204 research outputs found
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, , as a function of
the system size satisfies . 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 . 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 are located within small balls around the
points , where denotes the -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 -partioning problem
The randomized -number partitioning problem is the task to distribute
i.i.d. random variables into groups in such a way that the sums of the
variables in each group are as similar as possible. The restricted
-partitioning problem refers to the case where the number of elements in
each group is fixed to . In the case 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 in the restricted
problem and show that the vector of differences between the sums converges
to a -dimensional Poisson point process.Comment: 31pp, AMSTe
A tomography of the GREM: beyond the 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. This was proven in a large class of
models for energies that do not grow too fast with the system size. Considering
the example of the generalized random energy model, we show that the conjecture
breaks down for energies proportional to the volume of the system, and describe
the far more complex behavior that then sets in
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