637 research outputs found
On the localized phase of a copolymer in an emulsion: supercritical percolation regime
In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, and , each occurring with
density 1/2. The emulsion is a random mixture of liquids of two types, and
, organised in large square blocks occurring with density and ,
respectively, where . The copolymer in the emulsion has an energy
that is minus times the number of -matches minus times the
number of -matches, where without loss of generality the interaction
parameters can be taken from the cone . To make the model mathematically tractable, we assume that the
copolymer is directed and can only enter and exit a pair of neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical percolation regime , with the critical probability for directed bond percolation on
the square lattice, the free energy has a phase transition along a curve in the
cone that is independent of . At this critical curve, there is a transition
from a phase where the copolymer is fully delocalized into the -blocks to a
phase where it is partially localized near the -interface. In the present
paper we prove three theorems that complete the analysis of the phase diagram :
(1) the critical curve is strictly increasing; (2) the phase transition is
second order; (3) the free energy is infinitely differentiable throughout the
partially localized phase.Comment: 43 pages and 10 figure
The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics
We consider the continuous time version of the Random Walk Pinning Model
(RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on
Z^d with jump rate rho (that plays the role of the random medium), we modify
the law of a random walk X on Z^d with jump rate 1 by reweighting the paths,
giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t
\ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta
L_t(X,Y)), beta in R. As beta increases, the system exhibits a
delocalization/localization transition: there is a critical value beta_c, such
that if beta>beta_c the two walks stick together for almost-all Y realizations.
A natural question is that of disorder relevance, that is whether the quenched
and annealed systems have the same behavior. In this paper we investigate how
the disorder modifies the shape of the free energy curve: (1) We prove that, in
dimension d larger or equal to three 3, the presence of disorder makes the
phase transition at least of second order. This, in dimension larger or equal
to 4, contrasts with the fact that the phase transition of the annealed system
is of first order. (2) In any dimension, we prove that disorder modifies the
low temperature asymptotic of the free energy.Comment: 18 page
Hierarchical pinning models, quadratic maps and quenched disorder
We consider a hierarchical model of polymer pinning in presence of quenched
disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which
can be re-interpreted as an infinite dimensional dynamical system with random
initial condition (the disorder). It is defined through a recurrence relation
for the law of a random variable {R_n}_{n=1,2,...}, which in absence of
disorder (i.e., when the initial condition is degenerate) reduces to a
particular case of the well-known Logistic Map. The large-n limit of the
sequence of random variables 2^{-n} log R_n, a non-random quantity which is
naturally interpreted as a free energy, plays a central role in our analysis.
The model depends on a parameter alpha>0, related to the geometry of the
hierarchical lattice, and has a phase transition in the sense that the free
energy is positive if the expectation of R_0 is larger than a certain threshold
value, and it is zero otherwise. It was conjectured by Derrida et al. (1992)
that disorder is relevant (respectively, irrelevant or marginally relevant) if
1/2<alpha<1 (respectively, alpha<1/2 or alpha=1/2), in the sense that an
arbitrarily small amount of randomness in the initial condition modifies the
critical point with respect to that of the pure (i.e., non-disordered) model if
alpha is larger or equal to 1/2, but not if alpha is smaller than 1/2. Our main
result is a proof of these conjectures for the case alpha different from 1/2.
We emphasize that for alpha>1/2 we find the correct scaling form (for weak
disorder) of the critical point shift.Comment: 26 pages, 2 figures. v3: Theorem 1.6 improved. To appear on Probab.
Theory Rel. Field
Force--induced depinning of directed polymers
We present an approach to studying directed polymers in interaction with a
defect line and subject to a force, which pulls them away from the line. We
consider in particular the case of inhomogeneous interactions. We first give a
formula relating the free energy of these models to the free energy of the
corresponding ones in which the force is switched off. We then show how to
detect the presence of a re-entrant transition without fully solving the model.
We discuss some models in detail and show that inhomogeneous interaction, e.g.
disordered interactions, may induce the re-entrance phenomenon.Comment: 15 pages, 2 figure
Critical properties and finite--size estimates for the depinning transition of directed random polymers
We consider models of directed random polymers interacting with a defect
line, which are known to undergo a pinning/depinning (or
localization/delocalization) phase transition. We are interested in critical
properties and we prove, in particular, finite--size upper bounds on the order
parameter (the {\em contact fraction}) in a window around the critical point,
shrinking with the system size. Moreover, we derive a new inequality relating
the free energy \tf and an annealed exponent which describes extreme
fluctuations of the polymer in the localized region. For the particular case of
a --dimensional interface wetting model, we show that this implies an
inequality between the critical exponents which govern the divergence of the
disorder--averaged correlation length and of the typical one. Our results are
based on on the recently proven smoothness property of the depinning transition
in presence of quenched disorder and on concentration of measure ideas.Comment: 15 pages, 1 figure; accepted for publication on J. Stat. Phy
Smoothening of Depinning Transitions for Directed Polymers with Quenched Disorder
We consider disordered models of pinning of directed polymers on a defect
line, including (1+1)-dimensional interface wetting models, disordered
Poland--Scheraga models of DNA denaturation and other (1+d)-dimensional
polymers in interaction with columnar defects. We consider also random
copolymers at a selective interface. These models are known to have a
(de)pinning transition at some critical line in the phase diagram. In this work
we prove that, as soon as disorder is present, the transition is at least of
second order: the free energy is differentiable at the critical line, and the
order parameter (contact fraction) vanishes continuously at the transition. On
the other hand, it is known that the corresponding non-disordered models can
have a first order (de)pinning transition, with a jump in the order parameter.
Our results confirm predictions based on the Harris criterion.Comment: 4 pages, 1 figure. Version 2: references added, minor changes made.
To appear on Phys. Rev. Let
The free energy in the Derrida--Retaux recursive model
We are interested in a simple max-type recursive model studied by Derrida and
Retaux (2014) in the context of a physics problem, and find a wide range for
the exponent in the free energy in the nearly supercritical regime
Copolymer with pinning: variational characterization of the phase diagram
This paper studies a polymer chain in the vicinity of a linear interface
separating two immiscible solvents. The polymer consists of random monomer
types, while the interface carries random charges. Both the monomer types and
the charges are given by i.i.d. sequences of random variables. The
configurations of the polymer are directed paths that can make i.i.d.
excursions of finite length above and below the interface. The Hamiltonian has
two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface
interaction ("pinning"). The quenched and the annealed version of the model
each undergo a transition from a localized phase (where the polymer stays close
to the interface) to a delocalized phase (where the polymer wanders away from
the interface). We exploit the approach developed in [5] and [3] to derive
variational formulas for the quenched and the annealed free energy per monomer.
These variational formulas are analyzed to obtain detailed information on the
critical curves separating the two phases and on the typical behavior of the
polymer in each of the two phases. Our main results settle a number of open
questions.Comment: 46 pages, 9 figure
Fractional moment bounds and disorder relevance for pinning models
We study the critical point of directed pinning/wetting models with quenched
disorder. The distribution K(.) of the location of the first contact of the
(free) polymer with the defect line is assumed to be of the form
K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a
(de)-localization phase transition: the free energy (per unit length) is zero
in the delocalized phase and positive in the localized phase. For \alpha<1/2 it
is known that disorder is irrelevant: quenched and annealed critical points
coincide for small disorder, as well as quenched and annealed critical
exponents. The same has been proven also for \alpha=1/2, but under the
assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that
is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs
et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant.
Here we prove that, if 1/21, then quenched and annealed
critical points differ whenever disorder is present, and we give the scaling
form of their difference for small disorder. In agreement with the so-called
Harris criterion, disorder is therefore relevant in this case. In the marginal
case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at
infinity, we prove that the difference between quenched and annealed critical
points, which is known to be smaller than any power of the disorder strength,
is positive: disorder is marginally relevant. Again, the case considered by
Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and
remains open.Comment: 20 pages, 1 figure; v2: few typos corrected, references revised. To
appear on Commun. Math. Phy
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