The critical-fugacity asymptotic conjecture for the hard-core model on the lattice

Let $u^{\text{hc}}$ denote the hard-core model on $\mathbb{Z}^{d}$, and let $\lambda_{c}^{+}(d)$ be its upper critical fugacity. The dimension is $d$.

Critical-fugacity asymptotic conjecture.

$$\lambda_{c}^{+}(d)\sim\frac{e}{2d}\quad\text{as }d\to\infty.$$

This conjecture asserts that the lower bound supplied by the tree threshold gives the correct leading asymptotics for the onset of multiple Gibbs measures on $\mathbb{Z}^{d}$. The paper's theorem gives multiple Gibbs measures for $\lambda>C\frac{\log d}{d}$, leaving the sharper asymptotic open.