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 $r^{-(d+\sigma)}$. 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 $\nu$, and the anomalous dimension $\eta$. For $\sigma>1$, the critical fugacity varies smoothly with $\sigma$, while the Binder ratio and the critical exponents remain consistent with the short-range universality class. For $\sigma \le 1$, by contrast, the results clearly depart from short-range behavior, identifying $\sigma=1$ as the boundary between long-range and short-range regimes. In the long-range Wilson-Fisher regime with $1/2<\sigma \leq 1$, $\eta$ agrees with the long-range Gaussian-fixed-point prediction $\eta_{\mathrm{GFP}}=2-\sigma$, whereas $Q_N^c$ and $\nu$ vary nontrivially with $\sigma$ and exhibit discontinuous jumps at $\sigma=1$. These findings are in good agreement with the recently proposed universality diagram in the $(d,\sigma)$ plane for long-range O$(n)$ models, with the self-avoiding walk corresponding to the $n\to0$ limit.