Percolation-like phase transition in a non-equilibrium steady state

Abstract

We study the Gierer-Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let $p$ be the site occupation probability of the square lattice. For $p$ greater than a critical value $p_c$, the steady state consists of stripe-like patterns with long-range connectivity. For $p < p_c$, the connectivity is lost. The value of $p_c$ is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of $p_c$, the cluster-related quantities exhibit power-law scaling behaviour. The method of finite-size scaling is used to determine the values of the fractal dimension $d_f$, the ratio, $\frac{\gamma}{\nu}$, of the average cluster size exponent $\gamma$ and the correlation length exponent $\nu$ and also $\nu$ itself. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.