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Random flights on regular graphs

Published online by Cambridge University Press:  01 July 2016

Lajos Takács*
Affiliation:
Case Western Reserve University

Abstract

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

This is an invited address delivered at the Central Regional Meeting of the Institute of Mathematical Statistics, Nashville, Tennessee, March 1983.

References

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[3] Takács, L. (1982) Random walks on groups. Linear Algebra Appl. 43, 4967.CrossRefGoogle Scholar
[4] Takács, L. Harmonic analysis on Schur algebras.Google Scholar