Stochastic Loewner evolution relates anomalous diffusion and anisotropic percolation

We disclose the origin of anisotropic percolation perimeters in terms of the stochastic Loewner evolution (SLE) process. Precisely, our results from extensive numerical simulations indicate that the perimeters of multilayered and directed percolation clusters at criticality are the scaling limits of the Loewner evolution of an anomalous Brownian motion, being superdiffusive and subdiffusive, respectively. The connection between anomalous diffusion and fractal anisotropy is further tested by using long-range power-law correlated time series (fractional Brownian motion) as the driving functions in the evolution process. The fact that the resulting traces are distinctively anisotropic corroborates our hypothesis. Under the conceptual framework of SLE, our study therefore reveals different perspectives for mathematical and physical interpretations of non-Markovian processes in terms of anisotropic paths at criticality and vice versa.

Figure 1

Percolation models used to generate the SLE curves. (a) Regular percolation, where each site is occupied with the same probability $p$ [16]. (b) Multilayered percolation, where some rows are occupied with probability $p+\mathrm{\Delta }$ (light gray rows) and others with $p-\mathrm{\Delta }$ (dark gray rows) [17]. (c) Directed percolation is a spreading process which starts at the bottom of the tilted lattice and can only advance upwards with probability $p$ [18].

Figure 2

Examples of cluster perimeters at the critical point of (a) isotropic percolation on a triangular lattice ($p_c=0.5$), (d) multilayered percolation also on a triangular lattice ($\mathrm{\Delta }=0.2$ and $p_c=0.5$), and (g) directed percolation on a square lattice ($p_c\approx 0.644$). A detail of each curve in (a), (d), and (g) (pink square) can be seen in (b), (e), and (h), respectively. The driving functions obtained by applying the zipper algorithm [19] to the curves (a), (d), and (g) are shown in (c), (f), and (i), respectively.

Figure 3

Mean squared displacement of the driving functions for the three percolation models studied. The curves are the results of the numerical procedure described in the text applied to ${10}^4$ realizations of each type of percolation model. The $95\%$ confidence intervals were bootstrapped over 400 resamplings [30], but, being smaller than the symbols, are not shown. As expected, in the case of isotropic percolation, the displacement scales linearly with time, while it shows instead a distinctive subdiffusive behavior for directed percolation, with an exponent $\alpha \approx 0.67$. In the case of multilayered percolation, a clear superdiffusive behavior, with an exponent $\alpha \approx 1.78$, can be observed for $\mathrm{\Delta }=0.4$. The inset shows how this anomalous diffusion regime is gradually achieved as we increase the degree of anisotropy $\mathrm{\Delta }$.

Figure 4

Root mean squared estimations of the displacements in the $X$ and $Y$ directions of SLE traces driven by long-range power-law correlated time series (fractional Brownian motion). In (a), (c), and (e) we show typical realizations of uncorrelated, correlated, and anticorrelated driving functions, respectively. The simulation parameters ($H, b$, and $t_f$) were chosen based on the results shown in Fig. 3 (see Table 1 for the numerical values). Good agreement is observed between the uncorrelated result (b) and isotropic percolation (inset on the bottom), as is expected. The correlated trails (d) are also compatible with multilayered percolation (inset on the bottom). In the anticorrelated case (f), the same kind of anisotropy present in the directed percolation is observed (inset on the bottom). These results support our hypothesis that long-term correlations in the driving functions, i.e., the presence of anomalous diffusion, are responsible for the anisotropic behavior of the traces. The insets on the top of (b), (d), and (f) show examples of the traces generated from the simulations with the corresponding driving function shown in (a), (c), and (e), respectively.