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Self-Avoiding Walks on Hyperbolic Graphs

Published online by Cambridge University Press:  21 July 2005

NEAL MADRAS  and
C. CHRIS WU
NEAL MADRAS
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3 Canada (e-mail: madras@mathstat.yorku.ca)
C. CHRIS WU
Affiliation:
Department of Mathematics, Penn State University, Beaver Campus, 100 University Drive, Monaca, PA 15061, USA (e-mail: ccw3@psu.edu)
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Abstract

We study self-avoiding walks (SAWs) on non-Euclidean lattices that correspond to regular tilings of the hyperbolic plane (‘hyperbolic graphs’). We prove that on all but at most eight such graphs, (i) there are exponentially fewer $N$-step self-avoiding polygons than there are $N$-step SAWs, (ii) the number of $N$-step SAWs grows as $\mu_w^N$ within a constant factor, and (iii) the average end-to-end distance of an $N$-step SAW is approximately proportional to $N$. In terms of critical exponents from statistical physics, (ii) says that $\gamma=1$ and (iii) says that $\nu=1$. We also prove that $\gamma$ is finite on all hyperbolic graphs, and we prove a general identity about non-reversing walks that had previously been discovered for certain special cases.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 14 , Issue 4 , July 2005 , pp. 523 - 548
Copyright
© 2005 Cambridge University Press

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