112 research outputs found

    The Resistance of Feynman Diagrams and the Percolation Backbone Dimension

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    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 DBD_B of the percolation backbone to three loop order. Using renormalization group methods we obtain DB=2+ϵ/21−172ϵ2/9261+2ϵ3(−74639+22680ζ(3))/4084101D_B = 2 + \epsilon /21 - 172\epsilon^2 /9261 + 2 \epsilon^3 (- 74639 + 22680 \zeta (3))/4084101, where ϵ=6−d\epsilon = 6-d with dd being the spatial dimension and ζ(3)=1.202057..\zeta (3) = 1.202057...Comment: 10 pages, 2 figure

    Wilson renormalization of a reaction-diffusion process

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    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

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    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

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    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 θ\theta as well as the exponents zz and 2β/ν2\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

    The short-time behaviour of a kinetic Ashkin-Teller model on the critical line

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    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 θ\theta vary along the critical line. Another dynamical exponent zz is also obtained in the process.Comment: 14 pages LaTeX with 4 figures in postscrip

    Persistence in an antiferromagnetic Ising model with conserved magnetisation

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    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

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    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

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

    Microscopic Non-Universality versus Macroscopic Universality in Algorithms for Critical Dynamics

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    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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