187 research outputs found
Limiting dynamics for spherical models of spin glasses at high temperature
We analyze the coupled non-linear integro-differential equations whose
solutions is the thermodynamical limit of the empirical correlation and
response functions in the Langevin dynamics for spherical p-spin disordered
mean-field models. We provide a mathematically rigorous derivation of their FDT
solution (for the high temperature regime) and of certain key properties of
this solution, which are in agreement with earlier derivations based on
physical grounds
Transition from the annealed to the quenched asymptotics for a random walk on random obstacles
In this work we study a natural transition mechanism describing the passage
from a quenched (almost sure) regime to an annealed (in average) one, for a
symmetric simple random walk on random obstacles on sites having an identical
and independent law. The transition mechanism we study was first proposed in
the context of sums of identical independent random exponents by Ben Arous,
Bogachev and Molchanov in [Probab. Theory Related Fields 132 (2005) 579--612].
Let be the survival probability at time of the random walk,
starting from site , and let be some increasing function of time. We
show that the empirical average of over a box of side has
different asymptotic behaviors depending on . T here are constants
such that if , with
, a law of large numbers is satisfied and the empirical
survival probability decreases like the annealed one; if , with , also a central limit theorem is
satisfied. If , the averaged survival probability decreases like
the quenched survival probability. If and we obtain an intermediate regime. Furthermore, when the dimension
it is possible to describe the fluctuations of the averaged survival
probability when with : it is
shown that they are infinitely divisible laws with a L\'{e}vy spectral function
which explodes when as stable laws of characteristic exponent
. These results show that the quenched and annealed survival
probabilities correspond to a low- and high-temperature behavior of a
mean-field type phase transition mechanism.Comment: Published at http://dx.doi.org/10.1214/009117905000000404 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Transition asymptotics for reaction-diffusion in random media
We describe a universal transition mechanism characterizing the passage to an
annealed behavior and to a regime where the fluctuations about this behavior
are Gaussian, for the long time asymptotics of the empirical average of the
expected value of the number of random walks which branch and annihilate on
, with stationary random rates. The random walks are
independent, continuous time rate , simple, symmetric, with . A random walk at , binary branches at rate ,
and annihilates at rate . The random environment has coordinates
which are i.i.d. We identify a natural way to describe
the annealed-Gaussian transition mechanism under mild conditions on the rates.
Indeed, we introduce the exponents
, and assume
that for
small enough, where and
denotes the average of the expected value of the number of particles
at time and an environment of rates , given that initially there was
only one particle at 0. Then the empirical average of over a box of
side has different behaviors: if for some and large enough , a law of large
numbers is satisfied; if for some
and large enough , a CLT is satisfied. These statements are
violated if the reversed inequalities are satisfied for some negative
. Applications to potentials with Weibull, Frechet and double
exponential tails are given.Comment: To appear in: Probability and Mathematical Physics: A Volume in Honor
of Stanislav Molchanov, Editors - AMS | CRM, (2007
Slow relaxation, dynamic transitions and extreme value statistics in disordered systems
We show that the dynamics of simple disordered models, like the directed Trap
Model and the Random Energy Model, takes place at a coexistence point between
active and inactive dynamical phases. We relate the presence of a dynamic phase
transition in these models to the extreme value statistics of the associated
random energy landscape
Universality of REM-like aging in mean field spin glasses
Aging has become the paradigm to describe dynamical behavior of glassy
systems, and in particular spin glasses. Trap models have been introduced as
simple caricatures of effective dynamics of such systems. In this Letter we
show that in a wide class of mean field models and on a wide range of time
scales, aging occurs precisely as predicted by the REM-like trap model of
Bouchaud and Dean. This is the first rigorous result about aging in mean field
models except for the REM and the spherical model.Comment: 4 page
Principal component-based image segmentation: a new approach to outline in vitro cell colonies
The in vitro clonogenic assay is a technique to study the ability of a cell
to form a colony in a culture dish. By optical imaging, dishes with stained
colonies can be scanned and assessed digitally. Identification, segmentation
and counting of stained colonies play a vital part in high-throughput screening
and quantitative assessment of biological assays. Image processing of such
pictured/scanned assays can be affected by image/scan acquisition artifacts
like background noise and spatially varying illumination, and contaminants in
the suspension medium. Although existing approaches tackle these issues, the
segmentation quality requires further improvement, particularly on noisy and
low contrast images. In this work, we present an objective and versatile
machine learning procedure to amend these issues by characterizing, extracting
and segmenting inquired colonies using principal component analysis, k-means
clustering and a modified watershed segmentation algorithm. The intention is to
automatically identify visible colonies through spatial texture assessment and
accordingly discriminate them from background in preparation for successive
segmentation. The proposed segmentation algorithm yielded a similar quality as
manual counting by human observers. High F1 scores (>0.9) and low
root-mean-square errors (around 14%) underlined good agreement with ground
truth data. Moreover, it outperformed a recent state-of-the-art method. The
methodology will be an important tool in future cancer research applications
Flows driven by Banach space-valued rough paths
We show in this note how the machinery of C^1-approximate flows devised in
the work "Flows driven by rough paths", and applied there to reprove and extend
most of the results on Banach space-valued rough differential equations driven
by a finite dimensional rough path, can be used to deal with rough differential
equations driven by an infinite dimensional Banach space-valued weak geometric
Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force
in a simple way.Comment: 8 page
A matrix interpolation between classical and free max operations: I. The univariate case
Recently, Ben Arous and Voiculescu considered taking the maximum of two free
random variables and brought to light a deep analogy with the operation of
taking the maximum of two independent random variables. We present here a new
insight on this analogy: its concrete realization based on random matrices
giving an interpolation between classical and free settings.Comment: 14 page
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