Ergodicity and Mixing Properties of the Northeast Model

Abstract

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 $p_c=1-{\beta }_c$, where ${\beta }_c$ is the critical parameter for oriented percolation. For $p<p_c$, the trivial measure ${\delta }_0$ that puts mass one on the configuration with all spins set at $0$ is the unique ergodic, translation invariant, stationary measure. For $p\ge p_c$, 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.