Loewner driving functions for off-critical percolation clusters

We numerically study the Loewner driving function $U_t$ of a site percolation cluster boundary on the triangular lattice for $p<p_c$. It is found that $U_t$ shows a drifted random walk with a finite crossover time. Within this crossover time, the averaged driving function $⟨U_t⟩$ shows a scaling behavior $-\left(p_c-p\right)t^{\left(\nu +1\right)/2\nu }$ with a superdiffusive fluctuation whereas, beyond the crossover time, the driving function $U_t$ undergoes a normal diffusion with Hurst exponent 1/2 but with the drift velocity proportional to ${\left(p_c-p\right)}^{\nu }$, where $\nu =4/3$ is the critical exponent for two-dimensional percolation correlation length. The crossover time diverges as ${\left(p_c-p\right)}^{-2\nu }$ as $p\to p_c$.

Figure 1

(Color online) A schematic diagram of conformal mapping. $g_t$ maps $\mathbb{H}\backslash {\gamma }_{\left(0,t\right]}$ to $\mathbb{H}$ with ${\gamma }_t$ to $U_t$ on the real axis. The deformed grid in the left panel can be regarded as equipotential and electric force lines, and is mapped to the straight grid in the right panel.

Figure 2

(Color online) Percolation on the triangular lattice. The left panel shows a whole system with left (right) half of the system boundary being occupied (unoccupied). The right panel shows only the cluster boundary that starts from the origin.

Figure 3

(Color online) Examples of cluster boundaries (a)-(c) and their driving functions (d) for $p=0.5$, 0.49, and 0.48. The boundaries are shown up to $N=20000$ steps. The lattice spacing is set to be unity. The correlation lengths are $\xi =400$ and 150 for $p=0.49$ and 0.48, respectively.

Figure 4

(Color online) Drift velocity $v_d$ vs $\left(p_c-p\right)$. The drift velocities of the driving function averaged up to $t=21000$ are plotted in the logarithmic scale as a function of $\left(p_c-p\right)$. There is a crossover around $\left(p_c-p\right)\sim 0.01$ between the two regimes. The solid (dashed) line with the slope 1(4/3) shows the limiting behavior in the small (large) $p_c-p$ regime. The inset shows $t$ vs $⟨U_t⟩$ for $p=0.5$, 0.4975, 0.495, 0.4925, and 0.49 (from right to left). Each line represents average behavior over 4000 samples.

Figure 5

(Color online) $\text{Var}\left[U_t\right]$ vs $t$ in the logarithmic scale. (a) The plot for $p=0.42$. The behavior changes from superdiffusive to diffusive around the crossover time $t_c$ indicated by the arrow. The behavior for $p=p_c$, $\text{Var}\left[U_t\right]=6t$, is plotted for comparison. (b) The plots for several values of $p$. The crossover time $t_c$ becomes larger as we approach $p_c$.

Figure 6

(Color online) (a)$t_c$, $\xi$, and ${\xi }^2$ vs $\left(p_c-p\right)$ in the logarithmic scale. The horizontal solid line at $t=21000$ indicates the simulation range. (b) $-⟨x_t⟩$ vs $\left(p_c-p\right)$ for $t=\Delta t,2\Delta t$, and $3\Delta t$ with $\Delta t=5200$, in the logarithmic scale. Each data point represents average over 3000 samples.