38 problems

Consider the six-vertex model with domain-wall boundary conditions on an $N$ by $N$ lattice in the disordered region, and let $Z_N$ be its partition function. Let $C$ and $\kappa$…

Orthogonality conjecture. In the limit $c\to\infty$,

Let the ${\cal Z}_2$-invariant inhomogeneous six-vertex model have anisotropy parameter $q$ satisfying … Let the Euclidean black-hole nonlinear sigma model be viewed as a conformal…

Let $n$ be the fixed number of Bethe roots, let $H$ be the parameter in the transfer matrix, and let $I_j$ denote the corresponding Bethe quantum numbers. Bethe quantum-number conj…

Let $N\geq 3$ be the order of a primitive root of unity, let $z_j^S(\mu)$ denote the centres of complete strings, and let $z_j^S$ be their values as $\mu\to1$. Let $n_S$ be the num…

Let $N\geq 3$ be an integer, let $p_\mu=(0,0,\mu^{-N'},\mu+\mu^{-1})\in\operatorname{Spec}Z$ for $\mu\in\mathbb{C}^{\times}$, and write $Q_\mu(z)=Q_{p_\mu}(z)$ for the correspondin…

Bounded-boundary-variation conjecture. Only boundary conditions for which the variation of the function $\phi$ along the boundary remains bounded will lead to the usual thermodynam…

Six-vertex quadrant Gibbs-measure conjecture. We conjecture that the possible Gibbs measures for the six-vertex model in the quadrant should be very different depending on the valu…

Admissibility conjecture. If $G$ includes no oriented faces, then it is admissible.

Uniform bounds conjecture. There exist constants $0<c<C$, depending on all the data, such that $$

Let ${\rm H}_{L|{\cal S},S^z}^{({\rm cont})}$ and ${\rm H}_{L|{\cal S},S^z}^{({\rm disc})}$ be the continuous and discrete subspaces associated with a lattice model of…

Let $\mathcal M^{n,t,\gamma}$ be the measure from Theorem, and let $\widehat{\mathcal M}^{n,t,\gamma}$ be the probability measure on ordered $n$-tuples of negative even integers wi…

Exponential-boundedness conjecture. There exist constants $c_1,c_2>0$ such that

Let $a_1,a_2,b_1,b_2,c_1,c_2$ be the six-vertex weights, let $\Delta$ be the associated interaction parameter, and consider the six-vertex model with domain wall boundary condition…

Let ${\tt L}$ be fixed, and let $p$ and $n$ take generic values. Consider the algebraic system for the unordered set $\bm{w}=\{w_a\}_{a=1}^{\tt L}$ given by … for…

Consider the six-vertex model with integrable inhomogeneities in the horizontal direction and homogeneous weights in the vertical direction. Integrability conjecture. The equations…

Fix the ratio $n/N$, where $n$ is the number of Bethe roots and $N$ is the system size, and let $C$ denote the contour described in the paper. Thermodynamic Bethe-root contour conj…

Limiting-rectangle conjecture. The enclosed region tends, in the thermodynamic limit, to the rectangle

The F-model is a one-parameter sub-family of the $6$-vertex model indexed by $c>0$. It weights a homomorphism height function $f$ by $c^{-N(f)}$, where $N(f)$ is the number of diag…

The six-vertex model has a disordered phase whose fluctuations are considered as random height-function fluctuations. Nienhuis's conjecture. The fluctuations in the disordered phas…

The odd tacnode process is the limiting process with kernel $K^{\operatorname{tac},\operatorname{odd}}$. The six-vertex model with domain wall boundary conditions is the model of i…

Let $\Lambda$ be a generic domain for the six-vertex model, and consider a south side ending at a corner, with the relevant external portion of the Arctic curve lying above that si…

Let $r(z)$ be the scaled logarithmic derivative of the boundary correlation function $h_N(z)$, namely … For the six-vertex model with domain wall boundary conditions, let $t$ and…

The six-vertex model limit shape equations define a nonlinear system for the macroscopic height function, with infinitely many conserved quantities arising from commuting transfer…

Alternating-parity exponent conjecture. The corresponding exponents satisfy