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Random spanning trees of Cayley graphs and an associated compactification of semigroups

Published online by Cambridge University Press:  20 January 2009

Steven N. Evans
Steven N. Evans
Affiliation:
Department of Statistics #3860, University of California, 367 Evans Hall, Berkeley, Ca 94720–3860, U.S.A.
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A sequential construction of a random spanning tree for the Cayley graph of a finitely generated, countably infinite subsemigroup V of a group G is considered. At stage n, the spanning tree T isapproximated by a finite tree Tn rooted at the identity.The approximation Tn+1 is obtained by connecting edges to the points of V that are not already vertices of Tn but can be obtained from vertices of Tn via multiplication by a random walk step taking values in the generating set of V. This construction leads to a compactification of the semigroup V inwhich a sequence of elements of V that is not eventually constant is convergent if the random geodesic through the spanning tree T that joins the identity to the nth element of the sequence converges in distribution as n→∞. The compactification is identified in a number of examples. Also, it is shown that if h(Tn) and #(Tn) denote, respectively, the height and size of the approximating tree Tn, then there are constants 0<ch≤1 and 0≥c# ≤log2 such that limn→∞ n–1 h(Tn)= ch and limn→∞n–1 log# (Tn)= c# almost surely.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 611 - 620
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

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