Skip to main content

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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 34 *
Loading metrics...

Abstract views

Total abstract views: 115 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th March 2018. This data will be updated every 24 hours.