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

TL;DR

This paper extends the Loewner exploration process to include self-similar, correlated random forces modeled by fractional Brownian motion, resulting in scale-invariant traces that are not conformally invariant, with implications for understanding fractal dimensions and passage probabilities.

Contribution

It introduces a scale-invariant, non-conformally invariant generalization of Loewner evolution driven by fractional Brownian motion, exploring its properties and deviations from classical SLE predictions.

Findings

Traces are scale-invariant but not conformally invariant.

Fractal dimension decreases monotonically with Hurst exponent H.

Effective diffusivity parameter approximates SLE predictions near uncorrelated case.

Abstract

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.