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 $z_c$, 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 $σ\le 1$, 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 $1/2 < σ\leq 1$, $η$ agrees with the long-range Gaussian-fixed-point prediction $η_{\mathrm{GFP}}=2-σ$, 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 $n\to0$ limit.