112 problems

Strong Schatten convergence conjecture. The convergence above is strong in every $\mathfrak{S}^p(\mathfrak{H}_k)$ with $p>1/s+1/2$.

Häggström's uniqueness conjecture. For $p>p_{\mathsf{s}}$, the random-cluster model has a unique infinite-volume Gibbs measure on the wired tree.

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

Consider the antiferromagnetic hard-core Potts model with $q$ spins, equivalently the proper vertex-coloring model with $q$ colors, on an infinite graph of maximum degree $D$ satis…

Let $\mathcal{M}$ be the dimer-configuration space associated with the triangular quadri-tiling construction, and let $\mathbb{P}$ be the marginal on $\mathcal{M}$ of the probabili…

Uniqueness and dynamical characterization conjecture. The isometry-invariant thermodynamic limit ${\cal A}_{\mathbb{R}^3}$ should be unique, and should coincide with the polyhedral…

Gibbs-measure existence conjecture. The family

Consider a general ferromagnetic SOS model with coupling $J<0$, temperature $T>0$, and an integer height parameter $m>1$. A translation-invariant splitting Gibbs measure is a Gibbs…

Let $m,k\geq 2$, $J>0$, and $\beta\geq 0$. A translation-invariant splitting Gibbs measure is a Gibbs measure invariant under the tree translations. Antiferromagnetic uniqueness co…

Let $m,k\geq 2$ and $J<0$. Let $h=(h_0,h_1,\ldots,h_{m-1})$ be a vector of boundary-law parameters satisfying equation (2.6), with $h_0=0$. Symmetric-solution conjecture. For all…

Let $k\geq 2$, $J<0$, and $\beta>\beta^1_{\rm cr}$. Let $\mu^*_+$ be the top symmetric translation-invariant splitting Gibbs measure, and let $\omega^{(n)}$ denote the boundary con…

Let $m=2$, $J<0$, and $\beta\leq\beta^1_{\rm cr}$. Let $\mu^*$ denote the symmetric Gibbs measure associated with the unique solution in the ferromagnetic SOS model. Uniqueness con…

Let $m=2$, $k\geq 2$, and $J<0$. Let $\mu^*_{-}$, $\mu^*_{\rm m}$, and $\mu^*_{+}$ be the three symmetric TISGMs, and let $\delta_\omega$ denote the Dirac measure at a configuratio…

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…

Projection conjecture. Nice factor maps should also map Markov measures to BGM's.

Let $\gamma^+$ be the specification in the paper, let $\mathcal{G}(\gamma^+)$ denote its Gibbs measures, and let $C1$ be the condition defined in the paper. A measure $\mu$ is ergo…

Consider an ergodic, translation-invariant tiling-valued random process, invariant under color-preserving translations. Let $p_n,p_s,p_w,p_e$ be its placement probabilities for nor…

Let the Farey fraction spin chain be the one-dimensional ferromagnetic spin system described in the paper, with critical temperature $T_c=\frac{1}{2}$. An extremal Gibbs measure is…

Consider the magnetostriction model described above, with transition inverse temperature $\beta_{\text{t}}$ and the three distinct Gibbs states at $\beta=\beta_{\text{t}}$: one exp…

First-discontinuity-time conjecture. Since this configuration should be the first point of discontinuity to appear, we conjecture that

Consider the two-dimensional Ising model with boundary condition … denote the set of infinite-volume translationally invariant Gibbs measures, and consider the finite-volume Gibbs…

A spin glass model is considered with symmetric boundary conditions, and its finite-volume Gibbs measures are studied as the volume tends to infinity. Newman–Stein's conjecture. So…

Let such measures be invariant measures of systems with a stochastic dynamics, and suppose that they have no detailed balance. In the spin-flip setting, consider non-reversible pro…