NDLI: First Passage Percolation on $$\mathbb {Z}^2$$ : A Simulation Study

Content Provider	Springer Nature Link
Author	Alm, Sven Erick Deijfen, Maria
Copyright Year	2015
Abstract	First passage percolation on $$\mathbb {Z}^2$$ is a model for describing the spread of an infection on the sites of the square lattice. The infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges. In this paper, the speed of the growth and the shape of the infected set is studied by aid of large-scale computer simulations, with focus on continuous passage time distributions. It is found that the most important quantity for determining the value of the time constant, which indicates the inverse asymptotic speed of the growth, is $${\mathbf E}[\min \{\tau _1,\ldots ,\tau _4\}]$$ , where $$\tau _1,\ldots ,\tau _4$$ are i.i.d. passage time variables. The relation is linear for a large class of passage time distributions. Furthermore, the directional time constants are seen to be increasing when moving from the axis towards the diagonal, so that the limiting shape is contained in a circle with radius defined by the speed along the axes. The shape comes closer to the circle for distributions with larger variability.
Starting Page	657
Ending Page	678
Page Count	22
File Format	PDF
ISSN	00224715
Journal	Journal of Statistical Physics
Volume Number	161
Issue Number	3
e-ISSN	15729613
Language	English
Publisher	Springer US
Publisher Date	2015-08-30
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	First passage percolation Growth model Time constant Asymptotic shape Computer simulation Statistical Physics, Dynamical Systems and Complexity Theoretical, Mathematical and Computational Physics Physical Chemistry Quantum Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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