16,962 research outputs found
Finite-Size Scaling in Two-dimensional Continuum Percolation Models
We test the universal finite-size scaling of the cluster mass order parameter
in two-dimensional (2D) isotropic and directed continuum percolation models
below the percolation threshold by computer simulations. We found that the
simulation data in the 2D continuum models obey the same scaling expression of
mass M to sample size L as generally accepted for isotropic lattice problems,
but with a positive sign of the slope in the ln-ln plot of M versus L. Another
interesting aspect of the finite-size 2D models is also suggested by plotting
the normalized mass in 2D continuum and lattice bond percolation models, versus
an effective percolation parameter, independently of the system structure (i.e.
lattice or continuum) and of the possible directions allowed for percolation
(i.e. isotropic or directed) in regions close to the percolation thresholds.
Our study is the first attempt to map the scaling behaviour of the mass for
both lattice and continuum model systems into one curve.Comment: 9 pages, Revtex, 2 PostScript figure
Two-Dimensional Scaling Limits via Marked Nonsimple Loops
We postulate the existence of a natural Poissonian marking of the double
(touching) points of SLE(6) and hence of the related continuum nonsimple loop
process that describes macroscopic cluster boundaries in 2D critical
percolation. We explain how these marked loops should yield continuum versions
of near-critical percolation, dynamical percolation, minimal spanning trees and
related plane filling curves, and invasion percolation. We show that this
yields for some of the continuum objects a conformal covariance property that
generalizes the conformal invariance of critical systems. It is an open problem
to rigorously construct the continuum objects and to prove that they are indeed
the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure
Theory of continuum percolation I. General formalism
The theoretical basis of continuum percolation has changed greatly since its
beginning as little more than an analogy with lattice systems. Nevertheless,
there is yet no comprehensive theory of this field. A basis for such a theory
is provided here with the introduction of the Potts fluid, a system of
interacting -state spins which are free to move in the continuum. In the limit, the Potts magnetization, susceptibility and correlation functions
are directly related to the percolation probability, the mean cluster size and
the pair-connectedness, respectively. Through the Hamiltonian formulation of
the Potts fluid, the standard methods of statistical mechanics can therefore be
used in the continuum percolation problem.Comment: 26 pages, Late
The random geometry of equilibrium phases
This is a (long) survey about applications of percolation theory in
equilibrium statistical mechanics. The chapters are as follows:
1. Introduction
2. Equilibrium phases
3. Some models
4. Coupling and stochastic domination
5. Percolation
6. Random-cluster representations
7. Uniqueness and exponential mixing from non-percolation
8. Phase transition and percolation
9. Random interactions
10. Continuum modelsComment: 118 pages. Addresses: [email protected]
http://www.mathematik.uni-muenchen.de/~georgii.html [email protected]
http://www.math.chalmers.se/~olleh [email protected]
Theory of continuum percolation II. Mean field theory
I use a previously introduced mapping between the continuum percolation model
and the Potts fluid to derive a mean field theory of continuum percolation
systems. This is done by introducing a new variational principle, the basis of
which has to be taken, for now, as heuristic. The critical exponents obtained
are , and , which are identical with the mean
field exponents of lattice percolation. The critical density in this
approximation is \rho_c = 1/\ve where \ve = \int d \x \, p(\x) \{ \exp [-
v(\x)/kT] - 1 \}. p(\x) is the binding probability of two particles
separated by \x and v(\x) is their interaction potential.Comment: 25 pages, Late
The Judicial Expansion of American Exceptionalism
The percolation theory is established as a useful tool in the field of pharmaceutical materials science.It is shown that percolation theory, developed for analyzing insulator–conductor transitions, can beapplied to describe imperfect dc conduction in pharmaceutical microcrystalline cellulose duringdensification. The system, in fact, exactly reproduces the values of the percolation threshold andexponent estimated for a three-dimensional random continuum. Our data clearly show a crossoverfrom a power-law percolation theory region to a linear effective medium theory region at a celluloseporosity of ;0.7
Continuum percolation for Cox point processes
We investigate continuum percolation for Cox point processes, that is,
Poisson point processes driven by random intensity measures. First, we derive
sufficient conditions for the existence of non-trivial sub- and super-critical
percolation regimes based on the notion of stabilization. Second, we give
asymptotic expressions for the percolation probability in large-radius,
high-density and coupled regimes. In some regimes, we find universality,
whereas in others, a sensitive dependence on the underlying random intensity
measure survives.Comment: 21 pages, 5 figure
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