Scaling of Long-Range Loop-Erased Random Walks

TL;DR

This study investigates the scaling behavior of long-range loop-erased random walks with Lévy-flight jumps across dimensions, revealing a crossover from long-range to short-range behavior and identifying critical points with logarithmic corrections.

Contribution

The paper provides the first systematic numerical analysis of the geometric exponent for LR-LERW across different dimensions and jump distributions, confirming theoretical predictions.

Findings

Identified the relation $d_N=\sigma$ for $\sigma<d/2$, indicating Lévy-flight scaling.

Observed continuous variation of $d_N$ between long-range and short-range regimes.

Detected logarithmic corrections at marginal points $\sigma=d/2$ and $\sigma=2$.

Abstract

We study the scaling properties of long-range loop-erased random walks (LR-LERW), where the underlying random walker performs L\'evy-flight-like jumps with a power-law step-length distribution $P(\mathbf{r})\sim |\mathbf{r}|^{-(d+\sigma)}$. 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 L\'evy-flight scaling. For $d/2 < \sigma < 2$, loop erasure becomes relevant and $d_N$ varies continuously toward the SR-LERW value. At the marginal points with $\sigma=d/2$ or $\sigma=2$, clear logarithmic corrections are observed. At and above the upper critical dimension, $d \geq 4$, the scaling at $\sigma=2$ is found to be $N \sim R^2/\ln R$, consistent with that of the corresponding L\'evy flight. Our results provide a systematic numerical determination of $d_N(\sigma)$ 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.