60 research outputs found
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
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
Statistical Approach to Deterministic Modelling of Air Pollution and its Applications in Russia
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