We study the scaling properties of long-range loop-erased random walks (LR-LERW), where the underlying random walker performs Lévy-flight-like jumps with a power-law step-length distribution $P\left(\mathbf{r}\right)\sim {\left|\mathbf{r}\right|}^{-(d+\sigma )}$. Using extensive Monte Carlo simulations, we measure the scaling relation $N\sim R^{d_N}$ between the loop-erased step number $N$ and the spatial extent $R$, and determine the geometric exponent $d_N$ for various values of $\sigma$ in spatial dimensions $d=1,2,$ and 3, as well as at the marginal point $\sigma =2$ in $d=4$ and 5. We observe a continuous crossover from long-range (LR) to short-range (SR) behavior as $\sigma$ increases. Below the upper critical dimension $d<d_c=4$, for $\sigma <d/2$, loop erasure is asymptotically irrelevant and $d_N=\sigma$, consistent with Lévy-flight scaling. For $d/2<\sigma <2$, loop erasure becomes relevant and $d_N$ varies continuously toward the short-range loop-erased random walks value. At the marginal points with $\sigma =d/2$ or $\sigma =2$, clear logarithmic corrections are observed. At and above the upper critical dimension, $d\ge 4$, the scaling at $\sigma =2$ is found to be $N\sim R^2/lnR$, consistent with that of the corresponding Lévy flight. Our results provide a systematic numerical determination of $d_N\left(\sigma \right)$ for the LR-LERW across dimensions, and are consistent with ${\sigma }_{*}=2$ as the boundary between LR and SR critical behaviors recently established in a broad variety of statistical models.

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• Classical statistical mechanics
• Levy flights
• Long-range interactions
• Random walks