29 problems

Let $a,b,c,d$ be non-zero vertex weights satisfying … For each $n\geqslant 0$, consider the transfer matrix of the periodic eight-vertex model with an odd number $L=2n+1$ of vertic…

Let $L\geqslant 1$, let $u_1,\dots,u_L$ be inhomogeneity parameters, and define the inhomogeneous open-boundary transfer matrix … where $K^-(u)=K(u)$, $K^+(u)=K(u+2\eta)$, … … Set…

Fendley's conjecture. All eigenvalues of $\mathbb{O}_K$ are roots of unity.

Let $T_{L,L}$ be the diagonal transfer-matrix sub-block for a triangular-lattice strip of width $L$ with fully toroidal boundary conditions, and let $b^{(L)}_1$ denote the stated a…

Phase-diagram conjecture. Properties A–B hold throughout the entire Berker–Kadanoff phase, with $q_0(L)$ replaced by the boundary of the Berker–Kadanoff phase, both at finite size…

Let $L$ be the width of a triangular-lattice strip with cylindrical boundary conditions, let $B_k$ denote the Beraha numbers, and let $D(q)$ be the amplitude matrix. Cylindrical-st…

Quasi-everywhere convergence conjecture. Quasi-everywhere convergence in the displayed limit may fail; equivalently, the actual non-Lyapunov exceptional set may have positive capac…

Transfer-matrix commutativity conjecture. The transfer matrices are mutually commuting and commute with all these local conserved quantities:

Transfer-matrix commutativity conjecture. Transfer matrices with different spectral parameters mutually commute:

Let $T(u)$ denote the transfer matrix of the spin-chain model, depending on the spectral parameter $u$. The authors seek a factorization of $T(u)$ for generic values of $u$. Generi…

Let an $n$-state spin model and an $n$-state vertex model be related by the proposed mapping of Boltzmann weights, and let $L$ denote the lattice size. For the vertex model, write…

Bethe strap conjecture. The T-function satisfies

Let $c=e^{-2\beta H}$ and $d=e^{-2\beta J}$, and consider the one-dimensional 3-permaspin model with couplings $J$ and $H$. Let $z$ be the unique root of the cubic … whose argument…

Hidden-symmetry conjecture. All paths and cycles on at least four vertices have a hidden symmetry. The conjecture concerns the apparent common factors arising in the transfer-matri…

Periodicity conjecture. For the general XXZ case, the above periodicity statement holds. Its proof is not known in general, although numerical computations support it in the Ising…

Let $N$ be odd and let $\mathbf T(u)$ be the inhomogeneous eight-vertex-model transfer matrix on $V^{\otimes N}$, with $\phi(u)=\prod_{j=1}^{N}\theta_1(u-u_j|\tau)$. Razumov–Stroga…

Let $T^\mathrm{F}(x)$ be the weighted transfer matrix for full $s$-blocks, and let $T^\mathrm{H}(x)$ be its restriction to the strongly connected component generated by Hamiltonian…

Let $J$ be a Jacobi matrix, let $T_E(n)$ be the transfer matrix from the first site to site $n$, and let $\Sigma_{\rm ac}(J)$ denote the essential support of the absolutely continu…

For an $L_v\times L_h$ lattice at $z=-1$, let $G^{CC}_{L_h}$ and $G^{CF}_{L_h}$ be the generating functions along the periodic direction for the torus and cylinder, respectively, a…

Let $L_v$ denote the length in the periodic direction of an $L_v\times L_h$ lattice, with possible twists for Möbius-band and Klein-bottle boundary conditions. For each fixed $L_h$…

Spectral-boundary conjecture. The boundary of the critical domain coincides with the locus of points $(\beta,\mu)$ where $\mathbf\Lambda$ loses either the property of positivity or…

Let $L=2n+1$ be odd, let $\eta=\pi/3$, and let $T_L(u|x_1,\ldots,x_L)$ be the inhomogeneous eight-vertex transfer matrix with spectral parameters $u,x_1,\ldots,x_L$. Let $a(x)$ and…

Let $T(\lambda)$ denote the Perk–Schultz transfer matrix, and let $\varphi$ and $\psi$ denote the identification and conjugation operators introduced for the multispecies exclusion…

Let $T(\lambda)$ be the transfer matrix, let $V_{\mathfrak s}$ be the sector with fixed species content, and write $T_{\mathfrak s}(\lambda)$ for its restriction to…

Let $Q_0<\infty$ be such that the functions $\mathcal{F}_m(q)$ associated with the difference of the two dominant eigenvalues are analytic for $|q|>Q_0$, and let $\mathcal{F}_m$ de…