Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions

Understand

For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/non-uniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$.

• The conjectured value of the critical activity is approximately $3.796$.
• Until recently, the best lower bound followed from algorithmic results of Weitz (2006).
• Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$ where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$.