An almost sure large deviation principle for the Hopfield model

We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic `overlap'-parameters in the Hopfield model in the case where the number of random patterns, $M$, as a function of the system size $N$ satisfies $\limsup M(N)/N=0$. In this case the rate function (or free energy as a function of the overlap parameters) is independent of the disorder for almost all realization of the patterns and given by an explicit variational formula.Comment: 31pp; Plain-TeX, hardcopy available on request from [email protected]

Spectral characterization of aging: the rem-like trap model

We review the aging phenomenon in the context of the simplest trap model, Bouchaud's REM-like trap model, from a spectral theoretic point of view. We show that the generator of the dynamics of this model can be diagonalized exactly. Using this result, we derive closed expressions for correlation functions in terms of complex contour integrals that permit an easy investigation into their large time asymptotics in the thermodynamic limit. We also give a ``grand canonical'' representation of the model in terms of the Markov process on a Poisson point process. In this context we analyze the dynamics on various time scales.Comment: Published at http://dx.doi.org/10.1214/105051605000000359 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Poisson convergence in the restricted kk-partioning problem

The randomized $k$-number partitioning problem is the task to distribute $N$ i.i.d. random variables into $k$ groups in such a way that the sums of the variables in each group are as similar as possible. The restricted $k$-partitioning problem refers to the case where the number of elements in each group is fixed to $N/k$. In the case $k=2$ it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case $k>2$ in the restricted problem and show that the vector of differences between the $k$ sums converges to a $k-1$-dimensional Poisson point process.Comment: 31pp, AMSTe

Energy statistics in disordered systems: The local REM conjecture and beyond

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered. In the case of the Generalised Random energy model, we give a complete analysis for the behaviour of the local energy statistics at all energy scales. In particular, we show that, in this case, the REM conjecture holds exactly up to energies E_N<\b_c N, where \b_c is the critical temperature. We also explain the more complex behaviour that sets in at higher energies.Comment: to appear in Proceedings of Applications of random matrices to economics and other complex system

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

Local energy statistics in disordered systems: a proof of the local REM conjecture

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. Here we give necessary conditions for this hypothesis to be true, which we show to hold in wide classes of examples: short range spin glasses and mean field spin glasses of the SK type. We also show that, under certain conditions, the conjecture holds even if energy levels that grow moderately with the volume of the system are considered