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

TL;DR

This paper investigates a reaction-diffusion model on disordered lattices, revealing a percolation-like phase transition characterized by long-range connectivity and critical scaling behavior near a specific occupation probability.

Contribution

It demonstrates that the disordered Gierer-Meinhardt model exhibits a percolation-like transition with critical exponents consistent with ordinary percolation universality class.

Findings

Critical occupation probability p_c is much higher than standard percolation threshold.

Near p_c, cluster properties follow power-law scaling.

The model's critical exponents match those of ordinary percolation.

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.