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 \(H\geq \frac{1}{2}\equiv H_{\text{BM}}\), where \(H_{\text{BM}}\) 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" \(\kappa\), which coincides with the one of the ordinary Schramm-Loewner evolution (SLE) at \(H=H_{\text{BM}}\). In our numerical investigation, we focus on the scaling properties of the traces generated for \(\kappa=2,3\), \(\kappa=4\) and \(\kappa=6,8\) 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 \(\kappa\) values. The left passage probability (LPP) test demonstrates that, for \(H\) values not far from the uncorrelated case (small \(\epsilon_H\equiv \frac{H-H_{\text{BM}}}{H_{\text{BM}}}\)) the prediction of the ordinary SLE is applicable with an effective diffusivity parameter \(\kappa_{\text{eff}}\). Not surprisingly, the \(\kappa_{\text{eff}}\)'s do not fulfill the prediction of SLE for the relation between \(D_f(H)\) and the diffusivity parameter.