Self-avoiding random loops

A random loop, or polygon, is a simple random walk whose trajectory is a simple Jordan curve. The study of random loops is extended in two ways. First, the probability P/sub n/(x,y) that a random n-step loop contains a point (x,y) in the interior of the loop is studied, and (1/2, 1/2) is shown to be...

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Bibliographic Details
Published in:IEEE transactions on information theory 1988-11, Vol.34 (6), p.1509-1516
Main Authors: Dubins, L.E., Orlitsky, A., Reeds, J.A., Shepp, L.A.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Information theory
Markov processes
Mathematics
Probability
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Stochastic processes
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Summary:A random loop, or polygon, is a simple random walk whose trajectory is a simple Jordan curve. The study of random loops is extended in two ways. First, the probability P/sub n/(x,y) that a random n-step loop contains a point (x,y) in the interior of the loop is studied, and (1/2, 1/2) is shown to be (1/2)-(1/n). It is plausible that P/sub n/(x,y) tends toward 1/2 for all (x,y), but this is not proved even for (x,y)=(3/2,1/2) A way is offered to simulate random n-step self-avoiding loops. Numerical evidence obtained with this simulation procedure suggests that the probability P/sub n/(3/2,1/2) approximately=(1/2)-(c/n), for some fixed c.< >
ISSN:0018-9448
1557-9654
DOI:10.1109/18.21290