202 research outputs found
Scaling of loop-erased walks in 2 to 4 dimensions
We simulate loop-erased random walks on simple (hyper-)cubic lattices of
dimensions 2,3, and 4. These simulations were mainly motivated to test recent
two loop renormalization group predictions for logarithmic corrections in
, simulations in lower dimensions were done for completeness and in order
to test the algorithm. In , we verify with high precision the prediction
, where the number of steps after erasure scales with the number
of steps before erasure as . In we again find a power law,
but with an exponent different from the one found in the most precise previous
simulations: . Finally, we see clear deviations from the
naive scaling in . While they agree only qualitatively with the
leading logarithmic corrections predicted by several authors, their agreement
with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure
Percolation Threshold, Fisher Exponent, and Shortest Path Exponent for 4 and 5 Dimensions
We develop a method of constructing percolation clusters that allows us to
build very large clusters using very little computer memory by limiting the
maximum number of sites for which we maintain state information to a number of
the order of the number of sites in the largest chemical shell of the cluster
being created. The memory required to grow a cluster of mass s is of the order
of bytes where ranges from 0.4 for 2-dimensional lattices
to 0.5 for 6- (or higher)-dimensional lattices. We use this method to estimate
, the exponent relating the minimum path to the
Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site
and bond percolation, we find (4D) and
(5D). In order to determine
to high precision, and without bias, it was necessary to
first find precise values for the percolation threshold, :
(4D) and (5D) for site and
(4D) and (5D) for bond
percolation. We also calculate the Fisher exponent, , determined in the
course of calculating the values of : (4D) and
(5D)
Continuous Percolation Phase Transitions of Two-dimensional Lattice Networks under a Generalized Achlioptas Process
The percolation phase transitions of two-dimensional lattice networks under a
generalized Achlioptas process (GAP) are investigated. During the GAP, two
edges are chosen randomly from the lattice and the edge with minimum product of
the two connecting cluster sizes is taken as the next occupied bond with a
probability . At , the GAP becomes the random growth model and leads
to the minority product rule at . Using the finite-size scaling analysis,
we find that the percolation phase transitions of these systems with are always continuous and their critical exponents depend on .
Therefore, the universality class of the critical phenomena in two-dimensional
lattice networks under the GAP is related to the probability parameter in
addition.Comment: 7 pages, 14 figures, accepted for publication in Eur. Phys. J.
Competitive random sequential adsorption of point and fixed-sized particles: analytical results
We study the kinetics of competitive random sequential adsorption (RSA) of
particles of binary mixture of points and fixed-sized particles within the
mean-field approach. The present work is a generalization of the random car
parking problem in the sense that it considers the case when either a car of
fixed size is parked with probability q or the parking space is partitioned
into two smaller spaces with probability (1-q) at each time event. This allows
an interesting interplay between the classical RSA problem at one extreme
(q=1), and the kinetics of fragmentation processes at the other extreme (q=0).
We present exact analytical results for coverage for a whole range of q values,
and physical explanations are given for different aspects of the problem. In
addition, a comprehensive account of the scaling theory, emphasizing on
dimensional analysis, is presented, and the exact expression for the scaling
function and exponents are obtained.Comment: 7 pages, latex, 3 figure
Universal crossing probability in anisotropic systems
Scale-invariant universal crossing probabilities are studied for critical
anisotropic systems in two dimensions. For weakly anisotropic standard
percolation in a rectangular-shaped system, Cardy's exact formula is
generalized using a length-rescaling procedure. For strongly anisotropic
systems in 1+1 dimensions, exact results are obtained for the random walk with
absorbing boundary conditions, which can be considered as a linearized
mean-field approximation for directed percolation. The bond and site directed
percolation problem is itself studied numerically via Monte Carlo simulations
on the diagonal square lattice with either free or periodic boundary
conditions. A scale-invariant critical crossing probability is still obtained,
which is a universal function of the effective aspect ratio r_eff=c r where
r=L/t^z, z is the dynamical exponent and c is a non-universal amplitude.Comment: 7 pages, 4 figure
Proposal for a CFT interpretation of Watts' differential equation for percolation
G. M. T. Watts derived that in two dimensional critical percolation the
crossing probability Pi_hv satisfies a fifth order differential equation which
includes another one of third order whose independent solutions describe the
physically relevant quantities 1, Pi_h, Pi_hv.
We will show that this differential equation can be derived from a level
three null vector condition of a rational c=-24 CFT and motivate how this
solution may be fitted into known properties of percolation.Comment: LaTeX, 20p, added references, corrected typos and additional content
The critical amplitude ratio of the susceptibility in the random-site two-dimensional Ising model
We present a new way of probing the universality class of the site-diluted
two-dimensional Ising model. We analyse Monte Carlo data for the magnetic
susceptibility, introducing a new fitting procedure in the critical region
applicable even for a single sample with quenched disorder. This gives us the
possibility to fit simultaneously the critical exponent, the critical amplitude
and the sample dependent pseudo-critical temperature. The critical amplitude
ratio of the magnetic susceptibility is seen to be independent of the
concentration of the empty sites for all investigated values of . At the same time the average effective exponent is found
to vary with the concentration , which may be argued to be due to
logarithmic corrections to the power law of the pure system. This corrections
are canceled in the susceptibility amplitude ratio as predicted by theory. The
central charge of the corresponding field theory was computed and compared well
with the theoretical predictions.Comment: 6 pages, 4 figure
Gaussian fluctuations in an ideal bose-gas -- a simple model
Based on the canonical ensemble, we suggested the simple scheme for taking
into account Gaussian fluctuations in a finite system of ideal boson gas.
Within framework of scheme we investigated the influence of fluctuations on the
particle distribution in Bose -gas for two cases - with taking into account the
number of particles in the ground state and without this assumption. The
temperature and fluctuation parameter dependences of the modified Bose-
Einstein distribution have been determined. Also the dependence of the
condensation temperature on the fluctuation distribution parameter has been
obtained.Comment: arXiv admin note: text overlap with arXiv:cond-mat/040859
Dynamic Scaling in One-Dimensional Cluster-Cluster Aggregation
We study the dynamic scaling properties of an aggregation model in which
particles obey both diffusive and driven ballistic dynamics. The diffusion
constant and the velocity of a cluster of size follow
and , respectively. We determine the dynamic exponent and
the phase diagram for the asymptotic aggregation behavior in one dimension in
the presence of mixed dynamics. The asymptotic dynamics is dominated by the
process that has the largest dynamic exponent with a crossover that is located
at . The cluster size distributions scale similarly in all
cases but the scaling function depends continuously on and .
For the purely diffusive case the scaling function has a transition from
exponential to algebraic behavior at small argument values as changes
sign whereas in the drift dominated case the scaling function decays always
exponentially.Comment: 6 pages, 6 figures, RevTeX, submitted to Phys. Rev.
Kinetics of fragmentation-annihilation processes
We investigate the kinetics of systems in which particles of one species
undergo binary fragmentation and pair annihilation. In the latter, nonlinear
process, fragments react at collision to produce an inert species, causing loss
of mass. We analyse these systems in the reaction-limited regime by solving a
continuous model within the mean-field approximation. The rate of
fragmentation, for a particle of mass to break into fragments of masses
and , has the form (), and the annihilation
rate is constant and independent of the masses of the reactants. We find that
the asymptotic regime is characterized by the annihilation of small-mass
clusters. The results are compared with those for a model with linear mass-loss
(i.e.\ with a sink). We also study more complex models, in which the processes
of fragmentation and annihilation are controlled by mutually-reacting
catalysts. Both pair- and linear-annihilation are considered. Depending on the
specific model and initial densities of the catalysts, the time-decay of the
cluster-density can now be very unconventional and even non-universal. The
interplay between the intervening processes and the existence of a scaling
regime are determined by the asymptotic behaviour of the average-mass and of
the mass-density, which may either decay indefinitely or tend to a constant
value. We discuss further developments of this class of models and their
potential applications.Comment: 16 pages(LaTeX), submitted to Phys. Rev.
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