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Self-Avoiding Walks on Hyperbolic Graphs

  • NEAL MADRAS (a1) and C. CHRIS WU (a2)

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.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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