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A relatively quick way to simulate local random processes on a lattice

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Daniel Richardson  and
Wayne Burton
Daniel Richardson*
Affiliation:
University of Bath
Wayne Burton*
Affiliation:
University of Bath
*
∗Postal address: School of Mathematical Sciences, University of Bath, Claverdon Down, Bath BA2 7AY, UK.
∗Postal address: School of Mathematical Sciences, University of Bath, Claverdon Down, Bath BA2 7AY, UK.
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Abstract

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

Keywords

simulationlatticepercolationheap

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 3 , September 1998 , pp. 770 - 775
Copyright
Copyright © Applied Probability Trust 1998 

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References

Kingston, J. (1990). Algorithms and Data Structures. Addison Wesley, Reading, MA.Google Scholar
Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhauser, Boston.CrossRefGoogle Scholar
Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar
Smythe, R. T., and Wierman, J. C. (1978). First Passage Percolation on the Square Lattice. Lecture Notes in Mathematics 671, Springer-Verlag, Berlin.Google Scholar