The Retrieval Phase of the Hopfield Model: A Rigorous Analysis of the Overlap Distribution

Standard large deviation estimates or the use of the Hubbard-Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function \Phi_{N,\b} on $\R^M$. In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m^*(\b) the modulus of the spontaneous magnetization in the Curie-Weiss model and by \a the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0<\g_a<\g_c such that 1) If \sqrt\a\leq \g_a (m^*(\b))^2, then the absolute minima of $\Phi$ are located within small balls around the points $\pm m^*e^\mu$, where $e^\mu$ denotes the $\mu$-th unit vector while 2) if \sqrt\a\leq \g_c (m^*(\b))^2 at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. KEYWORDS: Hopfield model, neural networks, storage capacity, Gibbs measures, self-averaging, random matricesComment: 43 pages, uuencoded, Z-compressed Postscrip

Metastates in the Hopfield model in the replica symmetric regime

We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, $M$, is proportional to the volume with a sufficiently small proportionality constant \a>0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not converge almost surely, but only in law. The corresponding limiting law is constructed explicitly. We fit our result in the recently proposed language of ``metastates'' which we discuss in some length. As a byproduct, in a certain regime of the parameters \a and \b (the inverse temperature), we also give a simple proof of Talagrand's [T1] recent result that the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky [AGS] can be rigorously justified.Comment: 41pp, plain TE

Convergence of clock processes in random environments and ageing in the p-spin SK model

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829-846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Cerny [Comm. Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in law, with respect to the random environment.Comment: Published in at http://dx.doi.org/10.1214/11-AOP705 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Aging in the GREM-like trap model

The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume $L$-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit $L\to\infty$ of the two-time correlation function of the infinite volume $L$-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any $L$, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.Comment: 30 pages, 1 figur

Sample path large deviations for a class of Markov chains related to disordered mean field models

We prove a large deviation principle on path space for a class of discrete time Markov processes whose state space is the intersection of a regular domain \L\subset \R^d with some lattice of spacing \e. Transitions from $x$ to $y$ are allowed if \e^{-1}(x-y)\in \D for some fixed set of vectors \D. The transition probabilities p_\e(t,x,y), which themselves depend on \e, are allowed to depend on the starting point $x$ and the time $t$ in a sufficiently regular way, except near the boundaries, where some singular behaviour is allowed. The rate function is identified as an action functional which is given as the integral of a Lagrange function. %of time dependent relativistic classical mechanics. Markov processes of this type arise in the study of mean field dynamics of disordered mean field models.Comment: 56pp, AMS-Te