40 research outputs found
One-dimensional random field Kac's model: weak large deviations principle
We prove a quenched weak large deviations principle for the Gibbs measures of
a Random Field Kac Model (RFKM) in one dimension. The external random magnetic
field is given by symmetrically distributed Bernoulli random variables. The
results are valid for values of the temperature, , and magnitude,
, of the field in the region where the free energy of the corresponding
random Curie Weiss model has only two absolute minima and .
We give an explicit representation of the rate functional which is a positive
random functional determined by two distinct contributions. One is related to
the free energy cost to undergo a phase change (the surface
tension). The is the cost of one single phase change and depends
on the temperature and magnitude of the field. The other is a bulk contribution
due to the presence of the random magnetic field. We characterize the
minimizers of this random functional. We show that they are step functions
taking values and . The points of discontinuity are
described by a stationary renewal process related to the extrema for a
bilateral Brownian motion studied by Neveu and Pitman, where in our context
is a suitable constant depending on the temperature and on magnitude of the
random field. As an outcome we have a complete characterization of the typical
profiles of RFKM (the ground states) which was initiated in [14] and extended
in [16]
Large deviations from a macroscopic scaling limit for particle systems with Kac interaction and random potential
Spectral properties of integral operators in bounded, large intervals
We study the spectrum of one dimensional integral operators in bounded real
intervals of length , for value of large. The integral operators are
obtained by linearizing a non local evolution equation for a non conserved
order parameter describing the phases of a fluid. We prove a Perron-Frobenius
theorem showing that there is an isolated, simple minimal eigenvalue strictly
positive for finite, going to zero exponentially fast in . We lower
bound, uniformly on , the spectral gap by applying a generalization of the
Cheeger's inequality. These results are usefulfor deriving spectral properties
for non local Cahn-Hilliard type of equations in problems of interface
dynamics.Comment: An serious error has been corrected and an author has been adde
Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states
A particle system with a single locally-conserved field (density) in a
bounded interval with different densities maintained at the two endpoints of
the interval is under study here. The particles interact in the bulk through a
long range potential parametrized by and evolve according to an
exclusion rule. It is shown that the empirical particle density under the
diffusive scaling solves a quasi-linear integro-differential evolution equation
with Dirichlet boundary conditions. The associated dynamical large deviation
principle is proved. Furthermore, for small enough, it is also
demonstrated that the empirical particle density obeys a law of large numbers
with respect to the stationary measures (hydrostatic). The macroscopic particle
density solves a non local, stationary, transport equation
One-dimensional random field Kac's model: localization of the phases
We study the typical profiles of a one dimensional random field Kac model,
for values of the temperature and magnitude of the field in the region of the
two absolute minima for the free energy of the corresponding random field Curie
Weiss model. We show that, for a set of realizations of the random field of
overwhelming probability, the localization of the two phases corresponding to
the previous minima is completely determined. Namely, we are able to construct
random intervals tagged with a sign, where typically, with respect to the
infinite volume Gibbs measure, the profile is rigid and takes, according to the
sign, one of the two values corresponding to the previous minima. Moreover, we
characterize the transition from one phase to the other
Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations
In this paper we address the questions of perfectly sampling a Gibbs measure
with infinite range interactions and of perfectly sampling the measure together
with its finite range approximations. We solve these questions by introducing a
perfect simulation algorithm for the measure and for the coupled measures. The
algorithm works for general Gibbsian interaction under requirements on the
tails of the interaction. As a consequence we obtain an upper bound for the
error we make when sampling from a finite range approximation instead of the
true infinite range measure