Schramm-Loewner evolution in the random scatterer Henon-percolation landscapes

TL;DR

This paper investigates how environmental imperfections modeled by Henon map scatterers affect the conformal invariance and fractal properties of SLE curves, revealing power-law deviations in fractal dimension related to the Henon coupling strength.

Contribution

It introduces a novel approach using the Henon map to generate random scatterers in SLE, demonstrating how these imperfections alter the fractal and conformal properties of the curves.

Findings

Fractal dimension varies with the square of the Henon coupling parameter z.

Deviations from conformal symmetry are observed as z increases.

Effective diffusivity parameters change with the strength of environmental scatterers.

Abstract

The Shcramm-Loewner evolution (SLE) is a correlated exploration process, in which for the chordal set up, the tip of the trace evolves in a self-avoiding manner towards the infinity. The resulting curves are named SLE$_{\kappa}$, emphasizing that the process is controlled by one parameter $\kappa$ which classifies the conformal invariant random curves. This process when experiences some environmental imperfections, or equivalently some scattering random points (which can be absorbing or repelling) results to some other effective scale-invariant curves, which are described by the other effective fractal dimensions and equivalently the other effective diffusivity parameters $\kappa_{\text{effective}}$. In this paper we use the classical Henon map to generate scattering (absorbing/repelling) points over the lattice in a random way, that realizes the percolation lattice with which the SLE trace interact. We find some meaningful power-law changes of the fractal dimension (and also the effective diffusivity parameter) in terms of the strength of the Henon coupling, namely the $z$ parameter. For this, we have tested the fractal dimension of the curves as well as the left passage probability. Our observations are in support of the fact that this deviation (or equivalently non-zero $z$s) breaks the conformal symmetry of the curves. Also the effective fractal dimension of the curves vary with the second power of $z$, i.e. $D_F(z)-D_F(z=0)\sim z^2$.