Exploration of Schramm Loewner evolution in coastlines and invasion percolation trees

Abstract

The Schramm–Loewner Evolution (SLE) provides a rigorous framework for characterizing the scaling limits of fractal curves, encoding their geometric properties through the diffusion parameter of a Brownian motion and enabling a universal classification of stochastic growth processes. In this work, we explore SLE in two distinct classes of geometric structures: coastlines and nontrapping invasion percolation trees. For coastlines, we investigate their fractal properties by analyzing the roughness and fractal dimensions of isoheight lines in both real and artificial landscapes. Through rigorous numerical tests, we show that coastlines violate SLE, as evidenced by the nonconvergence of the diffusion parameters. For tree-like structures, we explore SLE in the context of branching processes, focusing on trees generated by nontrapping invasion percolation (NTIP). We examine three strategies for exploring trees: NTIP itself, Depth-First Search (DFS), and Breadth-First Search (BFS). We focus on the statistics of the corresponding driving functions, revealing their non-Brownian behavior and lack of conformal invariance. We also address the inverse problem of reconstructing tree-like structures from driving functions, achieving accurate reconstructions of sites using BFS and NTIP. These results contribute to advancing our understanding of SLE’s applicability to complex natural and artificial structures, highlighting its limitations in description and suggesting potential generalizations.