45 problems

Let $G$ be a random $\Delta$-regular graph, and fix an activity parameter $\lambda>0$. The hard-core model on $G$ is the distribution on independent sets $I$ with probability propo…

Critical-fugacity asymptotic conjecture.

Let $d\geq 3$, let $\ell>\ell_c(d)$, and let $\mathcal{G}(n,d)$ denote the random $d$-regular bipartite graph model. For a graph $G$ with bipartition $V_1\cup V_2$, let…

Let $G$ be a graph of maximum degree $Delta$, let $delta\in(0,1)$, and consider the paper's particle-system algorithm for sampling independent sets with prescribed marginals. Exten…

Consider the single-site nonlinear dynamics for the hard-core model on a graph of maximum degree $Delta$, in the density regime where the paper proves qualitative convergence. Exte…

Let $G$ be a graph and consider the mean-field and single-site nonlinear dynamics for the hard-core model in the irreducibility region covered by the paper's qualitative convergenc…

Threshold conjecture for extremal-graph uniqueness. Based on experimental results, there exists a threshold $\lambda_0$ such that, for $\lambda<\lambda_0$, the extremal graph attai…

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 $d\ge d_{0}$,

Consider the sparse Ising and hard-core models on graphs of maximum degree $\Delta$, for which the Glauber dynamics is polynomially mixing at criticality. The polynomial-mixing win…

Davies–Kang's conjecture. The same bound holds for all positive $\lambda$.

Consider the hard-core model on the hypercube $Q^d$, with fugacity $\lambda$, and let an independent set be sampled from this model. For sufficiently small fugacity, specifically w…

Galvin's conjecture. Such a container lemma should hold down to

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

Let $G$ be the nearest-neighbour graph on $\mathbb{Z}^d$, let $\mathcal{I}(G)$ denote its independent sets, and let $\operatorname{hc}(\lambda)$ be the set of hard-core Gibbs measu…

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…

Suppose that $D^2=2\ell^2$ with $\ell\in\mathbb N$ and $\ell=0\mod 3$. There are at least two $\mathbb Z^3$-symmetry classes of $D\cdot\mathbb A_3$-sub-lattices, and each such sub-…

Let $D^2=2\ell^2$ with $\ell\in\mathbb N$, $\ell\neq 0\mod 3$, and $\ell\neq 2^n$. In this case the perfect configurations consist of $D$-FCC sub-lattices and their $\mathbb Z^3$-s…

Let $D^2=2\ell^2$ with $\ell$ divisible by $3$, and let a given $D\cdot\mathbb A_3$-sub-lattice admit a continuum family of layered dense-packings. A perturbation analogous to the…

Critical-fugacity tightness conjecture. The bound $\lambda_c(d)$ on the fugacity is tight in the sense that, for $\lambda>\lambda_c(d)$, efficient approximate sampling may still be…

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…

Unique dominant HCP class conjecture. Among the classes of $D$-HCP configurations, only one class is dominant.

Unique dominant class conjecture. There is always a unique dominant class among these classes.

Uniqueness conjecture. For every such $D$, the EPGM is unique, at least for sufficiently large $u$; this unique EPGM is called disordered.

The paper studies a hierarchy of sufficient conditions for positivity of the avoidance probability, with bounds on the negative-fugacity singularity obtained by choosing an integer…