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Random walks on a dodecahedron

Published online by Cambridge University Press:  14 July 2016

G. Letac  and
L. Takács
G. Letac*
Affiliation:
Université Paul-Sabatier
L. Takács*
Affiliation:
Case Western Reserve University
*
∗Postal address: Université Paul-Sabatier, Mathématiques, 118 route de Narbonne, 31400 Toulouse, France.
∗∗Postal address: Department of Mathematics and Statistics, Case Western Reserve University, University Circle, Cleveland, OH 44106, U.S.A.
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Abstract

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn+1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X0 = A] and P[In = A | I0 = A]. The methods used are applicable to other solids.

Keywords

MARKOV CHAINSREGULAR DODECAHEDRON
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 373 - 384
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

The paper was prepared while this author was in residence at Case Western Reserve University as Visiting Professor of Mathematics and Statistics.

References

[1] Arnaud, J. P. (1977) Sur quelques propriétés des librairies. Thèse 3eme cycle, Université Paul-Sabatier, Toulouse.Google Scholar
[2] Callahan, F. P. (1966) Problem E1897. Amer. Math. Monthly 73, 665. Solution by Kuttler, J. R. and Rubenstein, N., Amer. Math. Monthly 74 (1977), 1008–1010.Google Scholar
[3] Engel, A. (1965) Problem E1752. Amer. Math. Monthly 72, 75. Solution by Teichmann, T., Amer. Math. Monthly 73 (1966), 200.Google Scholar
[4] Heller, A. (1965) On stochastic processes derived from Markov chains. Ann. Math. Statist. 36, 1286–1291.Google Scholar
[5] Letac, G. (1977) Problem 6149. Amer. Math. Monthly 84, 301.Google Scholar
[6] Letac, G. (1978) Chaînes de Markov sur les permutations. Presses de l'Universite de Montréal, Montréal.Google Scholar