Loop-erased random walk on a percolation cluster: Crossover from Euclidean to fractal geometry

2013/08/26 by E. Daryaei, S. Rouhani · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics #Complex Network Analysis Techniques #Critical exponent #Crossover #Euclidean geometry #Fractal #Fractal dimension #Geometry #Lattice (music) #Mathematical analysis #Mathematics #Percolation (cognitive psychology) #Percolation critical exponents #Percolation theory #Percolation threshold #Physics #Quantum mechanics #Random walk #Scaling #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topology (electrical circuits) #cond-mat.stat-mech

paper · pdf · doi:10.1103/physreve.89.062101

published as Physical Review E 2014 · 5 pages and 4 figures

arxiv created 2013/08/26 · openalex publication_date 2014/06/02 · arxiv updated 2015/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ≥ p_c, in two and three dimensions. We find that the fractal dimensions of LERW_p are close to normal LERW in a Euclidean lattice, for all p>p_c. However, our results reveal that LERW on critical incipient percolation clusters is fractal with d_f=1.217 ± 0.002 for d=2 and 1.43 ± 0.02 for d=3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior of the LERW_p crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p_c. For finite systems, two crossover exponents and a scaling relation can be derived. This work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.

Citations

• Dimension of the loop-erased random walk in three dimensions
• Dilute Fermi Gas in Quasi-One-Dimensional Traps: From Weakly Interacting Fermions via Hard Core Bosons to a Weakly Interacting Bose Gas
• Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold
• A self-avoiding random walk
• Scaling limits of loop-erased random walks and uniform spanning trees
• Effect of random defects on the critical behaviour of Ising models
• Loop-erased random walk and Poisson kernel on planar graphs
• Diffusion in disordered media

Cited by

• Loop erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution