Long-range self-avoiding walk in one dimension: a Monte Carlo study
Abstract
We study the one-dimensional long-range self-avoiding walk in the grand-canonical ensemble where the statistical weight of a jump of length r decays algebraically as r−(d+σ). Using large-scale Monte Carlo simulations with an efficient irreversible update scheme, we obtain high-precision estimates of the critical fugacity zc, the universal Binder ratio QNc, the correlation-length exponent ν, and the anomalous dimension η. For σ>1, the critical fugacity varies smoothly with σ, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For σ≤ 1, by contrast, the results clearly depart from short-range behavior, identifying σ=1 as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with 1/2 < σ≤ 1, η agrees with the long-range Gaussian-fixed-point prediction ηGFP=2−σ, whereas QNc and ν vary nontrivially with σ and exhibit discontinuous jumps at σ=1. These findings are in good agreement with the recently proposed universality diagram in the (d,σ) plane for long-range O(n) models, with the self-avoiding walk corresponding to the n→0 limit.
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BibTeX
@misc{zhou2026longrange,
title = {{Long-range self-avoiding walk in one dimension: a Monte Carlo study}},
author = {Jiang Zhou and Ziyu Liu and Pengcheng Hou and Zhijie Fan and Youjin Deng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23346},
eprint = {2609.23346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23346},
}
RIS
TY - UNPB
TI - Long-range self-avoiding walk in one dimension: a Monte Carlo study
AU - Jiang Zhou
AU - Ziyu Liu
AU - Pengcheng Hou
AU - Zhijie Fan
AU - Youjin Deng
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23346
UR - https://arxiv.org/abs/2609.23346
AB - We study the one-dimensional long-range self-avoiding walk in the grand-canonical ensemble where the statistical weight of a jump of length r decays algebraically as r^−(d+σ). Using large-scale Monte Carlo simulations with an efficient irreversible update scheme, we obtain high-precision estimates of the critical fugacity z_c, the universal Binder ratio Q_N^c, the correlation-length exponent ν, and the anomalous dimension η. For σ>1, the critical fugacity varies smoothly with σ, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For σ≤ 1, by contrast, the results clearly depart from short-range behavior, identifying σ=1 as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with 1/2 < σ≤ 1, η agrees with the long-range Gaussian-fixed-point prediction η_GFP=2−σ, whereas Q_N^c and ν vary nontrivially with σ and exhibit discontinuous jumps at σ=1. These findings are in good agreement with the recently proposed universality diagram in the (d,σ) plane for long-range O(n) models, with the self-avoiding walk corresponding to the n→0 limit.
ER -