Some aspects of the critical behavior of the Two-Neighbor Stochastic Cellular Automata

TL;DR

This paper investigates the phase transition behavior of Two-Neighbor Stochastic Cellular Automata using simulations and approximations, revealing a parabolic transition line near a critical point and confirming DP universality class away from it.

Contribution

It provides a detailed analysis of the phase diagram, especially near a special critical point, and validates bounds and interpolation formulas for the transition line.

Findings

The phase transition line near the special point is asymptotically parabolic.

System belongs to DP universality class except at the special point.

Monte Carlo data supports rigorous bounds and interpolation formulas.

Abstract

Using Pade approximations and Monte Carlo simulations, we study the phase diagram of the Two-Neighbor Stochastic Cellular Automata, which have two parameters $p_{1}$ and $p_{2}$ and include the mixed site-bond directed percolation (DP) as a special case. The phase transition line $p_{1}=p_{1 {\rm c}}(p_{2})$ has endpoints at $(p_{1}, p_{2})=(1/2,1)$ and at (0.8092, 0). The former point (1/2,1) is a special point at which Compact DP transition occurs and its critical exponents are known exactly. Results of time-dependent simulation show that in the whole range of parameters, excluding this point (1/2,1), the system belongs to the DP universality class. It is first shown that the shape of the phase transition line near this special point has, asymptotically, a parabolic shape, {\it i.e.}, $p_{1{\rm c}}(p_{2})-1/2 \sim (1-p_{2})^{\theta}$ with $\theta = 1/2$ for $0 < 1-p_{2} \ll 1$. We use the Monte Carlo data to assess the accuracy of rigorous bounds for the line recently reported by Liggett and by Katori and Tsukahara. It is also shown that outside the vicinity of the special point (1/2,1), the curve is well approximated by an interpolation formula, similar to the one proposed by Yanuka and Englman.