11 research outputs found
Metastable states in Brownian energy landscape
Random walks and diffusions in symmetric random environment are known to
exhibit metastable behavior: they tend to stay for long times in wells of the
environment. For the case that the environment is a one-dimensional two-sided
standard Brownian motion, we study the process of depths of the consecutive
wells of increasing depth that the motion visits. When these depths are looked
in logarithmic scale, they form a stationary renewal cluster process. We give a
description of the structure of this process and derive from it the almost sure
limit behavior and the fluctuations of the empirical density of the process.Comment: 21 pages, 6 figure
Patterns in Sinai's walk
Sinai's random walk in random environment shows interesting patterns on the
exponential time scale. We characterize the patterns that appear on infinitely
many time scales after appropriate rescaling (a functional law of iterated
logarithm). The curious rate function captures the difference between one-sided
and two-sided behavior.Comment: Published in at http://dx.doi.org/10.1214/11-AOP724 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Variational characterization of the critical curve for pinning of random polymers
In this paper we look at the pinning of a directed polymer by a
one-dimensional linear interface carrying random charges. There are two phases,
localized and delocalized, depending on the inverse temperature and on the
disorder bias. Using quenched and annealed large deviation principles for the
empirical process of words drawn from a random letter sequence according to a
random renewal process [Birkner, Greven and den Hollander, Probab. Theory
Related Fields 148 (2010) 403-456], we derive variational formulas for the
quenched, respectively, annealed critical curve separating the two phases.
These variational formulas are used to obtain a necessary and sufficient
criterion, stated in terms of relative entropies, for the two critical curves
to be different at a given inverse temperature, a property referred to as
relevance of the disorder. This criterion in turn is used to show that the
regimes of relevant and irrelevant disorder are separated by a unique inverse
critical temperature. Subsequently, upper and lower bounds are derived for the
inverse critical temperature, from which sufficient conditions under which it
is strictly positive, respectively, finite are obtained. The former condition
is believed to be necessary as well, a problem that we will address in a
forthcoming paper. Random pinning has been studied extensively in the
literature. The present paper opens up a window with a variational view. Our
variational formulas for the quenched and the annealed critical curve are new
and provide valuable insight into the nature of the phase transition. Our
results on the inverse critical temperature drawn from these variational
formulas are not new, but they offer an alternative approach, that is, flexible
enough to be extended to other models of random polymers with disorder.Comment: Published in at http://dx.doi.org/10.1214/11-AOP727 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The random pinning model with correlated disorder given by a renewal set
We investigate the effect of correlated disorder on the localization
transition undergone by a renewal sequence with loop exponent > 0,
when the correlated sequence is given by another independent renewal set with
loop exponent > 0. Using the renewal structure of the disorder
sequence, we compute the annealed critical point and exponent. Then, using a
smoothing inequality for the quenched free energy and second moment estimates
for the quenched partition function, combined with decoupling inequalities, we
prove that in the case > 2 (summable correlations), disorder is
irrelevant if 1/2, which extends the
Harris criterion for independent disorder. The case (1, 2)
(non-summable correlations) remains largely open, but we are able to prove that
disorder is relevant for > 1/ , a condition that is expected
to be non-optimal. Predictions on the criterion for disorder relevance in this
case are discussed. Finally, the case (0, 1) is somewhat special
but treated for completeness: in this case, disorder has no effect on the
quenched free energy, but the annealed model exhibits a phase transition
The spectrum of the random environment and localization of noise
We consider random walk on a mildly random environment on finite transitive
d- regular graphs of increasing girth. After scaling and centering, the
analytic spectrum of the transition matrix converges in distribution to a
Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph
changes from a regular tree to the integers, the noise becomes localized.Comment: 18 pages, 1 figur
Localization of favorite points for diffusion in a random environment
For a diffusion Xt in a one-dimensional Wiener medium W, it is known that there is a certain process (br(W))r>=0 that depends only on the environment, so that Xt-blogt(W) converges in distribution as t-->[infinity]. The paths of b are step functions. Denote by FX(t) the point with the most local time for the diffusion at time t. We prove that, modulo a relatively small time change, the paths of the processes (br(W))r>=0, (FX(er))r>=0 are close after some large r.Diffusion in random environment Favorite point Local time Ray-Knight theorem Localization