Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

Hidden Symmetries of Weighted Lozenge Tilings

We study the weighted partition function for lozenge tilings, with weights given by multivariate rational functions originally defined by Morales, Pak and Panova (2019) in the context of the factorial Schur functions. We prove that this partition function is symmetric for large families of regions. We employ both combinatorial and algebraic proofs.

Sampling biased monotonic surfaces using exponential metrics

Monotonic surfaces spanning finite regions of ℤd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ2 and for bias λ > d in ℤd when d > 2. In ℤ2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting.

Lozenge Tilings of Hexagons with Central Holes and Dents

Ciucu proved a simple product formula for the tiling number of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `fern', has been removed from the center (Adv. Math. 2017). In this paper, we present a multi-parameter generalization of this work by giving an explicit tiling enumeration for a hexagon with three ferns removed, besides the central fern as in Ciucu's region, we remove two new ferns from two sides of the hexagon. Our result also implies a new `dual' of MacMahon's classical formula of boxed plane partitions, corresponding to the exterior of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.