Scaling of Long-Range Loop-Erased Random Walks

Abstract

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 $P(\mathbf{r})\sim |\mathbf{r}{|}^{-(d+\ddot{I}ƒ)}$. 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 $\ddot{I}ƒ$ in spatial dimensions $d=1,2,$ and $3$, as well as at the marginal point $\ddot{I}ƒ=2$ in $d=4$ and $5$. We observe a continuous crossover from long-range (LR) to short-range (SR) behavior as $\ddot{I}ƒ$ increases. Below the upper critical dimension $d<d_c=4$, for $\ddot{I}ƒ<d/2$, loop erasure is asymptotically irrelevant and $d_N=\ddot{I}ƒ$, consistent with Lévy-flight scaling. For $d/2<\ddot{I}ƒ<2$, loop erasure becomes relevant and $d_N$ varies continuously toward the SR-LERW value. At the marginal points with $\ddot{I}ƒ=d/2$ or $\ddot{I}ƒ=2$, clear logarithmic corrections are observed. At and above the upper critical dimension, $d\ge 4$, the scaling at $\ddot{I}ƒ=2$ is found to be $N\sim R^2/\ln R$, consistent with that of the corresponding Lévy flight. Our results provide a systematic numerical determination of $d_N(\ddot{I}ƒ)$ for the LR-LERW across dimensions, and are consistent with $\ddot{I}{ƒ}_{\ast }=2$ as the boundary between LR and SR critical behaviors recently established in a broad variety of statistical models.