Compact directed percolation with movable partial reflectors

We study a version of compact directed percolation (CDP) in one dimension in which occupation of a site for the first time requires that a "mine" or antiparticle be eliminated. This process is analogous to the variant of directed percolation with a long-time memory, proposed by Grassberger, Chate and Rousseau [Phys. Rev. E 55, 2488 (1997)] in order to understand spreading at a critical point involving an infinite number of absorbing configurations. The problem is equivalent to that of a pair of random walkers in the presence of movable partial reflectors. The walkers, which are unbiased, start one lattice spacing apart, and annihilate on their first contact. Each time one of the walkers tries to visit a new site, it is reflected (with probability r) back to its previous position, while the reflector is simultaneously pushed one step away from the walker. Iteration of the discrete-time evolution equation for the probability distribution yields the survival probability S(t). We find that S(t) \sim t^{-delta}, with delta varying continuously between 1/2 and 1.160 as the reflection probability varies between 0 and 1.Comment: 12 pages, 4 figure

Small-scale behaviour in deterministic reaction models

In a recent paper published in this journal [J. Phys. A: Math. Theor. 42 (2009) 495004] we studied a one-dimensional particles system where nearest particles attract with a force inversely proportional to a power \alpha of their distance and coalesce upon encounter. Numerics yielded a distribution function h(z) for the gap between neighbouring particles, with h(z)=z^{\beta(\alpha)} for small z and \beta(\alpha)>\alpha. We can now prove analytically that in the strict limit of z\to 0, \beta=\alpha for \alpha>0, corresponding to the mean-field result, and we compute the length scale where mean-field breaks down. More generally, in that same limit correlations are negligible for any similar reaction model where attractive forces diverge with vanishing distance. The actual meaning of the measured exponent \beta(\alpha) remains an open question.Comment: Six pages. Section 2 has been rewritten. Accepted for publication in Journal of Physics A: Mathematical and Theoretica

On the Google-Fame of Scientists and Other Populations

We study the fame distribution of scientists and other social groups as measured by the number of Google hits garnered by individuals in the population. Past studies have found that the fame distribution decays either in power-law [arXiv:cond-mat/0310049] or exponential [Europhys. Lett., 67, (4) 511-516 (2004)] fashion, depending on whether individuals in the social group in question enjoy true fame or not. In our present study we examine critically Google counts as well as the methods of data analysis. While the previous findings are corroborated in our present study, we find that, in most situations, the data available does not allow for sharp conclusions.Comment: 6 pages, 1 figure, to appear in the proceedings of the 8th Granada seminar on Computational Physic