141 research outputs found
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
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 -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
of the two-time correlation function of the infinite volume
-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 , 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
Convergence of clock processes on infinite graphs and aging in Bouchaud's asymmetric trap model on
Using a method developed by Durrett and Resnick [22] we establish general
criteria for the convergence of properly rescaled clock processes of random
dynamics in random environments on infinite graphs. This complements the
results of [26], [19], and [20]: put together these results provide a unified
framework for proving convergence of clock processes. As a first application we
prove that Bouchaud's asymmetric trap model on exhibits a normal
aging behavior for all . Namely, we show that certain two-time
correlation functions, among which the classical probability to find the
process at the same site at two time points, converge, as the age of the
process diverges, to the distribution function of the arcsine law. As a
byproduct we prove that the fractional kinetics process ages
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]
Emergence of near-TAP free energy functional in the SK model at high temperature
We study the SK model at inverse temperature and strictly positive
field in the region of where the replica-symmetric formula is
valid. An integral representation of the partition function derived from the
Hubbard-Stratonovitch transformation combined with a duality formula is used to
prove that the infinite volume free energy of the SK model can be expressed as
a variational formula on the space of magnetisations, . The resulting free
energy functional differs from that of Thouless, Anderson and Palmer (TAP) by
the term where
is the Edwards-Anderson parameter and is the minimiser
of the replica-symmetric formula. Thus, both functionals have the same critical
points and take the same value on the subspace of magnetisations satisfying
. This result is based on an in-depth study of the global
maximum of this near-TAP free energy functional using Bolthausen's solutions of
the TAP equations, Bandeira & van Handel's bounds on the spectral norm of
non-homogeneous Wigner-type random matrices, and Gaussian comparison
techniques. It holds for in a large subregion of the de Almeida and
Thouless high-temperature stability region
Convergence of Clock Processes and Aging in Metropolis Dynamics of a Truncated REM
International audienceWe study the aging behavior of a truncated version of the Random Energy Model evolving under Metropolis dynamics. We prove that the natural time-time correlation function defined through the overlap function converges to an arcsine law distribution function, almost surely in the random environment and in the full range of time scales and temperatures for which such a result can be expected to hold. This establishes that the dynamics ages in the same way as Bouchaud's REM-like trap model, thus extending the universality class of the latter model. The proof relies on a clock process convergence result of a new type where the number of summands is itself a clock process. This reflects the fact that the exploration process of Metropolis dynamics is itself an aging process, governed by its own clock. Both clock processes are shown to converge to stable subor-dinators below certain critical lines in their timescale and temperature domains, almost surely in the random environment
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