84 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
$D_B$ of the percolation backbone to three loop order. Using renormalization
group methods we obtain $D_B = 2 + \epsilon /21 - 172\epsilon^2 /9261 + 2
\epsilon^3 (- 74639 + 22680 \zeta (3))/4084101$, where $\epsilon = 6-d$ with
$d$ being the spatial dimension and $\zeta (3) = 1.202057..$.Comment: 10 pages, 2 figure

### 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

### 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

### 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 $\theta$ as well
as the exponents $z$ and $2\beta/\nu$ 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

### 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 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

### Microscopic Deterministic Dynamics and Persistence Exponent

Numerically we solve the microscopic deterministic equations of motion with
random initial states for the two-dimensional $\phi^4$ 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.

### Generalized Dynamic Scaling for Critical Relaxations

The dynamic relaxation process for the two dimensional Potts model at
criticality starting from an initial state with very high temperature and
arbitrary magnetization is investigated with Monte Carlo methods. The results
show that there exists universal scaling behaviour even in the short-time
regime of the dynamic evolution. In order to describe the dependence of the
scaling behaviour on the initial magnetization, a critical characteristic
function is introduced.Comment: Latex, 8 pages, 3 figures, to appear in Phys. Rev. Let

### Non Markovian persistence in the diluted Ising model at criticality

We investigate global persistence properties for the non-equilibrium critical
dynamics of the randomly diluted Ising model. The disorder averaged persistence
probability $\bar{{P}_c}(t)$ of the global magnetization is found to decay
algebraically with an exponent $\theta_c$ that we compute analytically in a
dimensional expansion in $d=4-\epsilon$. Corrections to Markov process are
found to occur already at one loop order and $\theta_c$ is thus a novel
exponent characterizing this disordered critical point. Our result is
thoroughly compared with Monte Carlo simulations in $d=3$, which also include a
measurement of the initial slip exponent. Taking carefully into account
corrections to scaling, $\theta_c$ is found to be a universal exponent,
independent of the dilution factor $p$ along the critical line at $T_c(p)$, and
in good agreement with our one loop calculation.Comment: 7 pages, 4 figure

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