72 research outputs found
Localization-delocalization phenomena for random interfaces
We consider d-dimensional random surface models which for d=1 are the
standard (tied-down) random walks (considered as a random ``string''). In
higher dimensions, the one-dimensional (discrete) time parameter of the random
walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it.
The random surface is represented by real-valued random variables \phi_i, where
i is in \Z^d. A class of natural generalizations of the standard random walk
are gradient models whose laws are (formally) expressed as
P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i,
V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been
introduced in theoretical physics as (simplified) models for random
interfaces separating different phases. Of particular interest are
localization-delocalization phenomena, for instance for a surface interacting
with a wall by attracting or repulsive interactions, or both together. Another
example are so-called heteropolymers which have a noise-induced interaction.
Recently, there had been developments of new probabilistic tools for such
problems. Among them are: o Random walk representations of Helffer-Sj\"ostrand
type, o Multiscale analysis, o Connections with random trapping problems and
large deviations We give a survey of some of these developments
On a nonhierarchical version of the generalized random energy model
We introduce a natural nonhierarchical version of Derrida's generalized
random energy model. We prove that, in the thermodynamical limit, the free
energy is the same as that of a suitably constructed GREM.Comment: Published at http://dx.doi.org/10.1214/105051605000000665 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Periodic copolymers at selective interfaces: A Large Deviations approach
We analyze a (1+1)-dimension directed random walk model of a polymer dipped
in a medium constituted by two immiscible solvents separated by a flat
interface. The polymer chain is heterogeneous in the sense that a single
monomer may energetically favor one or the other solvent. We focus on the case
in which the polymer types are periodically distributed along the chain or, in
other words, the polymer is constituted of identical stretches of fixed length.
The phenomenon that one wants to analyze is the localization at the interface:
energetically favored configurations place most of the monomers in the
preferred solvent and this can be done only if the polymer sticks close to the
interface. We investigate, by means of large deviations, the energy-entropy
competition that may lead, according to the value of the parameters (the
strength of the coupling between monomers and solvents and an asymmetry
parameter), to localization. We express the free energy of the system in terms
of a variational formula that we can solve. We then use the result to analyze
the phase diagram.Comment: Published at http://dx.doi.org/10.1214/105051604000000800 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Critical behavior of the massless free field at the depinning transition
We consider the d-dimensional massless free field localized by a
delta-pinning of strength e. We study the asymptotics of the variance of the
field, and of the decay-rate of its 2-point function, as e goes to zero, for
general Gaussian interactions. Physically speaking, we thus rigorously obtain
the critical behavior of the transverse and longitudinal correlation lengths of
the corresponding d+1-dimensional effective interface model in a non-mean-field
regime. We also describe the set of pinned sites at small e, for a broad class
of d-dimensional massless models
Fast decay of covariances under pinning in the critical and supercritical membrane model
We consider the membrane model, that is the centered Gaussian field on
whose covariance matrix is given by the inverse of the discrete
Bilaplacian. We impose a pinning condition, giving a reward of
strength for the field to be at any site of the lattice. In
this paper we prove that in dimensions covariances of the pinned
field decay at least stretched-exponentially, as opposed to the field without
pinning, where the decay is polynomial in and logarithmic in
The proof is based on estimates for certain discrete Sobolev norms, and on a
Bernoulli domination result.Comment: 16 pages, 1 figure. Comments are welcom
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