19 problems

Pemantle–Steif conjecture. For any Heisenberg model on any graph, SB is equivalent to SB+.

Fix $(s,t,p,q)\in{\mathbb R}^4$. Let ${\cal Q}_n^{(s,t,p,q)}$ be the quadri-tilings of ${\cal Q}_n$ with first height change $(\lfloor ns\rfloor,\lfloor nt\rfloor)$ and second heig…

Let ${\mathbb T}$ be a periodically tiled torus with lattice of periods $\Lambda$, and let ${\mathbb T}_n={\mathbb T}/n\Lambda$ be such that ${\mathbb T}_n^*$ is bipartite. Let…

Let $G$ be a countably infinite graph of maximum degree … , the hard-core lattice gas on $G$ has a unique Gibbs measure whenever … for all vertices $x$; perhaps this remains true w…

Non-degeneracy conjecture. For every $c\in (\kappa h_{\min},\kappa h_{\max})$, $U_n$ given $\operatorname{Conset}_n(h,c)$ lies asymptotically in $\operatorname{Gibbsnondeg}(c)$.

Let $G=(V,E)$ be a simple graph, let $\lambda>0$, and let $\mu$ be the Gibbs distribution of the monomer-dimer model on $G$ with fugacity $\lambda$. The quantity…

Let $G=(V,E)$ be a graph, let $\lambda>0$, and let $\mu$ be the Gibbs distribution of the monomer-dimer model on $G$ with fugacity $\lambda$. The quantity…

Let $q \geq 2$ be an integer, let $\beta \in [0,1]$, and consider the $q$-state antiferromagnetic Potts model on graphs of maximum degree $\Delta$. Define … Weak spatial mixing (WS…

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 $z\geq 0$ be the activity and let $\beta\geq 0$ be the inverse temperature of the area-interaction model. Phase transition means non-uniqueness of the area-interaction measures…

Critical-point threshold conjecture. There is a $q_0=q_0(d)$ such that for $q>q_0$ there is non-uniqueness of Gibbs measures

Uniqueness conjecture. For $p\neq p_c$ there is a unique Gibbs measure.

Soft-core transition-time conjecture. There exists a time $t_{NG}(\beta)$ such that the time-evolved measure is Gibbs for all $t<t_{NG}(\beta)$ and non-Gibbs for all…

Emergence of the renormalized Gibbs measure. In the limit $\lambda=T^{-1}\to0$, one obtains the renormalized Gibbs measure in the sense that

Let $k\geq 4$ and consider the ferromagnetic model with critical values $\theta'_{\rm c}(k)$, $\lambda'_{\rm cr}(k)$, $\theta_{\rm c}(k)$, and $\lambda_{\rm cr}(k)$ as defined in t…

Let $\mathcal{S}^N$ be the sphere, let $H_0$ be the pure-noise Hamiltonian, and for $\beta>0$ let $\pi_{0,\beta}$ be the Gibbs measure with density proportional to…

Negative-temperature equivalence conjecture. For every $\beta>\beta_0$, the following are equivalent: the partition functions $Z_{N_k,\beta}$ are finite for all sufficiently large…

Consider the nearest-neighbor $q$-state clock model in spatial dimension $d\geq 3$, at inverse temperature below the critical-temperature threshold and with $q$ sufficiently large.…

Let the Ghirlanda–Guerra identities be imposed on the laws of random overlap structures, and consider the convex set of laws of stochastically stable ROSts. Ghirlanda–Guerra extrem…