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Nested Monte Carlo study of random packing on the sphere

Published online by Cambridge University Press:  14 July 2016

R. P. C. Rodgers  and
A. J. Baddeley
R. P. C. Rodgers*
Affiliation:
University of California, San Francisco
A. J. Baddeley*
Affiliation:
Centre for Mathematics and Computer Science, Amsterdam
*
∗ Postal address: Depts of Pharmaceutical Chemistry and Laboratory Medicine, University of California, San Francisco, CA 94143, USA.
∗∗ Postal address: Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, Netherlands.
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Abstract

We consider two random sequential packing processes in which spheres of unit radius are randomly attached to the surface of a fixed unit sphere. Independent random spheres are generated and added successively, provided there is no overlap with previous spheres. In model 1, the process stops when a trial sphere intersects one of the previously-accepted spheres. In model 2, random sequential packing, any such overlapping trial sphere is discarded and the next random sphere is tried, until it is impossible to add any further spheres.

Previous workers have conjectured convincingly that no exact analytical solution is possible for this type of problem. We use Monte Carlo simulation methods to estimate transition probabilities for the two models. Because some probabilities are extremely small, a simulation using independent repetitions of the model would be inefficient. We designed a branching process of conditionally binomial trials, and performed over 108 trials on a supercomputer.

Keywords

BRANCHING PROCESSBRANCHING SIMULATIONCAR PARKING PROBLEMCRAY XMPMONTE CARLORANDOM AGGREGATIONRANDOM NUMBER GENERATORSRANDOM POINTS ON A SPHERESPHERE PACKINGSUPERCOMPUTINGTESTS OF UNIFORM DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 539 - 552
Copyright
Copyright © Applied Probability Trust 1991 

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