Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution

We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability $p\ge p_c$. We numerically demonstrate that the scaling limit of planar ${\mathrm{LERW}}_p$ curves, for all $p>p_c$, can be described by Schramm-Loewner evolution (SLE) with a single parameter $\kappa$ that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient $\kappa$. Several geometrical tests are applied to ascertain this. All calculations are consistent with ${\mathrm{SLE}}_{\kappa }$, where $\kappa =1.732\pm 0.016$. This value of the diffusivity coefficient is outside the well-known duality range $2\le \kappa \le 8$. We also investigate how the winding angle of the ${\mathrm{LERW}}_p$ crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter $p$ from 1 to $p_c$. For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.

Figure 1

Deviation of the winding angle variance of the ${\mathrm{LERW}}_p$ on the percolation cluster from the normal LERW variance, i.e., $V(L,p^{\prime })-(\kappa /4)\text{ln}L$ as a function of $p^{\prime }$. The inset shows the dependence of the winding angle variance on the lateral size of the lattice $L$ for the LERW on the critical percolation cluster (the statistical error bars are shown, but are quite shorter and appear as horizontal lines). The slope in the linear-logarithmic plot corresponds to $\kappa /4=0.433\pm 0.004$.

Figure 2

Crossover scaling and data collapse for ${\mathrm{LERW}}_p$ for different system sizes. (a) Deviation of the winding angle variance of the ${\mathrm{LERW}}_p$ on the percolation cluster from the variance of the LERW on the critical percolation cluster, i.e., $V(L,p^{\prime })-\left(0.433\right)\text{ln}L$ versus $\text{ln}\left(p^{\prime }{L}^{\theta }\right)$ for different system sizes. The scaling function given by Eq. (1) is applied, with $\theta =0.89\pm 0.05$. (b) Deviation of the winding angle variance of the ${\mathrm{LERW}}_p$ on the percolation cluster from the normal LERW variance, i.e., $V(L,p^{\prime })-\frac{1}{2}\text{ln}L$ versus $\text{ln}\left(p^{\prime }{L}^{\eta }\right)$, with $\eta =0.14\pm 0.03$, for different system sizes. For each finite lattice size $L$, three regimes can be obtained with two crossover exponents, i.e., $\theta$ and $\eta$. A more precise estimate for $\beta$ can be obtained by the data collapsing for different lattice sizes in the intermediate regime, which is $\beta =0.088\pm 0.004$. All results have been averaged over $4\times {10}^4$ samples.

Figure 3

Left-passage probability for ${\mathrm{LERW}}_p$ on percolation clusters as a function of polar angle. (a) The LERW on the critical percolation cluster (shown in black) on a $513\times 513$ lattice (the visited sites are shown in gray). (b) Schematic representation of the left-passage definition (details in the text) on the obtained curve after rotation and translation. (c) Plot of ${\wp }_p(\varphi ,R)-{\wp }_{p=1}(\varphi ,R)$ for the LERW in the upper half plane where ${\wp }_p(\varphi ,R)$ is the probability for the LERW on the percolation cluster to pass to the left of a point with polar coordinates $(R,\varphi )$. At $p=p_c$ the results are in good agreement with ${\mathcal{P}}_{1.73}-{\mathcal{P}}_2$, where ${\mathcal{P}}_{1.73}\left(\varphi \right)$ is the left-passage probability for ${\mathrm{SLE}}_{1.73}$ given by Schramm's formula (4). The magnitude of statistical errors (not shown) is consistent with the apparent fluctuations of the data lines. The results are averages over $5\times {10}^4$ curves on a square lattice with $L=513$.

Figure 4

Statistics of the obtained driving function for ${\mathrm{LERW}}_p$: the second moment of the driving function $\langle {\xi }^2\left(t\right)\rangle$ versus Loewner time $t$ for a normal LERW (i.e., $p=1$) and for a LERW on the critical percolation cluster. The obtained diffusion coefficients are $\kappa =1.68\pm 0.07$ and $\kappa =1.94\pm 0.07$ for $p=p_c$ and $p=1$, respectively. The inset shows the probability distribution of the driving function at two different Loewner times for LERWs on the critical percolation cluster. The rescaled parameter $X$ is defined as $X=\xi \left(t\right)/\sqrt{\kappa t}$, where we have taken $\kappa =1.68$. The solid line is the normal distribution of the vanishing mean value and unit dispersion. Results are averages over $4\times {10}^4$ realizations on a square lattice with $L=1025$ for both cases.