12 problems

High-temperature exponential clustering conjecture. There exist $\beta^*$, $C_\mathrm{Cov}$, $c_\mathrm{Cov}>0$, and $n\in\mathbb{N}$ such that, for every $\beta<\beta^*$ and every…

Consider the Ising model and let ${\mathcal B}\subset [\beta_c,\infty[$ be the at most countable set of inverse temperatures at which the pressure may fail to be differentiable. Is…

Let $p,q\geq 3$ satisfy … For the Ising model on the corresponding Lobachevsky lattice, let $\mathcal{G}_\beta$ denote the set of Gibbs states, let…

Poulin–Hastings conjecture. The convergence of the Markov Entropy Decomposition is connected to the approximate saturation of strong subadditivity, also known as the decay of the c…

Consider quasi-local quantum dynamics with a steady state and Gibbs states of local Hamiltonians. A Gibbs state of a Hamiltonian is strictly local when the Hamiltonian contains onl…

Let $V$ be a metric space, let $\beta>0$, and let $\boldsymbol{b}:\mathcal{P}_{f}(V)\to[0,\infty)$ be a subbaditive function. Let $\Phi$ be a local interaction satisfying … For eve…

High-degree free-boundary state nonequilibrium conjecture. For every $\varepsilon>0$, for all sufficiently large $r$, if $J\geq(1+\varepsilon)J_{uniq}(r)$, then

Strict boundary condition conjecture. The strict boundary condition is equivalent to low-temperature stability of non-periodic ground states, meaning the existence of non-periodic…

Let $\mathcal{G}$ be the set of Gibbs (DLR) states and let $\operatorname{tr}\mathcal{G}$ denote the set of translation-invariant Gibbs states. Let $\mu^+$ and $\mu^-$ be the plus…

In the three-dimensional Ising model, a Dobrushin state is a Gibbs state generated by an interface separating plus and minus phases; an extremal state is an extreme point of the co…

Periodic Gibbs-state conjecture. Two-dimensional models should always possess a finite number of extremal Gibbs states, all of which are periodic.

Critical Potts phase-coexistence conjecture. There is a unique Gibbs state when $q=3$ and $q=4$, whereas for $q\geq 5$ there is coexistence at $\beta_c(q)$ of $q+1$ pure phases: th…