Self-similar but not conformally invariant traces obtained by modified Loewner forces

The two-dimensional Loewner exploration process is generalized to the case where the random force is self-similar with positively correlated increments. We model this random force by a fractional Brownian motion with Hurst exponent H≥1/2≡H_{BM}, where H_{BM} stands for the one-dimensional Brownian motion. By manipulating the deterministic force, we design a scale-invariant equation describing self-similar traces which lack conformal invariance. The model is investigated in terms of the "input diffusivity parameter" κ, which coincides with the one of the ordinary Schramm-Loewner evolution (SLE) at H=H_{BM}. In our numerical investigation, we focus on the scaling properties of the traces generated for κ=2,3, κ=4, and κ=6,8 as the representatives, respectively, of the dilute phase, the transition point, and the dense phase of the ordinary SLE. The resulting traces are shown to be scale invariant. Using two equivalent schemes, we extract the fractal dimension, D_{f}(H), of the traces which decrease monotonically with increasing H, reaching D_{f}=1 at H=1 for all κ values. The left passage probability (LPP) test demonstrates that, for H values not far from the uncorrelated case (small ε_{H}≡H-H_{BM}/H_{BM}), the prediction of the ordinary SLE is applicable with an effective diffusivity parameter κ_{eff}. Not surprisingly, the κ_{eff}'s do not fulfill the prediction of SLE for the relation between D_{f}(H) and the diffusivity parameter.