We prove, in all dimensions, that for a stationary Gibbs state with finite range or rapidly decreasing interaction, there is at most one infinite percolation cluster. This implies that the connectivity function is bounded away from 0.
Alberto Gandolfi. "Uniqueness of the Infinite Cluster for Stationary Gibbs States." Ann. Probab. 17 (4) 1403 - 1415, October, 1989. https://doi.org/10.1214/aop/1176991161