Ergodicity and Mixing Properties of the Northeast Model

The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: Whenever a site’s southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p−coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at pc = 1 − βc, where βc is the critical parameter for oriented percolation. For p < pc, the trivial measure δ0 that puts mass one on the configuration with all spins set at 0 is the unique ergodic, translation invariant, stationary measure. For p ≥ pc, the product Bernoulli-p measure on configuration space is the unique nontrivial, ergodic, translation invariant, stationary measure for the system, and it is mixing. For p > 2/3 it is shown that there is exponential decay of correlations.