For each $d\geq 2$, we give examples of $d$-dimensional periodic lattices on which the hard-core and Widom-Rowlinson models exhibit a phase transition which is monotonic, in the sense that there exists a critical value $\lambda_c$ for the activity parameter $\lambda$, such that there is a unique Gibbs measure (resp. multiple Gibbs measures) whenever $\lambda$ is less than $\lambda_c$ (resp. $\lambda$ greater than $\lambda_c$). This contrasts with earlier examples of such lattices, where the phase transition failed to be monotonic. The case of the cubic lattice $Z^d$ remains an open problem.
"A Monotonicity Result for Hard-core and Widom-Rowlinson Models on Certain $d$-dimensional Lattices." Electron. Commun. Probab. 7 67 - 78, 2002. https://doi.org/10.1214/ECP.v7-1048