14 problems

Three-halves-unit shifted-hole tiling conjecture. The number of lozenge tilings of this hexagon with the triangular hole equals

Consider a lozenge tiling of a hexagon with normalized side lengths $\alpha,\beta,\gamma$, and define the arctic region to be the set of lozenges connected to the boundary by seque…

Let $\alpha,\beta,\gamma$ be the normalized side lengths of a hexagon, let $(x,y)$ be a normalized location in this hexagon, and let $\mathcal P_{\alpha,\beta,\gamma}(x,y)$ denote…

Let $q$ and $\kappa$ be the model parameters, let $\mathcal{F}_n^s(0)$ be the orthogonal functions appearing in the barcode kernel, and let $\mathcal{K}^{\mathrm{barcode}}(s,t)$ be…

Let $K^{\mathrm{barcode}}_{(L)}(s,t)$ be the pre-limit barcode kernel for $s,t\in\mathbb{Z}$, and let $q$ be the model parameter. Barcode-kernel limit conjecture. For all…

For $t\in\mathbb{Z}$ and $M\in\mathbb{Z}$, define the two truncation limits $\rho_0(t)$ and $\rho_1(t)$ by the displayed sums over $n\in\frac12\mathbb{Z}_{\ge0}$. Let…

Let $(\mathsf{t},\mathsf{x})$ be a macroscopic point in the waterfall region $\mathcal{W}$, and let $q$ and $\kappa$ be the model parameters. Let…

Let $\bar\Omega$ be the limiting region with red boundary segments corresponding to constrained boundary conditions and blue segments corresponding to free boundary conditions. Let…

Let $L_{n,l}$ denote the number of lozenge tilings of the equilateral triangle of side length $n$ containing $l$ lozenges, and let $M_{n-1}$ be the polynomial appearing in the uppe…

Let $R^{\odot}_{x,yz}(\textbf{a};\ \textbf{c};\ \textbf{b})$ be the region described in the paper, and consider its tilings invariant under $180^{\circ}$ rotations. Product formula…

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…

Consider the scaling regime in which $x+y=y+t=z=N$, all fern lobe sizes are equal to $1$, both ferns have an even number of lobes, and their combined length is $m+n=M$. Let the reg…

Let $U$, $\varphi$ and their discretizations $(U_L,\varphi_L)_{L\ge1}$ be as above, and let $h_t^\eta$ denote the height function under Glauber dynamics started from an arbitrary i…

The odd axial-puncture criterion.