8 problems

Let $G=\textit{wand}$ be the fertile graph for the three-state Hard-Core model on a Cayley tree of order $k$, and let $\lambda$ be the activity parameter with critical value…

Consider $k\times k\times\cdots\times k$ cubes on $\mathbb{Z}^d$ in the high-fugacity regime, with columns oriented along one of the $d$ coordinate directions. A periodic Gibbs mea…

Consider the three-dimensional hard-core models on $\mathbb{Z}^3$ with interaction range $D$, including the case $D=2$ corresponding to packings of $2\times2\times2$ cubes, and sup…

Let the hard-core model be defined on the lattice models described in the source, and consider the exceptional values of the interaction range $D$ for which there are infinitely ma…

Consider the $2\times2$ hard-square model, in which occupied sites on the square lattice must be separated according to the hard-square exclusion rule, and let the fugacity tend to…

Consider the hard cross model, a hard-core lattice particle system on $\mathbb Z^2$ corresponding to third-nearest-neighbor exclusion. Its low- and high-fugacity expansions are exp…

A pairwise interaction process is a hard-core stochastic process in which interactions occur between pairs of points, and $\rho$ denotes the positive parameter used in the model. D…

Let the hard-sphere model or hard-core model be at density $\alpha$. The hard-sphere model consists of nonoverlapping spheres, while the hard-core model consists of particles occup…