112 research outputs found
The Resistance of Feynman Diagrams and the Percolation Backbone Dimension
We present a new view of Feynman diagrams for the field theory of transport
on percolation clusters. The diagrams for random resistor networks are
interpreted as being resistor networks themselves. This simplifies the field
theory considerably as we demonstrate by calculating the fractal dimension
of the percolation backbone to three loop order. Using renormalization
group methods we obtain , where with
being the spatial dimension and .Comment: 10 pages, 2 figure
Wilson renormalization of a reaction-diffusion process
Healthy and sick individuals (A and B particles) diffuse independently with
diffusion constants D_A and D_B. Sick individuals upon encounter infect healthy
ones (at rate k), but may also spontaneously recover (at rate 1/\tau). The
propagation of the epidemic therefore couples to the fluctuations in the total
population density. Global extinction occurs below a critical value \rho_{c} of
the spatially averaged total density. The epidemic evolves as the
diffusion--reaction--decay process
A + B --> 2B, B --> A ,
for which we write down the field theory. The stationary state properties of
this theory when D_A=D_B were obtained by Kree et al. The critical behavior for
D_A<D_B is governed by a new fixed point. We calculate the critical exponents
of the stationary state in an \eps expansion, carried out by Wilson
renormalization, below the critical dimension d_{c}=4. We then go on to to
obtain the critical initial time behavior at the extinction threshold, both for
D_A=D_B and D_A<D_B. There is nonuniversal dependence on the initial particle
distribution. The case D_A>D_B remains unsolved.Comment: 26 pages, LaTeX, 6 .eps figures include
Global Persistence in Directed Percolation
We consider a directed percolation process at its critical point. The
probability that the deviation of the global order parameter with respect to
its average has not changed its sign between 0 and t decays with t as a power
law. In space dimensions d<4 the global persistence exponent theta_p that
characterizes this decay is theta_p=2 while for d<4 its value is increased to
first order in epsilon = 4-d. Combining a method developed by Majumdar and Sire
with renormalization group techniques we compute the correction to theta_p to
first order in epsilon. The global persistence exponent is found to be a new
and independent exponent. We finally compare our results with existing
simulations.Comment: 15 pages, LaTeX, one .eps figure include
Universality and Scaling in Short-time Critical Dynamics
Numerically we simulate the short-time behaviour of the critical dynamics for
the two dimensional Ising model and Potts model with an initial state of very
high temperature and small magnetization. Critical initial increase of the
magnetization is observed. The new dynamic critical exponent as well
as the exponents and are determined from the power law
behaviour of the magnetization, auto-correlation and the second moment.
Furthermore the calculation has been carried out with both Heat-bath and
Metropolis algorithms. All the results are consistent and therefore
universality and scaling are confirmed.Comment: 14 pages, 14 figure
The short-time behaviour of a kinetic Ashkin-Teller model on the critical line
We simulate the kinetic Ashkin-Teller model with both ordered and disordered
initial states, evolving in contact with a heat-bath at the critical
temperature. The power law scaling behaviour for the magnetic order and
electric order are observed in the early time stage. The values of the critical
exponent vary along the critical line. Another dynamical exponent
is also obtained in the process.Comment: 14 pages LaTeX with 4 figures in postscrip
Persistence in an antiferromagnetic Ising model with conserved magnetisation
We obtain the persistence exponents for an antiferromagnetic Ising system in
which the magnetisation is kept constant. This system belongs to Model C
(system with non-conserved order parameter with a conserved density) and is
expected to have persistence exponents different from that of Model A (system
with no conservation) but independent of the conserved density. Our numerical
results for both local persistence at zero temperature and global persistence
at the critical temperature however indicate that the exponents are dependent
on the conserved magnetisation in both two and three dimensions. This
nonuniversal feature is attributed to the presence of the conserved field and
is special to the persistence phenomena.Comment: 8 pages, to be published in Physica A (Proceedings of the
Statphys-Kolkata IV, 2002
Levy-flight spreading of epidemic processes leading to percolating clusters
We consider two stochastic processes, the Gribov process and the general
epidemic process, that describe the spreading of an infectious disease. In
contrast to the usually assumed case of short-range infections that lead, at
the critical point, to directed and isotropic percolation respectively, we
consider long-range infections with a probability distribution decaying in d
dimensions with the distance as 1/R^{d+\sigma}. By means of Wilson's momentum
shell renormalization-group recursion relations, the critical exponents
characterizing the growing fractal clusters are calculated to first order in an
\epsilon-expansion. It is shown that the long-range critical behavior changes
continuously to its short-range counterpart for a decay exponent of the
infection \sigma =\sigma_c>2.Comment: 9 pages ReVTeX, 2 postscript figures included, submitted to Eur.
Phys. J.
Microscopic Deterministic Dynamics and Persistence Exponent
Numerically we solve the microscopic deterministic equations of motion with
random initial states for the two-dimensional theory. Scaling behavior
of the persistence probability at criticality is systematically investigated
and the persistence exponent is estimated.Comment: to appear in Mod. Phys. Lett.
Microscopic Non-Universality versus Macroscopic Universality in Algorithms for Critical Dynamics
We study relaxation processes in spin systems near criticality after a quench
from a high-temperature initial state. Special attention is paid to the stage
where universal behavior, with increasing order parameter emerges from an early
non-universal period. We compare various algorithms, lattice types, and
updating schemes and find in each case the same universal behavior at
macroscopic times, despite of surprising differences during the early
non-universal stages.Comment: 9 pages, 3 figures, RevTeX, submitted to Phys. Rev. Let
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