Global properties of stochastic Loewner evolution driven by Lévy processes

Standard Schramm–Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication (Rushkin et al 2006 J. Stat. Mech. P01001 [cond-mat/0509187]) we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable Lévy process). We then discussed the small scale properties of the resulting Lévy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, α, which defines the shape of the stable Lévy distribution. We learn about this behavior by studying a Fokker–Planck equation which gives the probability distribution for end points of the trace as a function of time. As in the short time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at α = 1. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for α>1, goes as logt for α = 1, and saturates for α<1. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is X(t)∼t1/α. In the latter case the scale is Y (t)∼A+Bt1−1/α for α≠1, and Y (t)∼lnt for α = 1. Scaling functions for the probability density are given for various limiting cases