Critical dynamics of the directed percolation with L\'evy-driven temporally quenched disorder

Abstract

Quenched disorder in absorbing phase transitions can disrupt the structure and symmetry of reaction-diffusion processes, offering a more accurate mapping to real physical systems. We developed a temporally quenched disorder method in the (1+1)-dimensional direct percolation (DP) model, where the increment of conditional probability is determined by the cumulative distribution function (CDF) of the L\'evy distribution. Monte Carlo (MC) simulations reveal that the model has a critical region governing the transition between absorbing and active states, and this region changes as the parameter $\beta$, which influences distribution properties. Guided by dynamic scaling laws, we observe that significant variations in the L\'evy distribution parameter $\beta$ lead to notable changes in the particle density decay exponent $\alpha$, total particle number exponent $\theta$, and spreading exponent $\tilde{z}$. The quenching mechanism we introduced has broad potential applications in various theoretical and experimental studies of absorbing phase transitions.