Diffusion disorder in the contact process

Abstract

We study the effects of spatially inhomogeneous diffusion on the non-equilibrium phase transition in the contact process. The directed-percolation critical point in the contact process is known to be stable against the addition of a spatially uniform diffusion term. Correspondingly, we find quenched randomness in the diffusion rates to be irrelevant by power counting in the field-theory of the contact process. However, large-scale Monte Carlo simulations demonstrate that such diffusion disorder destabilizes the clean directed percolation critical point. Instead, the transition belongs to the same infinite-randomness universality class as the contact process with disorder in the infection or healing rates. To explain these results, we develop an effective model with an infinite diffusion rate; it shows that diffusion disorder generates an effective disorder in the healing rates. The same mechanism also appears in the field-theoretic description: Whereas diffusion disorder is irrelevant by power-counting, it generates standard random-mass disorder under renormalization. We discuss the validity of this mechanism for other absorbing state transitions and non-equilibrium phase transitions in general.

Figures (12)

• Figure 1: Number of active sites $N_{s}$ vs. survival probability $P_{s}$ for several infection rates $\lambda$. Data for the critical curve $\lambda_{c}=2.3444$ (shown with errorbars for select points) is fitted with Eq.\ref{['eq:activated_Ns_Ps']} with a floating exponent, yielding $\bar{\Theta}/\bar{\delta}=3.24(1)$. The fit is of good quality $(\chi^{2}\approx0.05)$ and shown with straight dashed line.
• Figure 2: Survival probability $P_{s}^{-1/\bar{\delta}}$ vs. $\ln t$ for several infection rates $\lambda$. Data for the critical curve $\lambda_{c}=2.3444$ (shown with errorbars for select points) is fitted with Eq.\ref{['eq:asymptotic_rho_Ps']} with fixed exponent ${1/\bar{\delta}}=2.618$ following. The fit is shown here with straight dashed line.
• Figure 3: Number of active sites $N_{s}^{1/\bar{\Theta}}$ vs. $\ln t$ for several infection rates $\lambda$. Data for the critical curve $\lambda_{c}=2.3444$ (shown with errorbars for select points) is fitted with Eq.\ref{['eq:asymptotic_R_Ns']}with fixed exponent $1/\bar{\Theta}=0.8091$. The fit is shown here with straight dashed line.
• Figure 4: Number of active sites $N_{s}$ vs. $\ln t$ for several infection rates $\lambda$ in the inactive phase and scaled critical curve $0.86\times N_{s}$ at $\lambda=\lambda_{c}$. Linear fits have been used to find the intersection points (shown here with error bars) between the scaled critical curve and the subcritical curves. Inset: Crossing time vs. $\lvert\lambda-\lambda_{c}\rvert^{-1}$. The solid line is a power-law fit, yielding $\nu_{\perp}\psi=0.96(2)$.
• Figure 5: a) Contact process (CP) with quenched binary diffusion disorder: $D_i=0$ with probability $p$ and $D_i=\infty$ otherwise, where adjacent $D_i=\infty$ sites form a pocket. b) Dynamical rules. c) Effective dynamics of the $D_i=0$ sites, resembling a disordered CP with site-dependent healing rates. d) Dynamical rules of the effective process; when more than one neighbor is active, the survival probability is non-Poissonian and depends on the number of $D_i=\infty$ sites in between. The effective rates $\tilde{\tau}_i$ and $\tilde{\tau}_{i-1,i}$ denote the late-time exponential decay rates of the survival probability.