Critical and quasicritical behavior in a three-species dynamical model of semi-directed percolation

Abstract

We investigate a one-dimensional three-species dynamical model whose dynamics naturally generate the semi-directed percolation cluster in time and show a non-equilibrium absorbing state phase transition from an active to inactive state. The critical threshold and exponents associated with the dynamic process are determined using Monte Carlo simulations. Critical behavior observed shows that the model belongs to the directed percolation (DP) universality class. Further, we consider the effect of spontaneous activity generation in the dynamical model. While, as expected, this destroys the usual critical behaviour, we find that the dynamic susceptibility shows a maximum at a specific value of the control parameter, indicating a quasi-critical behaviour, similar to the findings in the case of DP models and DP-inspired models of neuronal activity with spontaneous activity generation. Interestingly, in the presence of spontaneous activity, we find that spatial and temporal correlations exhibit power-law decays at a value of the control parameter different from the pseudo-threshold corresponding to the peak of the dynamic susceptibility, indicating that there are two pseudo-thresholds in such a case, one where the response function is maximum and another where the spatial and temporal correlations show scale-free behaviour.

Figures (14)

• Figure 1: (a) Schematic of the time evolution of sites of a one-dimensional lattice of length $L=10$ in the three species dynamical model of semi-directed percolation. For the sake of clarity, the two processes happening at each time step are represented separately (process P1 defined in Eqs. \ref{['a1']}, \ref{['a2']}, \ref{['a3']}, \ref{['a4']}, and P2 in Eq. \ref{['b1']}). (b) The active sites alone in a) constitute a semi-directed percolation cluster (sites marked X) on a $10 \times 10$ square lattice with occupation probability $p$. An isotropic cluster includes the additional occupied sites in a) (sites marked O here). Arrows are shown to indicate the ‘ forward’ and ‘ backward’ connections between successive rows on the square lattice.
• Figure 2: (a) Effect of spontaneous activity is illustrated. Even without the presence of an active neighboring site in state $2$, a $1-$ cluster spontaneously becomes $2-$ cluster with probability $\epsilon$. (b) Typical time evolution of clusters of active sites when spontaneous activity is present. All sites of the one-dimensional lattice are active initially. Lattice size is $L=512$, $\epsilon=0.005$ and $p=0.61$.
• Figure 3: Typical configurations of clusters of active sites generated in the model starting from a single active site (top panel) and $L$ active sites (bottom panel) in a one-dimensional lattice of size $L=512$. Periodic boundary condition is used, and the time evolution of the lattice up to $512$ time steps is shown. (a) When $p<< p_c$ active cluster dies out very soon. (b) Closer to the threshold, activity survives for longer and longer durations. (c) Well above the threshold $p_c$, activity spreads over the entire system.
• Figure 4: (a) Variation of density of active sites $\rho(t)$ with time $t$ for different values of the parameter $p$ near the threshold. At the threshold $p_c$, we expect a power-law of the form $\rho(t)\sim t^{-\alpha}$. (b) Evolution of survival probability $p_s(t)$ with $t$ for different $p$ values near the threshold. At $p_c$, survival probability obeys a power law of the form $p_s(t) \sim t^{-\delta}$ at threshold.
• Figure 5: Evolution of the number of active sites $N(t)$ with time $t$ for a system starting with a single active site for different values of the parameter $p$ near the threshold. At $p_c$, $N(t)$ obeys a power law of the form $N(t)\sim t^{\theta}$.