1,416 research outputs found
Universality in Two-Dimensional Enhancement Percolation
We consider a type of dependent percolation introduced by Aizenman and
Grimmett, who showed that certain "enhancements" of independent (Bernoulli)
percolation, called essential, make the percolation critical probability
strictly smaller. In this paper we first prove that, for two-dimensional
enhancements with a natural monotonicity property, being essential is also a
necessary condition to shift the critical point. We then show that (some)
critical exponents and the scaling limit of crossing probabilities of a
two-dimensional percolation process are unchanged if the process is subjected
to a monotonic enhancement that is not essential. This proves a form of
universality for all dependent percolation models obtained via a monotonic
enhancement (of Bernoulli percolation) that does not shift the critical point.
For the case of site percolation on the triangular lattice, we also prove a
stronger form of universality by showing that the full scaling limit is not
affected by any monotonic enhancement that does not shift the critical point.Comment: 36 pages, 4 figure
Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation
Consider a cellular automaton with state space
where the initial configuration is chosen according to a Bernoulli
product measure, 1's are stable, and 0's become 1's if they are surrounded by
at least three neighboring 1's. In this paper we show that the configuration
at time n converges exponentially fast to a final configuration
, and that the limiting measure corresponding to is in
the universality class of Bernoulli (independent) percolation.
More precisely, assuming the existence of the critical exponents ,
, and , and of the continuum scaling limit of crossing
probabilities for independent site percolation on the close-packed version of
(i.e., for independent -percolation on ), we
prove that the bootstrapped percolation model has the same scaling limit and
critical exponents.
This type of bootstrap percolation can be seen as a paradigm for a class of
cellular automata whose evolution is given, at each time step, by a monotonic
and nonessential enhancement.Comment: 15 page
Quantum disorder and Griffiths singularities in bond-diluted two-dimensional Heisenberg antiferromagnets
We investigate quantum phase transitions in the spin-1/2 Heisenberg
antiferromagnet on square lattices with inhomogeneous bond dilution. It is
shown that quantum fluctuations can be continuously tuned by inhomogeneous bond
dilution, eventually leading to the destruction of long-range magnetic order on
the percolating cluster. Two multicritical points are identified at which the
magnetic transition separates from the percolation transition, introducing a
novel quantum phase transition. Beyond these multicritical points a
quantum-disordered phase appears, characterized by an infinite percolating
cluster with short ranged antiferromagnetic order. In this phase, the
low-temperature uniform susceptibility diverges algebraically with
non-universal exponents. This is a signature that the novel quantum-disordered
phase is a quantum Griffiths phase, as also directly confirmed by the
statistical distribution of local gaps. This study thus presents evidence of a
genuine quantum Griffiths phenomenon in a two-dimensional Heisenberg
antiferromagnet.Comment: 14 pages, 17 figures; published versio
Experimental evidence of percolation phase transition in surface plasmons generation
Carrying digital information in traditional copper wires is becoming a major
issue in electronic circuits. Optical connections such as fiber optics offers
unprecedented transfer capacity, but the mismatch between the optical
wavelength and the transistors size drastically reduces the coupling
efficiency. By merging the abilities of photonics and electronics, surface
plasmon photonics, or 'plasmonics' exhibits strong potential. Here, we propose
an original approach to fully understand the nature of surface electrons in
plasmonic systems, by experimentally demonstrating that surface plasmons can be
modeled as a phase of surface waves. First and second order phase transitions,
associated with percolation transitions, have been experimentally observed in
the building process of surface plasmons in lattice of subwavelength apertures.
Percolation theory provides a unified framework for surface plasmons
description
Tunneling-percolation origin of nonuniversality: theory and experiments
A vast class of disordered conducting-insulating compounds close to the
percolation threshold is characterized by nonuniversal values of transport
critical exponent t, in disagreement with the standard theory of percolation
which predicts t = 2.0 for all three dimensional systems. Various models have
been proposed in order to explain the origin of such universality breakdown.
Among them, the tunneling-percolation model calls into play tunneling processes
between conducting particles which, under some general circumstances, could
lead to transport exponents dependent of the mean tunneling distance a. The
validity of such theory could be tested by changing the parameter a by means of
an applied mechanical strain. We have applied this idea to universal and
nonuniversal RuO2-glass composites. We show that when t > 2 the measured
piezoresistive response \Gamma, i. e., the relative change of resistivity under
applied strain, diverges logarithmically at the percolation threshold, while
for t = 2, \Gamma does not show an appreciable dependence upon the RuO2 volume
fraction. These results are consistent with a mean tunneling dependence of the
nonuniversal transport exponent as predicted by the tunneling-percolation
model. The experimental results are compared with analytical and numerical
calculations on a random-resistor network model of tunneling-percolation.Comment: 13 pages, 12 figure
Response of a catalytic reaction to periodic variation of the CO pressure: Increased CO_2 production and dynamic phase transition
We present a kinetic Monte Carlo study of the dynamical response of a
Ziff-Gulari-Barshad model for CO oxidation with CO desorption to periodic
variation of the CO presure. We use a square-wave periodic pressure variation
with parameters that can be tuned to enhance the catalytic activity. We produce
evidence that, below a critical value of the desorption rate, the driven system
undergoes a dynamic phase transition between a CO_2 productive phase and a
nonproductive one at a critical value of the period of the pressure
oscillation. At the dynamic phase transition the period-averged CO_2 production
rate is significantly increased and can be used as a dynamic order parameter.
We perform a finite-size scaling analysis that indicates the existence of
power-law singularities for the order parameter and its fluctuations, yielding
estimated critical exponent ratios and . These exponent ratios, together with theoretical symmetry
arguments and numerical data for the fourth-order cumulant associated with the
transition, give reasonable support for the hypothesis that the observed
nonequilibrium dynamic phase transition is in the same universality class as
the two-dimensional equilibrium Ising model.Comment: 18 pages, 10 figures, accepted in Physical Review
Origin of the Universal Roughness Exponent of Brittle Fracture Surfaces: Correlated Percolation in the Damage Zone
We suggest that the observed large-scale universal roughness of brittle
fracture surfaces is due to the fracture process being a correlated percolation
process in a self-generated quadratic damage gradient. We use the quasi-static
two-dimensional fuse model as a paradigm of a fracture model. We measure for
this model, that exhibits a correlated percolation process, the correlation
length exponent nu approximately equal to 1.35 and conjecture it to be equal to
that of uncorrelated percolation, 4/3. We then show that the roughness exponent
in the fuse model is zeta = 2 nu/(1+2 nu)= 8/11. This is in accordance with the
numerical value zeta=0.75. As for three-dimensional brittle fractures, a
mean-field theory gives nu=2, leading to zeta=4/5 in full accordance with the
universally observed value zeta =0.80.Comment: 4 pages RevTeX
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