26 problems

Kenyon–Okounkov conjecture. In the interior of $\mathcal{L}$, the fluctuations are governed by the pullback of the Gaussian free field from $\mathbb{H}$.

CKP thermodynamic-limit conjecture. The same formula for local correlations should hold in the thermodynamic limit for domino tilings of more general regions.

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…

For each interior non-extremal point $(x^*,y^*)$ arising as a normalized limit point of a sequence of large simply-connected domino-tileable regions, let the associated tiling-valu…

Let $R_n$, $R^*$, $(x_n,y_n)\to(x^*,y^*)$, and the asymptotic renormalized height function ${\mathcal H}$ be as in the scaling-robustness conjecture, with $(x^*,y^*)$ an interior p…

Let $R_1,R_2,\dots$ be finite, simply-connected, domino-tileable regions that grow without bound, with suitably rescaled copies converging to a compact subset $R^*$ of the plane. S…

Biased arctic ellipse conjecture. The analogue of the arctic circle theorem should hold for this biased distribution. This predicts that the boundary between the frozen and tempera…

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 the random tiling model on a plane bipartite lattice, with the lattice embedded in the plane in a suitable way. Universality conjecture. The validity of the model does not…

Let monomers be holes in the square lattice, with two-point correlations defined by the random tiling model. Krauth–Moessner conjecture. The two-point correlations of monomers beha…

Let $\alpha$ and $\beta$ be parameters satisfying … Let $t$ be the interface parameter, and let a smooth region mean a smooth region of the limit shape to the left of the interface…

Let $\mathsf{Ht}_\mathcal N$ be the random height function of the model at scale $\mathcal N$, let $\mathbb E[\mathsf{Ht}_\mathcal N]$ be its expectation, let $\pounds$ denote the…

Let a limit density have two support intervals meeting at a common boundary point $t_d$, where the local fluctuations of the correlation kernel are described by the Pearcey kernel.…

Universal edge fluctuation conjecture. The local statistics of uniform lozenge tilings for polygonal domains have a universal scaling limit at each particular type of points.

Bulk EGTI universality conjecture. Around a point inside the liquid region, the local statistics should be given by the EGTI measure with slope matching the gradient of the limitin…

The six interface processes are the three continuous-continuous limits—Airy, Pearcey, and soft tacnode processes—and the three discrete-continuous limits—GUE-minor, Cusp-Airy, and…

Let $\mathfrak{L}$ be a $\mathbb{Z}^2$-periodic lattice, let $R\subset\mathbb{R}^2$ be a bounded domain, and let $h_\epsilon$ be the normalized random height function arising from…

Circular limit-shape conjecture. As $n\to\infty$, the limit-shape curve is

Consider the cubiculated box … Let $\mathbf T,\mathbf T_0,\mathbf T_1$ be independent uniformly distributed random tilings of $\mathcal R$, and let $\operatorname{Tw}$ denote the t…

Harmonic-measure conjecture. For all such dimer models and liquid domains, the asymptotic tile or edge densities are given by the pullback, under $f$, of the appropriate harmonic m…

Plane partitions can be represented as lozenge tilings, and in the scaling limit their frozen and liquid regions are separated by an arctic curve. A turning point is a point where…

Fix an arbitrary tilable polygonal domain $Upsilon$ on the triangular grid. Let $H_L(x,y)$ be the random height function of a uniformly random lozenge tiling of the dilated domain…

For a uniformly random tiling of a polygonal domain, let the liquid region be equipped with the complex structure determined by the local complex slope, and let the Gaussian Free F…

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…

For a dimer model with fixed boundary, let $(s,t)$ be a possible slope of the limit shape, and let $\mu_{s,t}$ denote the measure on plane tilings defined for that slope. Local sta…