1,153 research outputs found
Metastable states, quasi-stationary distributions and soft measures
We establish metastability in the sense of Lebowitz and Penrose under
practical and simple hypothesis for (families of) Markov chains on finite
configuration space in some asymptotic regime, including the case of
configuration space size going to infinity. By comparing restricted ensemble
and quasi-stationary measures, we study point-wise convergence velocity of
Yaglom limits and prove asymptotic exponential exit law. We introduce soft
measures as interpolation between restricted ensemble and quasi-stationary
measure to prove an asymptotic exponential transition law on a generally
different time scale. By using potential theoretic tools, we prove a new
general Poincar\'e inequality and give sharp estimates via two-sided
variational principles on relaxation time as well as mean exit time and
transition time. We also establish local thermalization on a shorter time scale
and give mixing time asymptotics up to a constant factor through a two-sided
variational principal. All our asymptotics are given with explicit quantitative
bounds on the corrective terms.Comment: 41 page
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
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, , 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
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