202 research outputs found

### 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 $k$-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

### 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]

### 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

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