Crossover from Self--Similar to Self--Affine Structures in Percolation

TL;DR

This paper investigates the transition from isotropic to anisotropic structures in percolation, using a field-theoretical and renormalization group approach to predict crossover scales and effective exponents.

Contribution

It introduces a novel theoretical framework for analyzing crossover phenomena in percolation, bridging self-similar and self-affine regimes.

Findings

Calculated effective exponents for the crossover region.

Predicted the scale at which the anisotropic crossover occurs.

Demonstrated broad applicability of the method.

Abstract

We study the crossover from self--similar scaling behavior to asymptotically self--affine (anisotropic) structures. As an example, we consider bond percolation with one preferred direction. Our theory is based on a field--theoretical representation, and takes advantage of a renormalization group approach designed for crossover phenomena. We calculate effective exponents for the connectivity describing the entire crossover region from isotropic to directed percolation, and predict at which scale of the anisotropy the crossover should occur. We emphasize the broad range of applicability of our method.