Crossover from Isotropic to Directed Percolation

TL;DR

This paper investigates the transition from isotropic to directed percolation using a random cluster model, deriving the crossover exponent in 2D and discussing stability and universality across dimensions.

Contribution

It provides an exact crossover exponent in 2D and explores the stability and universality of percolation types across different dimensions.

Findings

Exact crossover exponent y_{DP}=y_T-1 at r=1 in 2D

Isotropic percolation is stable in 2D, confirmed by numerical results

Directed percolation stability varies with dimension, with universality at intermediate r

Abstract

Directed percolation is one of the generic universality classes for dynamic processes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model, with a parameter $r$ controlling the spontaneous birth of new forest fires. We obtain the exact crossover exponent $y_{DP}=y_T-1$ at $r=1$ using Coulomb gas methods in 2D. Isotropic percolation is stable, as is confirmed by numerical finite-size scaling results. For $D \geq 3$, the stability seems to change. An intuitive argument, however, suggests that directed percolation at $r=0$ is unstable and that the scaling properties of forest fires at intermediate values of $r$ are in the same universality class as isotropic percolation, not only in 2D, but in all dimensions.