1,065 research outputs found
A quenched large deviation principle in a continuous scenario
We prove the analogue for continuous space-time of the quenched LDP derived
in Birkner, Greven and den Hollander (2010) for discrete space-time. In
particular, we consider a random environment given by Brownian increments, cut
into pieces according to an independent continuous-time renewal process. We
look at the empirical process obtained by recording both the length of and the
increments in the successive pieces. For the case where the renewal time
distribution has a Lebesgue density with a polynomial tail, we derive the
quenched LDP for the empirical process, i.e., the LDP conditional on a typical
environment. The rate function is a sum of two specific relative entropies, one
for the pieces and one for the concatenation of the pieces. We also obtain a
quenched LDP when the tail decays faster than algebraic. The proof uses
coarse-graining and truncation arguments, involving various approximations of
specific relative entropies that are not quite standard.
In a companion paper we show how the quenched LDP and the techniques
developed in the present paper can be applied to obtain a variational
characterisation of the free energy and the phase transition line for the
Brownian copolymer near a selective interface
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
Phase diagram for a copolymer in a micro-emulsion
In this paper we study a model describing a copolymer in a micro-emulsion.
The copolymer consists of a random concatenation of hydrophobic and hydrophilic
monomers, the micro-emulsion consists of large blocks of oil and water arranged
in a percolation-type fashion. The interaction Hamiltonian assigns energy
to hydrophobic monomers in oil and energy to hydrophilic
monomers in water, where are parameters that without loss of
generality are taken to lie in the cone . Depending on the values of these
parameters, the copolymer either stays close to the oil-water interface
(localization) or wanders off into the oil and/or the water (delocalization).
Based on an assumption about the strict concavity of the free energy of a
copolymer near a linear interface, we derive a variational formula for the
quenched free energy per monomer that is column-based, i.e., captures what the
copolymer does in columns of different type. We subsequently transform this
into a variational formula that is slope-based, i.e., captures what the polymer
does as it travels at different slopes, and we use the latter to identify the
phase diagram in the -cone. There are two regimes:
supercritical (the oil blocks percolate) and subcritical (the oil blocks do not
percolate). The supercritical and the subcritical phase diagram each have two
localized phases and two delocalized phases, separated by four critical curves
meeting at a quadruple critical point. The different phases correspond to the
different ways in which the copolymer can move through the micro-emulsion. The
analysis of the phase diagram is based on three hypotheses of percolation-type
on the blocks. We show that these three hypotheses are plausible, but do not
provide a proof.Comment: 100 pages, 16 figures. arXiv admin note: substantial text overlap
with arXiv:1204.123
A general smoothing inequality for disordered polymers
This note sharpens the smoothing inequality of Giacomin and Toninelli for
disordered polymers. This inequality is shown to be valid for any disorder
distribution with locally finite exponential moments, and to provide an
asymptotically sharp constant for weak disorder. A key tool in the proof is an
estimate that compares the effect on the free energy of tilting, respectively,
shifting the disorder distribution. This estimate holds in large generality
(way beyond disordered polymers) and is of independent interest.Comment: 14 page
A large-deviation view on dynamical Gibbs-non-Gibbs transitions
We develop a space-time large-deviation point of view on Gibbs-non-Gibbs
transitions in spin systems subject to a stochastic spin-flip dynamics. Using
the general theory for large deviations of functionals of Markov processes
outlined in Feng and Kurtz [11], we show that the trajectory under the
spin-flip dynamics of the empirical measure of the spins in a large block in
Z^d satisfies a large deviation principle in the limit as the block size tends
to infinity. The associated rate function can be computed as the action
functional of a Lagrangian that is the Legendre transform of a certain
non-linear generator, playing a role analogous to the moment-generating
function in the Gartner-Ellis theorem of large deviation theory when this is
applied to finite-dimensional Markov processes. This rate function is used to
define the notion of "bad empirical measures", which are the discontinuity
points of the optimal trajectories (i.e., the trajectories minimizing the rate
function) given the empirical measure at the end of the trajectory. The
dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad
empirical measures: for short times no bad empirical measures occur, while for
intermediate and large times bad empirical mea- sures are possible. A future
research program is proposed to classify the various possible scenarios behind
this crossover, which we refer to as a "nature-versus-nurture" transition.Comment: Key words and phrases: Stochastic spin-flip dynamics, Gibbs-non-Gibbs
transition, empirical measure, non-linear generator, Nisio control semigroup,
large deviation principle, bad configurations, bad empirical measures, nature
versus nurture
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment
We continue our study of the parabolic Anderson equation for the space-time field , where is the diffusion constant,
is the discrete Laplacian, is the coupling
constant, and is a space-time random
environment that drives the equation. The solution of this equation describes
the evolution of a "reactant" under the influence of a "catalyst" ,
both living on . In earlier work we considered three choices for :
independent simple random walks, the symmetric exclusion process, and the
symmetric voter model, all in equilibrium at a given density. We analyzed the
\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of w.r.t.\ , and showed that these exponents
display an interesting dependence on the diffusion constant , with
qualitatively different behavior in different dimensions . In the present
paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential
growth rate of conditional on . We first prove existence and derive
some qualitative properties of the quenched Lyapunov exponent for a general
that is stationary and ergodic w.r.t.\ translations in and
satisfies certain noisiness conditions. After that we focus on the three
particular choices for mentioned above and derive some more detailed
properties. We close by formulating a number of open problems.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday,
33 pages. Final revised versio
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