Contact process with temporal disorder

We investigate the influence of time-varying environmental noise, i.e., temporal disorder, on the nonequilibrium phase transition of the contact process. Combining a real-time renormalization group, scaling theory, and large scale Monte-Carlo simulations in one and two dimensions, we show that the temporal disorder gives rise to an exotic critical point. At criticality, the effective noise amplitude diverges with increasing time scale, and the probability distribution of the density becomes infinitely broad, even on a logarithmic scale. Moreover, the average density and survival probability decay only logarithmically with time. This infinite-noise critical behavior can be understood as the temporal counterpart of infinite-randomness critical behavior in spatially disordered systems, but with exchanged roles of space and time. We also analyze the generality of our results, and we discuss potential experiments.

Figure 1

Schematic of the strong-noise RG. The RG step replaces the time evolution in the top panel with that in the bottom panel by eliminating the segment with the smallest change of $x=-ln\rho$, i.e, the segment with the smallest $|lna|$. Here, the segment $\mathrm{\Delta }{t}_3^{\mathrm{up}}$ with multiplier $a_3^{\mathrm{up}}$ is decimated by combining it with the segments $\mathrm{\Delta }{t}_2^{\mathrm{dn}}$ and $\mathrm{\Delta }{t}_3^{\mathrm{dn}}$. This results in the renormalized time evolution featuring the coarse-grained segment $\mathrm{\Delta }{\tilde{t}}^{\mathrm{dn}}$ with renormalized multiplier ${\tilde{a}}^{\mathrm{dn}}$, as given by Eqs. (7) and (5), respectively.

Figure 2

Time evolution of the density of an individual noise realization at criticality, plotted as $x=-ln\rho$ vs $t$. The one-dimensional (1D) data are for a system of ${10}^4$ sites using $\mu =1,\mathrm{\Delta }t=1$ and binary distributed $\lambda$ with ${\lambda }_h=8.1,{\lambda }_l=0.81$ and $p=0.8$. The 2D data are for ${1000}^2$ sites using $\mu =1,\mathrm{\Delta }t=2$ and binary distributed $\lambda$ with ${\lambda }_h=3.65,{\lambda }_l=0.365$, and $p=0.8$ [see Eq. (26) for the meaning of the parameters].

Figure 3

Schematic of the RG flow in the ${\mathcal{R}}_0\text{−}{\mathcal{P}}_0$ plane. For $C<C_c$, the flow asymptotically approaches a line of stable fixed points that represent the active phase. In the inactive phase, $C>C_c$, the flow is towards ${\mathcal{R}}_0\to 0$, and ${\mathcal{P}}_0\to \infty$. For $C=C_c$, one flows into the critical fixed point at ${\mathcal{R}}_0^{*}=1/d$, and ${\mathcal{P}}_0^{*}=0$.

Figure 4

Inverse survival probability $1/P_s$ vs time $t$ for several values of the infection rate ${\lambda }_h$. The data are averages over 30000–50000 disorder realizations, with one simulation run per realization. The statistical error of the data is about one symbol size.

Figure 5

Double logarithmic plot of the data of Fig. 4. Inset: Inverse effective exponent $1/{\delta }_{\mathrm{eff}}$ vs time $t$.

Figure 6

Number of sites in the active cloud and its radius at criticality, ${\lambda }_h=27.27$, plotted as ${(N_s/t)}^{-1/y_N}$ and ${(R/t)}^{-1/y_R}$ vs. time $t$ with $y_N=3.6$ and $y_R=1.7$. The data are averages over 50000 disorder realizations with one run per configuration, their errors are about one symbol size. Also shown is the inverse average density for decay runs at criticality (system of $200000$ sites, 50000 disorder configurations). Inset: Density $N_s/\left(P_{s}R\right)$ of active sites inside an active cloud in a critical spreading run.

Figure 7

Inverse survival probability $1/P_s$ vs time $t$ for infection rates ${\lambda }_h$ at and below the critical rate ${\lambda }_c=27.27$. The data are averages over $5\times {10}^4$ to ${10}^6$ disorder realizations with one run per configuration, their errors are about a symbol size. The dash-dotted line shows $1.1/P_s$ for $\lambda ={\lambda }_c$. Inset: Distance from criticality $r=({\lambda }_c-\lambda )/{\lambda }_c$ vs crossing time $t_x$, plotted as $r^{-1/2}$ vs $ln\left(t_x\right)$.

Figure 8

Probability distribution $P(x,t)$ with $x=-ln{P}_s$ at criticality for different times $t$. The argument $x$ is scaled by a factor $f_t$ such that the curves at different times coincide. The data are averages over 20000 disorder realizations with 1000 runs for each configuration. For comparison, the solid black squares show $P(x,t)$ for a system with a box disorder distribution (see text). Inset: Scale factor $f_t$ vs $lnt$.

Figure 9

Probability distribution $P(x,t)$ with $x=-ln\left(N_S{P}_S^{-1}{t}^{-1}\right)$ at criticality for different times $t$, scaled such that the curves coincide. The data are averages over $500000$ disorder realizations. Inset: Scale factor $f_t^{1/(y_N-1)}$ vs $lnt$ with $y_N=3.7$.

Figure 10

$P_s^{-1},{(N_s/t)}^{-1/y_N}$, and ${(R/t)}^{-1/y_R}$ vs time $t$ at criticality $\left({\lambda }_h=20.49\right)$ for a system with a shorter disorder time interval $\mathrm{\Delta }t=3$. The exponents are fixed at $y_N=3.6$ and $y_R=1.7$. The data are averages over $100000$ disorder realizations with one run per configuration, leading to errors of about one symbol size.

Figure 11

Inverse survival probability $1/P_s$ vs time $t$ for several values of the infection rate ${\lambda }_h$ in a system with weaker temporal disorder (see text). The data are averages over 300 disorder realizations, with 100 simulation runs per realization. Note the crossover to logarithmic behavior at $t\approx {10}^3$.

Figure 12

Inverse survival probability $1/P_s$ vs time $t$ in two space dimensions for several values of the infection rate ${\lambda }_h$. The data are averages over 60000 to $120000$ disorder realizations, with one simulation run per realization. The statistical error of the data is about one symbol size. Inset: Double logarithmic plot of $P_s$ vs $t$.

Figure 13

Number of sites in the active cloud and its radius in two dimensions at criticality, ${\lambda }_h=6.06$, plotted as ${(N_s/t)}^{-1/y_N}$ and ${(R/t)}^{-1/y_R}$ vs time $t$ with $y_N=2.5\left(3\right)$ and $y_R=0.29\left(5\right)$. (60000 disorder realizations with one run per configuration, the errors are about one symbol size.) Also shown is the inverse average density ${\rho }_{\mathrm{av}}^{-1}$ for decay runs (system of ${3200}^2$ sites, 60000 disorder configurations). Inset: Density $N_s/\left(P_s{R}^2\right)$ of active sites inside an active cloud in a critical spreading run.

Figure 14

Inverse survival probability $1/P_s$ vs. time $t$ for infection rates ${\lambda }_h$ at and below the critical rate ${\lambda }_c=6.06$. (60000 to $100000$ disorder realizations with one run per configuration, giving statistical errors of about a symbol size.) The dash-dotted line represents $1.09/P_s$ for $\lambda ={\lambda }_c$. Inset: Distance from criticality $r=({\lambda }_c-\lambda )/{\lambda }_c$ vs crossing time $t_x$, plotted as $r^{-1/2}$ vs $ln\left(t_x\right)$.

Figure 15

Log-log plots of the average lifetime $\tau$ of samples of linear size $L=N^{1/d}$ for several infection rates ${\lambda }_h$ at and above the critical point. The solid lines are power law fits. The data are averages over 10000–20000 disorder configurations. Left: One-dimensional system with critical infection rate ${\lambda }_h\approx 27.27$. Right: Two-dimensional system with critical infection rate ${\lambda }_h\approx 3.65$.

Figure 16

Directed bond percolation in (1+1) dimensions. Arrows represent occupied bonds while dashed lines indicate empty bonds. A density decay experiment starts with all sites being active (red dots in the bottom row). The red (thick) arrows show how the activity spreads with increasing time.