How to simulate the quasistationary state

For a large class of processes with an absorbing state, statistical properties of the surviving sample attain time-independent values in the quasistationary (QS) regime. We propose a practical simulation method for studying quasistationary properties, based on the equation of motion governing the QS distribution. In applications to the contact process, the method is shown to reproduce exact results (for the process on a complete graph) and known scaling behavior to high precision. At the critical point, our method is about an order of magnitude more efficient than conventional simulation.

Figure 1

QS distributions via recurrence relations (solid lines) and QS simulation (symbols) for the CP on a complete graph of 100 sites. $\lambda =0.5$, 1.0, and 1.5 (left to right).

Figure 2

Quasistationary density $\rho$ in the one-dimensional CP. Open symbols, QS simulation, $L=20$; filled symbols, QS simulation, $L=200$. Solid lines represent results of conventional simulations.

Figure 3

Quasistationary moment ratio $M$ in the one-dimensional CP. Symbols as in Fig. 2.

Figure 4

Quasistationary lifetime $\tau$ in the one-dimensional CP. Symbols as in Fig. 2.

Figure 5

Quasistationary lifetime $\tau$ vs system size $L$ in the critical one-dimensional CP. The inset shows the deviation from the pure power law, $\tau \propto L^{1.5815}$ (open symbols), and from the power law with a correction to scaling term decaying as $L^{-0.75}$ (filled symbols, shifted vertically for visibility).

Figure 6

Quasistationary order parameter $\rho$ vs system size $L$ in the critical one-dimensional CP. The inset shows the deviation from the pure power law, $\tau \propto L^{-0.2528}$ (open symbols), and from the power law with a correction to scaling term decaying as $L^{-0.75}$ (filled symbols, shifted vertically for visibility).