Uniqueness of Gibbs Measures and Absorption Probabilities

Gibbs measures are studied using a Markov chain on the nonnegative integers. Uniqueness of Gibbs measures follows from absorption of the chain at $\{0\}$. To this end, we derive a certain inequality. For one-dimensional systems this extends a well-known uniqueness result of Ruelle and for models near the $1/r^2$-interaction Ising model it is a natural improvement of some other results.