11 problems

Let $D$ denote the activity parameter of the high-density hard-core model on $\mathbb{Z}^3$, and let a Gibbs measure be an infinite-volume equilibrium measure for this model. Uniqu…

Davies et al.'s variance conjecture. For any $\lambda>0$,

Balanced-event probability conjecture. There exist $C,d_{0}>0$ such that for all dimensions $d\ge d_{0}$,

Critical-fugacity asymptotic conjecture.

Hard-core Caro–Wei conjecture. The expected size of $X$ satisfies

Let $d_X$ and $d_Y$ be sufficiently large integers, let $\delta\geq 1$ and $\delta',\delta”>0$, and let $\Sigma=X\sqcup Y$ be a $\delta$-approximately $(d_X,d_Y)$-biregular graph s…

Single-class HCP conjecture. All EPGMs are generated by PCs from a single equivalence class. This class has cardinality $16\ell^3$ and consists of $8$ $D$-HCP configurations and th…

Let $\alpha_G(\lambda)$ denote the occupancy fraction of a graph $G$ in the hard-core model, and let $H_{4,6}$ be the $(4,6)$-cage. (4,6)-cage occupancy-fraction conjecture. For ev…

Let $\alpha_G(\lambda)$ denote the occupancy fraction of a graph $G$ in the hard-core model, and let $H_{3,8}$ be the $(3,8)$-cage, also known as the Levi graph or Tutte–Coxeter gr…

Fugacity-uniform ratio conjecture. For every $4\epsilon>04$, there exists $4\lambda>04$ such that, for every triangle-free graph $G$,

Let $\mathbf{M}$ be a branching matrix, and for a specific value $\lambda<\operatorname{WSM}(T_{\mathbf{M}})$ let $V_\lambda$ be the fixed points of the recurrences of the marginal…