73 research outputs found
Ginzburg-Landau model with small pinning domains
We consider a Ginzburg-Landau type energy with a piecewise constant pinning
term in the potential . The function is different from
1 only on finitely many disjoint domains, called the {\it pinning domains}.
These pinning domains model small impurities in a homogeneous superconductor
and shrink to single points in the limit ; here, \v is the inverse of
the Ginzburg-Landau parameter. We study the energy minimization in a smooth
simply connected domain with Dirichlet boundary
condition on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d
>0. Our main result is that, for small \v, minimizers have distinct
zeros (vortices) which are inside the pinning domains and they have a degree
equal to 1. The question of finding the locations of the pinning domains with
vortices is reduced to a discrete minimization problem for a finite-dimensional
functional of renormalized energy. We also find the position of the vortices
inside the pinning domains and show that, asymptotically, this position is
determined by {\it local renormalized energy} which does not depend on the
external boundary conditions.Comment: 39 page
Are the Tails of Percolation Thresholds Gaussians ?
The probability distribution of percolation thresholds in finite lattices
were first believed to follow a normal Gaussian behaviour. With increasing
computer power and more efficient simulational techniques, this belief turned
to a stretched exponential behaviour, instead. Here, based on a further
improvement of Monte Carlo data, we show evidences that this question is not
yet answered at all.Comment: 7 pages including 3 figure
Efficient Monte Carlo algorithm and high-precision results for percolation
We present a new Monte Carlo algorithm for studying site or bond percolation
on any lattice. The algorithm allows us to calculate quantities such as the
cluster size distribution or spanning probability over the entire range of site
or bond occupation probabilities from zero to one in a single run which takes
an amount of time scaling linearly with the number of sites on the lattice. We
use our algorithm to determine that the percolation transition occurs at
occupation probability 0.59274621(13) for site percolation on the square
lattice and to provide clear numerical confirmation of the conjectured
4/3-power stretched-exponential tails in the spanning probability functions.Comment: 8 pages, including 3 postscript figures, minor corrections in this
version, plus updated figures for the position of the percolation transitio
Chiral tunneling in single and bilayer graphene
We review chiral (Klein) tunneling in single-layer and bilayer graphene and
present its semiclassical theory, including the Berry phase and the Maslov
index. Peculiarities of the chiral tunneling are naturally explained in terms
of classical phase space. In a one-dimensional geometry we reduced the original
Dirac equation, describing the dynamics of charge carriers in the single layer
graphene, to an effective Schr\"odinger equation with a complex potential. This
allowed us to study tunneling in details and obtain analytic formulas. Our
predictions are compared with numerical results. We have also demonstrated
that, for the case of asymmetric n-p-n junction in single layer graphene, there
is total transmission for normal incidence only, side resonances are
suppressed.Comment: submitted to Proceedings of Nobel Symposium on graphene, May 201
Universality of the Crossing Probability for the Potts Model for q=1,2,3,4
The universality of the crossing probability of a system to
percolate only in the horizontal direction, was investigated numerically by
using a cluster Monte-Carlo algorithm for the -state Potts model for
and for percolation . We check the percolation through
Fortuin-Kasteleyn clusters near the critical point on the square lattice by
using representation of the Potts model as the correlated site-bond percolation
model. It was shown that probability of a system to percolate only in the
horizontal direction has universal form for
as a function of the scaling variable . Here,
is the probability of a bond to be closed, is the
nonuniversal crossing amplitude, is the nonuniversal metric factor,
is the nonuniversal scaling index, is the correlation
length index.
The universal function . Nonuniversal scaling factors
were found numerically.Comment: 15 pages, 3 figures, revtex4b, (minor errors in text fixed,
journal-ref added
Crossing probability and number of crossing clusters in off-critical percolation
We consider two-dimensional percolation in the scaling limit close to
criticality and use integrable field theory to obtain universal predictions for
the probability that at least one cluster crosses between opposite sides of a
rectangle of sides much larger than the correlation length and for the mean
number of such crossing clusters.Comment: 13 pages, 5 figures. Published version with references, appendix and
comparison with numerics adde
A fast Monte Carlo algorithm for site or bond percolation
We describe in detail a new and highly efficient algorithm for studying site
or bond percolation on any lattice. The algorithm can measure an observable
quantity in a percolation system for all values of the site or bond occupation
probability from zero to one in an amount of time which scales linearly with
the size of the system. We demonstrate our algorithm by using it to investigate
a number of issues in percolation theory, including the position of the
percolation transition for site percolation on the square lattice, the
stretched exponential behavior of spanning probabilities away from the critical
point, and the size of the giant component for site percolation on random
graphs.Comment: 17 pages, 13 figures. Corrections and some additional material in
this version. Accompanying material can be found on the web at
http://www.santafe.edu/~mark/percolation
Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast
We consider divergence-form scalar elliptic equations and vectorial equations
for elasticity with rough (, )
coefficients that, in particular, model media with non-separated scales
and high contrast in material properties. We define the flux norm as the
norm of the potential part of the fluxes of solutions, which is equivalent to
the usual -norm. We show that in the flux norm, the error associated with
approximating, in a properly defined finite-dimensional space, the set of
solutions of the aforementioned PDEs with rough coefficients is equal to the
error associated with approximating the set of solutions of the same type of
PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial).
We refer to this property as the {\it transfer property}.
A simple application of this property is the construction of finite
dimensional approximation spaces with errors independent of the regularity and
contrast of the coefficients and with optimal and explicit convergence rates.
This transfer property also provides an alternative to the global harmonic
change of coordinates for the homogenization of elliptic operators that can be
extended to elasticity equations. The proofs of these homogenization results
are based on a new class of elliptic inequalities which play the same role in
our approach as the div-curl lemma in classical homogenization.Comment: Accepted for publication in Archives for Rational Mechanics and
Analysi
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