of a critical temperature $\\b_p$\ncharacterized as the first time quenched and annealed free energy differ. We\nprove that as $p\\uparrow\\infty$, $\\b_p\\to\\sqrt {\\a 2\\ln 2}$. Moreover, we show\nthat for any $\\a>0$ and for all inverse temperatures $\\b$, the free energy\nconverges to that of the REM at inverse temperature $\\b/\\sqrt\\a$. Moreover,\nabove the critical temperature the distribution of the replica overlap is\nconcentrated at zero. We show that for large enough $\\a$, there exists a\nnon-empty interval of in the low temperature regime where the distribution has\nmass both near zero and near $\\pm 1$. As was first shown by M. Talagrand in the\ncase of the $p$-spin SK model, this implies the the Gibbs measure at low\ntemperatures is concentrated, asymptotically for large $N$, on a countable\nunion of disjoint sets, no finite subset of which has full mass. Finally, we\nshow that there is $\\a_p\\sim 1/p!$ such that for $\\a>\\a_p$ the set carrying\nalmost all mass does not contain the original patterns. In this sense we\ndescribe a genuine spin glass transition.\n Our approach follows that of Talagrand's analysis of the $p$-spin SK-model.\nThe more complex structure of the random interactions necessitates, however,\nconsiderable technical modifications. In particular, various results that\nfollow easily in the Gaussian case from integration by parts fromulas have to\nbe derived by expansion techniques.\n

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