42 research outputs found
Interacting partially directed self avoiding walk. From phase transition to the geometry of the collapsed phase
In this paper, we investigate a model for a dimensional
self-interacting and partially directed self-avoiding walk, usually referred to
by the acronym IPDSAW. The interaction intensity and the free energy of the
system are denoted by and , respectively. The IPDSAW is known to
undergo a collapse transition at . We provide the precise asymptotic
of the free energy close to criticality, that is we show that
where is computed
explicitly and interpreted in terms of an associated continuous model. We also
establish some path properties of the random walk inside the collapsed phase
. We prove that the geometric conformation adopted by the
polymer is made of a succession of long vertical stretches that attract each
other to form a unique macroscopic bead, we identify the horizontal extension
of the random walk inside the collapsed phase and we establish the convergence
of the rescaled envelope of the macroscopic bead towards a deterministic Wulff
shape.Comment: Accepted for publication in the Annals of Probabilit
On the localized phase of a copolymer in an emulsion: supercritical percolation regime
Article / Letter to editorMathematisch Instituu
Critical exponents in zero dimensions
In the vicinity of the onset of an instability, we investigate the effect of
colored multiplicative noise on the scaling of the moments of the unstable mode
amplitude. We introduce a family of zero dimensional models for which we can
calculate the exact value of the critical exponents for all the
moments. The results are obtained through asymptotic expansions that use the
distance to onset as a small parameter. The examined family displays a variety
of behaviors of the critical exponents that includes anomalous exponents:
exponents that differ from the deterministic (mean-field) prediction, and
multiscaling: non-linear dependence of the exponents on the order of the
moment
On the magnetic fields generated by experimental dynamos
We review the results obtained by three successful fluid dynamo experiments
and discuss what has been learnt from them about the effect of turbulence on
the dynamo threshold and saturation. We then discuss several questions that are
still open and propose experiments that could be performed to answer some of
them.Comment: 40 pages, 13 figure
Effects of the low frequencies of noise on On-Off intermittency
A bifurcating system subject to multiplicative noise can exhibit on-off
intermittency close to the instability threshold. For a canonical system, we
discuss the dependence of this intermittency on the Power Spectrum Density
(PSD) of the noise. Our study is based on the calculation of the Probability
Density Function (PDF) of the unstable variable. We derive analytical results
for some particular types of noises and interpret them in the framework of
on-off intermittency. Besides, we perform a cumulant expansion for a random
noise with arbitrary power spectrum density and show that the intermittent
regime is controlled by the ratio between the departure from the threshold and
the value of the PSD of the noise at zero frequency. Our results are in
agreement with numerical simulations performed with two types of random
perturbations: colored Gaussian noise and deterministic fluctuations of a
chaotic variable. Extensions of this study to another, more complex, system are
presented and the underlying mechanisms are discussed.Comment: 13pages, 13 figure
Phase diagram for a copolymer in a micro-emulsion
Analysis and Stochastic
A mathematical model for a copolymer in an emulsion
Analysis and Stochastic
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
A simple mechanism for the reversals of Earth's magnetic field
We show that a model, recently used to describe all the dynamical regimes of
the magnetic field generated by the dynamo effect in the VKS experiment [1],
also provides a simple explanation of the reversals of Earth's magnetic field,
despite strong differences between both systems.Comment: update version, with new figure
Annealed scaling for a charged polymer
Analysis and Stochastic