Active Width at a Slanted Active Boundary in Directed Percolation

TL;DR

This paper investigates the divergence of the active region width near the percolation threshold in directed percolation, revealing a logarithmic correction to the critical scaling law.

Contribution

It introduces a detailed analysis of the active region width, showing a logarithmic correction to the known power-law divergence at the critical point.

Findings

The width diverges as W ^{- u_\u2113} \, ext{ln}( ext{}/) near the threshold.

Numerical data confirm the predicted scaling behavior.

The logarithmic factor results from screening effects in the active region.

Abstract

The width W of the active region around an active moving wall in a directed percolation process diverges at the percolation threshold p_c as W \simeq A \epsilon^{-\nu_\parallel} \ln(\epsilon_0/\epsilon), with \epsilon=p_c-p, \epsilon_0 a constant, and \nu_\parallel=1.734 the critical exponent of the characteristic time needed to reach the stationary state \xi_\parallel \sim \epsilon^{-\nu_\parallel}. The logarithmic factor arises from screening of statistically independent needle shaped sub clusters in the active region. Numerical data confirm this scaling behaviour.