137 research outputs found
Fluctuations for the Ginzburg-Landau Interface Model on a Bounded Domain
We study the massless field on , where is a bounded domain with smooth boundary, with Hamiltonian
\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y)). The interaction \CV is assumed
to be symmetric and uniformly convex. This is a general model for a
-dimensional effective interface where represents the height. We
take our boundary conditions to be a continuous perturbation of a macroscopic
tilt: for , , and
continuous. We prove that the fluctuations of linear
functionals of about the tilt converge in the limit to a Gaussian free
field on , the standard Gaussian with respect to the weighted Dirichlet
inner product for some explicit . In a subsequent article,
we will employ the tools developed here to resolve a conjecture of Sheffield
that the zero contour lines of are asymptotically described by , a
conformally invariant random curve.Comment: 58 page
Self-similar stable processes arising from high-density limits of occupation times of particle systems
We extend results on time-rescaled occupation time fluctuation limits of the
-branching particle system with Poisson initial condition. The earlier results in the homogeneous case
(i.e., with Lebesgue initial intensity measure) were obtained for dimensions
only, since the particle system becomes locally extinct if
. In this paper we show that by introducing high density
of the initial Poisson configuration, limits are obtained for all dimensions,
and they coincide with the previous ones if . We also give
high-density limits for the systems with finite intensity measures (without
high density no limits exist in this case due to extinction); the results are
different and harder to obtain due to the non-invariance of the measure for the
particle motion. In both cases, i.e., Lebesgue and finite intensity measures,
for low dimensions ( and
, respectively) the limits are determined by
non-L\'evy self-similar stable processes. For the corresponding high dimensions
the limits are qualitatively different: -valued L\'evy
processes in the Lebesgue case, stable processes constant in time on
in the finite measure case. For high dimensions, the laws of all
limit processes are expressed in terms of Riesz potentials. If , the
limits are Gaussian. Limits are also given for particle systems without
branching, which yields in particular weighted fractional Brownian motions in
low dimensions. The results are obtained in the setup of weak convergence of
S'(R^d)$-valued processes.Comment: 28 page
A numerical approach to copolymers at selective interfaces
We consider a model of a random copolymer at a selective interface which
undergoes a localization/delocalization transition. In spite of the several
rigorous results available for this model, the theoretical characterization of
the phase transition has remained elusive and there is still no agreement about
several important issues, for example the behavior of the polymer near the
phase transition line. From a rigorous viewpoint non coinciding upper and lower
bounds on the critical line are known.
In this paper we combine numerical computations with rigorous arguments to
get to a better understanding of the phase diagram. Our main results include:
- Various numerical observations that suggest that the critical line lies
strictly in between the two bounds.
- A rigorous statistical test based on concentration inequalities and
super-additivity, for determining whether a given point of the phase diagram is
in the localized phase. This is applied in particular to show that, with a very
low level of error, the lower bound does not coincide with the critical line.
- An analysis of the precise asymptotic behavior of the partition function in
the delocalized phase, with particular attention to the effect of rare atypical
stretches in the disorder sequence and on whether or not in the delocalized
regime the polymer path has a Brownian scaling.
- A new proof of the lower bound on the critical line. This proof relies on a
characterization of the localized regime which is more appealing for
interpreting the numerical data.Comment: accepted for publication on J. Stat. Phy
Bismut-Elworthy-Li formulae for Bessel processes
In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all δ ≥ 0 and T > 0, we compute the derivative of the function x↦PδTF(x), where (Pδt)t≥0 is the transition semi-group associated to the δ-dimensional Bessel process, and F is any bounded Borel function on R+. The obtained expression shows a nice interplay between the transition semi-groups of the δ—and the (δ + 2)-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula
Large deviations of lattice Hamiltonian dynamics coupled to stochastic thermostats
We discuss the Donsker-Varadhan theory of large deviations in the framework
of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We
derive a general formula for the Donsker-Varadhan large deviation functional
for dynamics which satisfy natural properties under time reversal. Next, we
discuss the characterization of the stationary state as the solution of a
variational principle and its relation to the minimum entropy production
principle. Finally, we compute the large deviation functional of the current in
the case of a harmonic chain thermostated by a Gaussian stochastic coupling.Comment: Revised version, published in Journal of Statistical Physic
Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle
Stochastic lattice gases with degenerate rates, namely conservative particle
systems where the exchange rates vanish for some configurations, have been
introduced as simplified models for glassy dynamics. We introduce two
particular models and consider them in a finite volume of size in
contact with particle reservoirs at the boundary. We prove that, as for
non--degenerate rates, the inverse of the spectral gap and the logarithmic
Sobolev constant grow as . It is also shown how one can obtain, via a
scaling limit from the logarithmic Sobolev inequality, the exponential decay of
a macroscopic entropy associated to a degenerate parabolic differential
equation (porous media equation). We analyze finally the tagged particle
displacement for the stationary process in infinite volume. In dimension larger
than two we prove that, in the diffusive scaling limit, it converges to a
Brownian motion with non--degenerate diffusion coefficient.Comment: 25 pages, 3 figure
Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems
We consider a simple class of fast-slow partially hyperbolic dynamical
systems and show that the (properly rescaled) behaviour of the slow variable is
very close to a Friedlin--Wentzell type random system for times that are rather
long, but much shorter than the metastability scale. Also, we show the
possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon
that turns out to be related to the lack of absolutely continuity of the
central foliation.Comment: To appear in Journal of Statistical Physic
Surface tension in the dilute Ising model. The Wulff construction
We study the surface tension and the phenomenon of phase coexistence for the
Ising model on \mathbbm{Z}^d () with ferromagnetic but random
couplings. We prove the convergence in probability (with respect to random
couplings) of surface tension and analyze its large deviations : upper
deviations occur at volume order while lower deviations occur at surface order.
We study the asymptotics of surface tension at low temperatures and relate the
quenched value of surface tension to maximal flows (first passage
times if ). For a broad class of distributions of the couplings we show
that the inequality -- where is the surface
tension under the averaged Gibbs measure -- is strict at low temperatures. We
also describe the phenomenon of phase coexistence in the dilute Ising model and
discuss some of the consequences of the media randomness. All of our results
hold as well for the dilute Potts and random cluster models
Thermal Conductivity for a Momentum Conserving Model
We introduce a model whose thermal conductivity diverges in dimension 1 and
2, while it remains finite in dimension 3. We consider a system of oscillators
perturbed by a stochastic dynamics conserving momentum and energy. We compute
thermal conductivity via Green-Kubo formula. In the harmonic case we compute
the current-current time correlation function, that decay like in
the unpinned case and like if a on-site harmonic potential is
present. This implies a finite conductivity in or in pinned cases, and
we compute it explicitly. For general anharmonic strictly convex interactions
we prove some upper bounds for the conductivity that behave qualitatively as in
the harmonic cases.Comment: Accepted for the publication in Communications in Mathematical
Physic
Rigorous Probabilistic Analysis of Equilibrium Crystal Shapes
The rigorous microscopic theory of equilibrium crystal shapes has made
enormous progress during the last decade. We review here the main results which
have been obtained, both in two and higher dimensions. In particular, we
describe how the phenomenological Wulff and Winterbottom constructions can be
derived from the microscopic description provided by the equilibrium
statistical mechanics of lattice gases. We focus on the main conceptual issues
and describe the central ideas of the existing approaches.Comment: To appear in the March 2000 special issue of Journal of Mathematical
Physics on Probabilistic Methods in Statistical Physic
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