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

Figures (7)

• Figure 1: Geometric exponent $d_N$ as a function of $\sigma$ in 1D (red), 2D (blue), and 3D (green). The estimates are obtained from finite-size scaling fits using Eq. (\ref{['eq:fss']}) and Eq. (\ref{['eq:log']}). The black dashed line shows the Lévy-flight reference behavior, $d_N=\sigma$ for $\sigma<2$ and $d_N=2$ for $\sigma>2$. The black solid lines mark the SR-LERW limits $d_{N,\text{SR}}^{\mathrm{1D}}=1$, $d_{N,\text{SR}}^{\rm{2D}}=5/4$, and $d_{N,\text{SR}}^{\rm{3D}}=1.62400(5)$PhysRevE.82.062102.
• Figure 2: Log--log plot of $N/R$ versus $R$ in 1D for $\sigma=0.7$, $1.2$, $1.6$, $2.0$, and $2.5$ from bottom to top. The slanted curves for $\sigma<2$ show deviations from the SR scaling. The curve at $\sigma=2.0$ is still clearly non-horizontal, implying a logarithmic correction at the marginal point, whereas the nearly horizontal curve at $\sigma=2.5$ indicates recovery of the SR limit $d_N=1$.
• Figure 3: Log--log plot of $N/R^{5/4}$ versus $R$ in 2D for $\sigma=1.2$, $1.4$, $1.8$, $2.0$, and $2.5$ from bottom to top. The non-horizontal behavior for $\sigma<2$ indicates deviations from the SR scaling. The curve at $\sigma=2.0$ is still visibly non-horizontal, implying a logarithmic correction at the marginal point, whereas the nearly horizontal curve at $\sigma=2.5$ indicates recovery of the SR limit $d_N=5/4$.
• Figure 4: Log--log plot of $N/R^{1.624}$ versus $R$ in 3D for $\sigma=1.6$, $1.7$, $1.8$, $2.0$, and $2.5$ from bottom to top. The curves for $\sigma<2$ deviate from horizontal behavior. The curves at $\sigma=2.0$ and $2.5$ are both close to horizontal, indicating that any logarithmic correction at $\sigma=2$ is very weak and that the scaling at $\sigma=2.5$ has already returned to the SR limit $d_N \approx 1.624$.
• Figure 5: Log-log plot of $N/R^{d/2}$ versus $\ln R$ at the marginal point $\sigma=d/2$ for 1D, 2D, and 3D. The dashed lines serve as guides to the eye and highlight the logarithmic scaling, with slopes $-0.40(1)$, $-0.37(2)$, and $-0.23(2)$ for 1D, 2D, and 3D, respectively.