Crossover from isotropic to directed percolation

Directed percolation is one of the generic dynamic universality classes. We study the crossover from isotropic to directed percolation by representing the combined problem as a random cluster model. A parameter r controls 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. Isotropic percolation is stable in 2D. Our numerical finite-size scaling results confirm this. An intuitive argument suggests that directed percolation at r = 0 does not change stability for D greater than or equal to 3. It remains unstable, such that forest fires at 0 < r < 1 have the same scaling properties as isotropic percolation all dimensions.