76 problems

Gaussian free field–discrete Gaussian conjecture. The fluctuations of the dimer-model height function in the liquid region of a graph of scale $N$ are approximated in distribution…

Kenyon–Okounkov conjecture. The limit of the height fluctuations of $h_{Omega_\delta}$ is the Gaussian Free Field in a non-trivial complex structure determined by the limit shape.

Let $\Delta$ be a convex lattice polygon containing the origin in its interior. A Laurent polynomial $W$ has Newton polygon $\Delta$, and a pair $(G,D)$ consists of a consistent di…

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…

Let $(s,t)$ satisfy $|s|+|t| \leq 2$, and let $a,b,c,d$ be weights satisfying $ab=cd$ such that the average height function for weighted torus tilings has tilt $(s,t)$. Let…

In the dimer model on an $n \times n$ torus, let $p_a$, $p_b$, $p_c$, and $p_d$ denote the limiting densities of the four edge types in the thermodynamic limit as…

Consider the plurimer correlation in the random tiling model, where the results established in the paper assume triangular plurimers, with all but one monomer having side-length…

Consider a weighted $L^\ast$-punctured planar bipartite graph with generic edge weights $\operatorname{wt}\in\mathbb{R}_{>0}^{|\mathbf{V}^\ast|-1}$, and let $H$ be the function spe…

$k$-marked GUE-corners conjecture. If the model is $k$-periodic in the vertical direction, then its scaling limit is a $k$-marked GUE-corners process.

Let a dimer model be defined on a simply connected bounded domain. At each location in the domain, consider a torus dimer model with weights $(a_0,a_1,a_2)$ chosen according to the…

Let $\rho_{\mathrm{hex}}$ and $\rho_{\mathrm{sq}}$ denote the expected numbers of loops per vertex in the double-dimer models on the infinite honeycomb graph and the square grid…

Let $\rho$ denote the expected density of double-dimer loops per vertex for the infinite honeycomb graph. The numerical evaluation of the exact infinite-series formula for $\rho$ g…

Let a limiting t-surface be described by the parametrization in the source, with cusp apexes $P_i$ and parameters $a$ and $t$. The boundary points determine the parameter $a$. Cusp…

Consider a dimer model exhibiting gaseous phases, with an associated t-surface and origami map. A maximal surface is a space-like maximal surface in the ambient Minkowski space. Ga…

Consider a doubly periodic dimer model, with a liquid region in its scaling limit and an associated Harnack curve for the underlying weighted lattice. Kenyon–Okounkov's conjecture.…

Let a dimer model have a scaling limit whose t-surface has cusp locations, and let the model's global fluctuations contain a discrete Gaussian component with a shift. Cusp-location…

Finite-{mathbb N} facet conjecture. Facets exist in the multinomial dimer model for every finite $N$, and their size shrinks as $N\to\infty$.

Universality conjecture. For a very large class of “higher genus” dimer models, the integral formula for height moments has the same form as the formula given in the paper's Coroll…

Conformal-coordinate conjecture. For more general boundary conditions and other periodic lattices, an appropriate analogue of the critical point mapping gives the correct coordinat…

Zigzag-mutation invariance conjecture. Zigzag mutation is an isomorphism

Monomial-reduction conjecture. There exists a mutation sequence

Decorated-reduction conjecture. Under certain conditions for the decorated Newton polygon, there exists an integrable system such that:

Reduced Goncharov–Kenyon integrability conjecture. The Hamiltonians $\mathcal{Z}_{a,b}(\mathbf{x})$ for $(a,b)$ in the interior of $N$ define an integrable system on the subvariety…

Torus-knot dimer-model conjecture. Every $(p,q)$ torus knot admits a dimer model.

Existence and uniqueness conjecture. Perfect t-embeddings of nice planar bipartite graphs should exist in general, and should be unique up to the natural group of symmetries.