57 research outputs found
A weakness in strong localization for Sinai's walk
Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in
random environment. It is known for a phenomenon of strong localization,
namely, the walk spends almost all time at or near the bottom of deep valleys
of the potential. Our main result shows a weakness of this localization
phenomenon: with probability one, the zones where the walk stays for the most
time can be far away from the sites where the walk spends the most time. In
particular, this gives a negative answer to a problem of Erd\H{o}s and
R\'{e}v\'{e}sz [Mathematical Structures--Computational
Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated
for the usual homogeneous random walk.Comment: Published at http://dx.doi.org/10.1214/009117906000000863 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field
We study the statistics of the extremes of a discrete Gaussian field with
logarithmic correlations at the level of the Gibbs measure. The model is
defined on the periodic interval , and its correlation structure is
nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm.
Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory
Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random
Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001)
026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008)
372001]. At low temperature, it is shown that the normalized covariance of two
points sampled from the Gibbs measure is either or . This is used to
prove that the joint distribution of the Gibbs weights converges in a suitable
sense to that of a Poisson-Dirichlet variable. In particular, this proves a
conjecture of Carpentier and Le Doussal that the statistics of the extremes of
the log-correlated field behave as those of i.i.d. Gaussian variables and of
branching Brownian motion at the level of the Gibbs measure. The method of
proof is robust and is adaptable to other log-correlated Gaussian fields.Comment: Published in at http://dx.doi.org/10.1214/13-AAP952 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime
We consider transient one-dimensional random walks in random environment with
zero asymptotic speed. An aging phenomenon involving the generalized Arcsine
law is proved using the localization of the walk at the foot of "valleys" of
height . In the quenched setting, we also sharply estimate the
distribution of the walk at time .Comment: 24 pages, accepted for publication in "Bulletin de la SMF
Limit laws for transient random walks in random environment on \z
We consider transient random walks in random environment on \z with zero
asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that
the hitting time of the level converges in law, after a proper
normalization, towards a positive stable law, but they do not obtain a
description of its parameter. A different proof of this result is presented,
that leads to a complete characterization of this stable law. The case of
Dirichlet environment turns out to be remarkably explicit.Comment: 31 pages, accepted for publication in "Annales de l'Institut Fourier
A probabilistic representation of constants in Kesten's renewal theorem
The aims of this paper are twofold. Firstly, we derive some probabilistic
representation for the constant which appears in the one-dimensional case of
Kesten's renewal theorem. Secondly, we estimate the tail of some related random
variable which plays an essential role in the description of the stable limit
law of one-dimensional transient sub-ballistic random walks in random
environment.Comment: 27 page
Scaling limit and aging for directed trap models
We consider one-dimensional directed trap models and suppose that the
trapping times are heavy-tailed. We obtain the inverse of a stable subordinator
as scaling limit and prove an aging phenomenon expressed in terms of the
generalized arcsine law. These results confirm the status of universality
described by Ben Arous and \v{C}ern\'y for a large class of graphs.Comment: 16 pages, accepted for publication in "Markov processes and Related
Fields
Universality and Sharpness in Absorbing-State Phase Transitions
We consider the Activated Random Walk model in any dimension with any sleep
rate and jump distribution and ergodic initial state. We show that the
stabilization properties depend only on the average density of particles,
regardless of how they are initially located on the lattice
Stable fluctuations for ballistic random walks in random environment on Z
We consider transient random walks in random environment on Z in the positive
speed (ballistic) and critical zero speed regimes. A classical result of
Kesten, Kozlov and Spitzer proves that the hitting time of level , after
proper centering and normalization, converges to a completely asymmetric stable
distribution, but does not describe its scale parameter. Following a previous
article by three of the authors, where the (non-critical) zero speed case was
dealt with, we give a new proof of this result in the subdiffusive case that
provides a complete description of the limit law. The case of Dirichlet
environment turns out to be remarkably explicit
Quenched limits for the fluctuations of transient random walks in random environment on Z
We consider transient nearest-neighbor random walks in random environment on
Z. For a set of environments whose probability is converging to 1 as time goes
to infinity, we describe the fluctuations of the hitting time of a level n,
around its mean, in terms of an explicit function of the environment. Moreover,
their limiting law is described using a Poisson point process whose intensity
is computed. This result can be considered as the quenched analog of the
classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975)
145-168].Comment: Published in at http://dx.doi.org/10.1214/12-AAP867 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note:
substantial text overlap with arXiv:1004.133
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