Anton Bovier, Beat Niederhauser
The spin-glass phase-transition in the Hopfield Model with $p$-spin interactions
(593K, gzipped postscript)
ABSTRACT. 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
$p\uparrow\infty$, $\b_p\rightarrow\sqrt {\a 2\ln 2}$. 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 {\it replica overlap} is concentrated at zero.
We show that for large enough $\a$, there exists a non-empty interval
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.