The European Physical Journal B (EPJ B)

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Eur. Phys. J. B 7, 137-145

Lévy-flight spreading of epidemic processes leading
to percolating clusters

H.K. Janssen¹ - K. Oerding¹ - F. van Wijland² - H.J. Hilhorst²

¹ Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
² Laboratoire de Physique Théorique et Hautes Énergies, Université de Paris-Sud, 91405 Orsay Cedex, France
janssen@thphy.uni-duesseldorf.de, oerding@thphy.uni-duesseldorf.de

Received: 17 July 1998 / Revised: 20 July 1998 / Accepted: 28 July 1998

Abstract
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 $\varepsilon$ -expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection $\sigma =\sigma _{c}\gt 2$ .

PACS
64.60.Ak Renormalization-group, fractal, and percolation studies of phase transitions - 64.60.Ht Dynamic critical phenomena - 05.40.+j Fluctuation phenomena, random processes, and Brownian motion

Contents

Lévy-flight spreading of epidemic processes leading to percolating clusters

Lévy-flight spreading of epidemic processes leading
to percolating clusters