# Percolation in Media with Columnar Disorder

## TL;DR

This paper investigates percolation on a cubic lattice with columnar disorder, revealing anisotropic clusters, a Griffiths phase with power-law distributions, and contrasting behaviors at criticality.

## Contribution

It introduces a generalized percolation model with columnar disorder, analyzing anisotropic scaling, Griffiths phase characteristics, and critical behaviors distinct from directed percolation.

## Key findings

- Clusters are highly anisotropic near the transition.
- A Griffiths phase with non-universal power-law tails exists.
- Critical clusters exhibit power-law growth, unlike directed percolation.

## Abstract

We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very anisotropic, with different scaling exponents for the sizes parallel and perpendicular to the columns. Below the critical point there is a Griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non-universal powers. This region is very similar to the Griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction, and the proof follows essentially the same arguments as in that case. But in contrast to directed percolation in disordered media, the number of active ("growth") sites in a growing cluster at criticality shows a power law, while the probability of a cluster to continue to grow shows logarithmic behavior.