Before revision:  As an application of our results, for the hard-core model on independent sets weighted by a fugacity \lambda, we establish O(nn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree \Delta when \lambda<\lambda_c(\Delta) where \lambda_c(\Delta) is the critical point for the uniqueness/non-uniqueness phase transition on the \Delta-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n\log{n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region.
After revision:  As an application of our results, for the hard-core model on independent sets weighted by a fugacity \lambda, we establish O(nn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree \Delta when ) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region.