4,759 research outputs found
Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field
We study the statistics of the extremes of a discrete Gaussian field with
logarithmic correlations at the level of the Gibbs measure. The model is
defined on the periodic interval , and its correlation structure is
nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm.
Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory
Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random
Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001)
026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008)
372001]. At low temperature, it is shown that the normalized covariance of two
points sampled from the Gibbs measure is either or . This is used to
prove that the joint distribution of the Gibbs weights converges in a suitable
sense to that of a Poisson-Dirichlet variable. In particular, this proves a
conjecture of Carpentier and Le Doussal that the statistics of the extremes of
the log-correlated field behave as those of i.i.d. Gaussian variables and of
branching Brownian motion at the level of the Gibbs measure. The method of
proof is robust and is adaptable to other log-correlated Gaussian fields.Comment: Published in at http://dx.doi.org/10.1214/13-AAP952 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Interface Collisions
We provide a theoretical framework to analyze the properties of frontal
collisions of two growing interfaces considering different short range
interactions between them. Due to their roughness, the collision events spread
in time and form rough domain boundaries, which defines collision interfaces in
time and space. We show that statistical properties of such interfaces depend
on the kinetics of the growing interfaces before collision, but are independent
of the details of their interaction and of their fluctuations during the
collision. Those properties exhibit dynamic scaling with exponents related to
the growth kinetics, but their distributions may be non-universal. These
results are supported by simulations of lattice models with irreversible
dynamics and local interactions. Relations to first passage processes are
discussed and a possible application to grain boundary formation in
two-dimensional materials is suggested.Comment: Paper with 12 pages and 2 figures; supplemental material with 4 pages
and 3 figure
Mean Field Games and Applications.
This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.Mean Field Games;
Thin film modeling of crystal dissolution and growth in confinement
We present a continuum model describing dissolution and growth of a crystal
contact confined against a substrate. Diffusion and hydrodynamics in the liquid
film separating the crystal and the substrate are modeled within the
lubrication approximation. The model also accounts for the disjoining pressure
and surface tension. Within this framework, we obtain evolution equations which
govern the non-equilibrium dynamics of the crystal interface. Based on this
model, we explore the problem of dissolution under an external load, known as
pressure solution. We find that in steady-state, diverging (power-law)
crystal-surface repulsions lead to flat contacts with a monotonic increase of
the dissolution rate as a function of the load. Forces induced by viscous
dissipation then surpass those due to disjoining pressure at large enough
loads. In contrast, finite repulsions (exponential) lead to sharp pointy
contacts with a dissolution rate independent on the load and on the liquid
viscosity. Ultimately, in steady-state the crystal never touches the substrate
when pressed against it, independently from the nature of the crystal-surface
interaction due to the combined effects of viscosity and surface tension
Confined Growth with slow surface kinetics: a Thin Film Model approach
Recent experimental and theoretical investigations of crystal growth from
solution in the vicinity of an impermeable wall have shown that: (i) growth can
be maintained within the contact region when a liquid film is present between
the crystal and the substrate; (ii) a cavity can form in the center of the
contact region due to insufficient supply of mass through the liquid film.
Here, we investigate the influence of surface kinetics on these phenomena using
a thin film model. First, we determine the growth rate within the confined
region in the absence of a cavity. Growth within the contact induces a drift of
the crystal away from the substrate. Our results suggest novel strategies to
measure surface kinetic coefficients based on the observation of this drift.
For the specific case where growth is controlled by surface kinetics outside
the contact, we show that the total displacement of the crystal due to the
growth in the contact is finite. As a consequence, the growth shape approaches
asymptotically the free growth shape truncated by a plane passing through the
center of the crystal. Second, we investigate the conditions under which a
cavity forms. The critical supersaturation above which the cavity forms is
found to be larger for slower surface kinetics. In addition, the critical
supersaturation decays as a power law of the contact size. The asymptotic value
of the critical supersaturation and the exponent of the decay are found to be
different for attractive and repulsive disjoining pressures. Finally, our
previous representation of the transition within a morphology diagram appears
to be uninformative in the limit of slow surface kinetics
Nonlinear wavelength selection in surface faceting under electromigration
We report on the control of the faceting of crystal surfaces by means of
surface electromigration. When electromigration reinforces the faceting
instability, we find perpetual coarsening with a wavelength increasing as
. For strongly stabilizing electromigration, the surface is stable.
For weakly stabilizing electromigration, a cellular pattern is obtained, with a
nonlinearly selected wavelength. The selection mechanism is not caused by an
instability of steady-states, as suggested by previous works in the literature.
Instead, the dynamics is found to exhibit coarsening {\it before} reaching a
continuous family of stable non-equilibrium steady-states.Comment: 5 pages, 4 figures, submitte
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