33 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 $n$ be a positive integer, and consider a semiregular hexagon with side lengths $2n-1$, $2n-1$, and $2n$, repeated in opposite positions. A central lozenge is the lozenge occup…

Let $H_{m,n}$ denote the finite torus graphs used to model periodic lozenge tilings, let $\nu_{m,n}$ be their uniform measures, and let $\nu$ be the unique weak limit of…

Let $X'$ be the space of lozenge tilings of the plane, and let $\nu$ be the measure obtained as the limit of uniform measures on periodic tilings. Its measure-theoretic entropy equ…

For lozenge tilings of a hexagon of scale $N$, let the inverse spectral gap denote the reciprocal of the spectral gap of the Glauber dynamics, and let the mixing time be the time n…

Let $\vec b=(b_1,b_2,\ldots,b_T)$ be a realization of the barcode process with $b_j\in\{0,1\}$, and define its height function by $h(t)=\sum_{j=K}^t b_j$, where $K$ is a fixed offs…

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…

In the $q^{\mathrm{Volume}}$ lozenge tiling model, let the height function be the function recording the tiling's local height, and let the fluctuation field denote its centered an…

Consider a homeomorphic solution $G:\mathcal{U}\to\mathcal{D}$ of the complex-linear Beltrami equation … The associated complex structure consists of the diffeomorphism $G$ and its…

GUE-corners universality conjecture. At such tangency points, the GUE-corners process is the universal scaling limit.

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…

Let $H_{n,x}(\mathbf{A},\mathbf{B})$ and $\overline{H}_{n,x}(\mathbf{A},\mathbf{B})$ be the two hexagonal regions with central triangular holes and additional holes on three crossi…

Discrete Gaussian hole-height conjecture. There is a sequence of integer shifts $\mu_L$ such that

Let $m$ be a nonnegative integer and let $N$ and $r$ be positive integers with $N\geq r$. Consider a hexagon with side lengths $2m,N,N,2m,N,N$ clockwise from the left, and let it c…

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…

A fern is a finite string of triangles of alternating orientations that touch at corners and are lined up along a common axis. Consider a hexagon with any finite number of collinea…

Square-root phenomenon. For an arbitrary centrally symmetric system of ferns, the suitably normalized number of centrally symmetric lozenge tilings should equal the square root of…

Ciucu–Eisenkölbl–Krattenthaler–Zare's conjecture. The number of lozenge tilings of a hexagon with side-lengths $x,y+m,z,x+m,y,z+m$, with an equilateral triangle of side-length $m$…