41 problems

Product formula conjecture. There exists an integer sequence $\{o_n\}_{n \geq 0}$ such that

Di Francesco's conjecture. The number of domino tilings of the Aztec triangle is given by a product formula similar to the product formula for alternating sign matrices.

Flip-and-trit connectivity conjecture. Any two tilings of a box can be connected by a finite sequence of flips and trits.

Let $(A,B)$ be chosen uniformly at random from the balanced ordered partitions of $\{1,2,\ldots,2k\}$, and let $s,t\in\{1,2,\ldots,2k\}$ have opposite parity. The events $s\in A$ a…

Consider the double domino interface in a domino tiling, and let the lattice mesh tend to zero. The associated domino tiling height function is expected to have the Gaussian free f…

Let $U_{\Phi}=\{u:|u_1|+|u_2|<\frac{1}{2}\}$ be the set of interior slopes, and let $\mu_u$ be the explicitly described ergodic gradient Gibbs measure of slope $u$. Uniqueness conj…

Consider the path arising from the double domino tiling model in plane strips. SLE4 scaling-limit conjecture. The scaling limit of this path should be the trace of…

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

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 a sequence of finite regions converging to a fixed shape, and let $(s,t)$ be the tilt given by the variational principle wherever that tilt is defined. Let $\mu_{s,t}$ be…

Let $\mu_{s,t}$ denote the measure on domino tilings of the plane associated with tilt $(s,t)$. A measure is conditionally uniform when, given a tiling outside any finite region, t…

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 domino tilings of the triangular lattice with periodic boundary conditions. ReCom moves define a metagraph on the tilings, and parity invariants partition it into four sec…

Electrostatic correlation conjecture. In the limit of large mutual separations between the holes,

Modulo-$4$ conjecture. Except sometimes when $r$ and $c$ are both congruent to $3$ modulo $4$, the count $a(r,c;h)$ is a multiple of $4$.

Kong's modulo-$8$ conjecture. For all $k$, the integer $c_k$ is congruent to $1$ modulo $8$.

Kong's conjecture. The number $a(2k+1)$ is an odd multiple of $2^k$.

Let $\bar d(r,c)$ denote the maximum number of non-bonding dominoes that can be placed on an $r\times c$ rectangular board. Maximum-filling conjecture. If one of $r$ and $c$ is an…

Let $D(r,c,d)$ denote the number of ways to place $d$ non-bonding dominoes on an $r\times c$ rectangular board. Quadratic formula conjecture. For integers $r,c\ge 3$, … The formula…

Let ${\mathcal T}_n$ be the Aztec triangle of order $n$, obtained from a lattice square of side length $2n$ by cutting it into congruent parts along a zig-zag path and adjoining th…

Let $AD(n)$ be the Aztec diamond of order $n$. Let $O(2n,[2n])$ be the set of off-diagonally symmetric domino tilings of $AD(2n)$ with no boundary defect, and let $D(2n-1)$ be the…