3,554 research outputs found
On the critical threshold for continuum AB percolation
Consider a bipartite random geometric graph on the union of two independent
homogeneous Poisson point processes in -space, with distance parameter
and intensities . For any we consider the percolation
threshold associated to the parameter . Denoting by
the percolation threshold for the standard Poisson
Boolean model with radii , we show the lower bound for any with a fixed
constant. In particular, tends to infinity when
tends to from above.Comment: 12 pages, 3 figure
Results for a critical threshold, the correction-to-scaling exponent and susceptibility amplitude ratio for 2d percolation
We summarize several decades of work in finding values for the percolation
threshold p_c for site percolation on the square lattice, the universal
correction-to-scaling exponent Omega, and the susceptibility amplitude ratio
C^+/C^-, in two dimensions. Recent studies have yielded the precise values p_c
= 0.59274602(4), Omega = 72/91 = 0.791, and C^+/C^- = 161.5(1.5), resolving
long-standing controversies about the last two quantities and verifying the
widely used value p_c = 0.592746 for the first.Comment: Talk presented at 24th Annual Workshop:"Recent Developments in
Computer Simulational Studies in Condensed Matter Physics," Center for
Computational Physics, University of Georgia, Athens, Georgia Feb. 21-25,
201
Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks
This article is a mini-review about electrical current flows in networks from
the perspective of statistical physics. We briefly discuss analytical methods
to solve the conductance of an arbitrary resistor network. We then turn to
basic results related to percolation: namely, the conduction properties of a
large random resistor network as the fraction of resistors is varied. We focus
on how the conductance of such a network vanishes as the percolation threshold
is approached from above. We also discuss the more microscopic current
distribution within each resistor of a large network. At the percolation
threshold, this distribution is multifractal in that all moments of this
distribution have independent scaling properties. We will discuss the meaning
of multifractal scaling and its implications for current flows in networks,
especially the largest current in the network. Finally, we discuss the relation
between resistor networks and random walks and show how the classic phenomena
of recurrence and transience of random walks are simply related to the
conductance of a corresponding electrical network.Comment: 27 pages & 10 figures; review article for the Encyclopedia of
Complexity and System Science (Springer Science
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
Percolation-induced exponential scaling in the large current tails of random resistor networks
There is a renewed surge in percolation-induced transport properties of
diverse nano-particle composites (cf. RSC Nanoscience & Nanotechnology Series,
Paul O'Brien Editor-in-Chief). We note in particular a broad interest in
nano-composites exhibiting sharp electrical property gains at and above
percolation threshold, which motivated us to revisit the classical setting of
percolation in random resistor networks but from a multiscale perspective. For
each realization of random resistor networks above threshold, we use network
graph representations and associated algorithms to identify and restrict to the
percolating component, thereby preconditioning the network both in size and
accuracy by filtering {\it a priori} zero current-carrying bonds. We then
simulate many realizations per bond density and analyze scaling behavior of the
complete current distribution supported on the percolating component. We first
confirm the celebrated power-law distribution of small currents at the
percolation threshold, and second we confirm results on scaling of the maximum
current in the network that is associated with the backbone of the percolating
cluster. These properties are then placed in context with global features of
the current distribution, and in particular the dominant role of the large
current tail that is most relevant for material science applications. We
identify a robust, exponential large current tail that: 1. persists above
threshold; 2. expands broadly over and dominates the current distribution at
the expense of the vanishing power law scaling in the small current tail; and
3. by taking second moments, reproduces the experimentally observed power law
scaling of bulk conductivity above threshold
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