Directed percolation conjecture for cellular automata revisited

TL;DR

This study revisits the directed percolation conjecture in cellular automata, revealing dimension-dependent phase transition behaviors and universality classes through simulations and theoretical analysis.

Contribution

It provides new insights into the nature of phase transitions in cellular automata with specific acceptance rules across different dimensions.

Findings

1D transitions are continuous for y<4 and discontinuous for y=4,5.

2D transitions for y=1 follow 2+1 dimensional DP universality.

Transitions become first order for y>1 in 2D.

Abstract

The directed percolation (DP) hypothesis for stochastic, range-4 cellular automata with acceptance rule $y \le\sum_{j=-4}^4 s_{i-j} \le 6$, in cases of $y < 6$ was investigated in one and two dimensions. Simulations, mean-field renormalization group and coherent anomaly calculations show that in one dimension the phase transitions for $y<4$ are continuous and belong to the DP class for $y=4,5$ they are discontinuous. The same rules in two dimensions for $y=1$ show $2+1$ dimensional DP universality; but in cases of $y > 1$ the transitions become first order.