Directed Percolation: An Overture and Three Easy Pieces
        Peter Grassberger 
        Höchstleistungsrechenzentrum HLRZ 
        Forschungszentrum Jülich 
        D-52425 Jülich 
        grass@hlrz.kfa-juelich.de

We present simulations of three variants of directed percolation in 2 dimensions, viewed as a model for the spreading of an epidemic in 1-dimensional space. The first variant concerns long term memory, as modeled previously by Grassberger, Chate and Rousseau [1]. While the phase diagram of this model is well understood, previous simulations could not give precise scaling laws for the case where the re-infection probability is critical, but the chance for a first infection is subcritical. The second variant deals with spatial disorder. As pointed out some time ago by Noest and others [2,3], one should expect there a Griffiths-like phase in the subcritical regime. For one particular version of this model, Bramson et al. [4] (see also [5]) proved also the existence of a supercritical phase with sublinear spreading. Finally, we consider also directed percolation with temporal
disorder as studied by I. Jensen [6].

These simulations are made with two novel algorithms which are most efficient for OFF-critical clusters: For supercritical clusters we use a contour-following depth-first algorithm, while an algorithm with Rosenbluth type bias and enrichment is used for subcritical clusters. In all three problems we find significant new results.

[1] P. Grassberger et al., Phys. Rev. E 55, 2488 (1997)
[2] A. Noest, Phys. Rev. Lett. 57, 90 (1986); Phys. Rev. B 38, 2715 (1988)
[3] R. Dickman and A.G. Moreira, Phys. Rev. E 54, R3090 (1996);
    Phys. Rev. E 57, 1263 (1998)
[4] M. Bramson et al., Ann. Prob. 19, 960 (1991)
[5] R. Cafiero et al., Phys. Rev. E 57, 5060 (1998)
[6] I. Jensen, Phys. Rev. Lett. 77, 4988 (1996)

Back