### Abstract

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponents v and 2 Δ_{4} -γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relation dv = 2 Δ_{4} -γ. In two dimensions, we confirm the predicted exponent v=3/4 and the hyperscaling relation; we estimate the universal ratios <R_{g}^{2}>/<R_{e}^{2}>=0.14026±0.00007, <R_{m}^{2}>/<R_{e}^{2}>=0.43961±0.00034, and Ψ^{*}=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimate v=0.5877±0.0006 with a correctionto-scaling exponent Δ_{1}=0.56±0.03 (subjective 68% confidence limits). This value for v agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy for Δ_{1}. Earlier Monte Carlo estimates of v, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R_{g}^{2}>/<R_{e}^{2}>=0.1599±0.0002 and Ψ^{*}=0.2471±0.0003; since Ψ^{*}>0, hyperscaling holds. The approach to Ψ^{*} is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relation dv = 2 Δ_{4} -γ for two-dimensional SAWs.

Original language | English (US) |
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Pages (from-to) | 661-754 |

Number of pages | 94 |

Journal | Journal of Statistical Physics |

Volume | 80 |

Issue number | 3-4 |

DOIs | |

State | Published - Aug 1 1995 |

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### Keywords

- Karp-Luby algorithm
- Monte Carlo
- Self-avoiding walk
- critical exponent
- hyperscaling
- interpenetration ratio
- pivot algorithm
- polymer
- renormalization group
- second virial coefficient
- two-parameter theory
- universal amplitude ratio

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*80*(3-4), 661-754. https://doi.org/10.1007/BF02178552