Singularities of the Green function of a random walk on a discrete group |
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Authors: | Donald I Cartwright |
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Institution: | (1) School of Mathematics and Statisties, University of Sydney, 2006 Sydney, NSW, Australia |
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Abstract: | LetX be a countable discrete group and let be an irreducible probability onX. The radius of convergence of the Green function
is finite, and independent ofx. Let
be the period of . We show that for eachx X the singularities of the analytic functionz G(x; z) on the circle {z![isin](/content/w71730h73jl21205/xxlarge8712.gif) :|z|= } are precisely the points e
2 ik/d
k=0, ...,d–1. In particular, is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when (e)>0). This affirms a conjecture ofLalley 5]. When is symmetric, i.e., (x
–1)= (x) for allx X, d is either 1 or 2. As another particular case of our result, we see that- is then a singularity ofz G (x; z) if and only ifd=2, in which caseX is bicolored . This answers a question ofde la Harpe, Robertson andValette 2]. |
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Keywords: | |
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