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