Universal scaling behaviour of directed percolation and the pair contact process in an external field

Abstract

We consider the scaling behavior of directed percolation and of the pair contact process with a conjugated field. In particular we determine numerically the equation of state and show that both models are characterized by the same universal scaling function. Furthermore we derive the equation of state for the pair contact process within a mean-field approach which again agrees with the mean-field equation of state of the directed percolation universality class.

High-precision Monte Carlo study of directed percolation in (d+1) dimensions

“…For both bond and site DP we have the following. If b v is the number of Bernoulli trials required to determine the state of v ∈ V t given the site configuration [29] 0.644 700 15 (5) [8] triangular 0.595 647 0(3) 0.595 646 75(10) [30] 0.478 025 0(4) 0.478 025 25 (5) [30] 0.595 646 8(5) [8] 0.478 025(1) [8] honeycomb 0.839 931 6(2) 0.839 933(5) [31] 0.822 856 9(2) 0.822 856 80 (6) at time t − 1, then…”

Tricritical Directed Percolation

“…The external field h is conjugated to the order parameter and is usually implemented as a spontaneous particle creation process (see e.g. [33]). Furthermore, η = η(x, t) denotes the noise which accounts for fluctuations of the particle density.…”

Finite-size scaling of directed percolation in the steady state

“…The control parameter of the transition is τ , and τ c denotes its critical value. In the infinite volume limit, a finite positive particle density occurs below the transition point (τ < τ c ) whereas the absorbing vacuum state (s = 0) is approached above the transition point if the source term h ≥ 0 (which can be implemented in simulations, e.g., as a spontaneous particle creation process [25]) is absent. In a finite system, the absorbing state is inevitably approached even for τ < τ c , if h = 0.…”