Uniqueness of locally stable Gibbs point processes via spatial birth-death dynamics

We prove that for every locally stable and tempered pair potential with bounded range, there exists a unique infinite-volume Gibbs point process on Rᵈ for every activity < (e^L C_) ^-1, where L is the local stability constant and C_: = supₗ ₑ^₃ ₑ^₃ 1 - e^-| (x, y) | dy is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least e, where the improvements becomes larger as the negative parts of the potential become more prominent (i. e. , for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.