Boxed Plane Partitions Research Articles

Arctic curves of the four-vertex model

We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. More generally, the considered boundary conditions relate vertex models to scalar products of off-shell Bethe states, boxed plane partitions, and fishnet diagrams in quantum field theory. In the scaling limit, the model exhibits the emergence of an arctic curve separating a central disordered region from six frozen ‘corners’ of ferroelectric or anti-ferroelectric type. We determine the analytic expression of the interface by means of the Tangent Method. We supplement this heuristic method with an alternative, rigorous derivation of the arctic curve. This is based on the exact evaluation of suitable correlation functions, devised to detect spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to the orthogonalizing measure of the Hahn polynomials. As a by-product, we also deduce that the arctic curve’s fluctuations are governed by the Tracy–Widom distribution.

Lozenge Tilings of a Hexagon with a Horizontal Intrusion

Motivated by a conjecture posed by Fulmek and Krattenthaler, we provide product formulas for the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion. As a direct corollary, we obtain a product formula for the number of boxed plane partitions with a certain restriction. We also investigate the asymptotic behavior of the ratio between the number of lozenge tilings of a semiregular hexagon containing a horizontal intrusion and that of a semiregular hexagon without an intrusion.

Correlations in totally symmetric self‐complementary plane partitions

Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of a hexagon with free boundary to express them as perfect matchings of a family of non-bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit shape of TSSCPPs in the scaling limit.

Determinant formulas for the five-vertex model

We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box of given size). The partition function of an inhomogeneous model is given in terms of a determinant. For the homogeneous model, it can be given in terms of a Hankel determinant. We also show that in the homogeneous case the partition function is a τ-function of the sixth Painlevé equation with respect to the rapidity variable of the weights.

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.

The partition function of the four-vertex model in inhomogeneous external field and trace statistics

The exactly solvable four-vertex model with the fixed boundary conditions in the presence of an inhomogeneous linearly growing external field is considered. The partition function of the model is calculated and represented in the determinantal form. The established connection with the boxed plane partitions allows us to calculate the generating function of plane partitions with the fixed sums of their diagonals. The obtained results are another example of the connection of integrable models with enumerative combinatorics.

The phase model and the norm-trace generating function of plane partitions

A quantum phase model is considered and the temporal evolution of the exponential of the first moment of particles distribution is studied. The calculation of the temporal evolution is based on the properties of Schur functions. The form-factor of the considered distribution function is expressed in the determinant form and it is proved that in the q-parametrization it is equal to the generating function of the norm-trace generating function of the boxed plane partitions. The asymptotics of the temporal evolution in the double scaling limit and decreasing temperature is obtained. It is shown that the amplitudes of the leading asymptotics depend on the generating function of plane partitions with the fixed sum of diagonal elements.

A q-enumeration of lozenge tilings of a hexagon with three dents

MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an a×b×c box is equal toHq(a)Hq(b)Hq(c)Hq(a+b+c)Hq(a+b)Hq(b+c)Hq(c+a), where Hq(n):=[0]q!⋅[1]q!…[n−1]q! and [n]q!:=∏i=1n(1+q+q2+…+qi−1). By viewing a boxed plane partition as a lozenge tiling of a semi-regular hexagon, MacMahon's theorem yields a natural q-enumeration of lozenge tilings of the hexagon. However, such q-enumerations do not appear often in the domain of enumeration of lozenge tilings. In this paper, we consider a new q-enumeration of lozenge tilings of a hexagon with three bowtie-shaped regions removed from three non-consecutive sides.The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths 2n, 2n+3, 2n, 2n+3, 2n, 2n+3 (in cyclic order) with the central unit triangles on the (2n+3)-sides removed. Moreover, our result also implies a q-enumeration of boxed plane partitions with certain constraints.

Combinatorial Aspects of Correlation Functions of the XXZ Heisenberg Chain in Limiting Cases

We discuss the connection between quantum integrable models and some aspects of enumerative combinatorics and the theory of partitions. As a basic example, we consider the spin XXZ Heisenberg chain in the limiting cases of zero and infinite anisotropy. The representation of the Bethe wave functions via Schur functions allows us to apply the theory of symmetric functions to calculating the thermal correlation functions as well as the form factors in the determinantal form. We provide a combinatorial interpretation of the correlation functions in terms of nests of self-avoiding lattice paths. The suggested interpretation is in turn related to the enumeration of boxed plane partitions. The asymptotic behavior of the thermal correlation functions is studied in the limit of small temperature provided that the characteristic parameters of the system are large enough. The leading asymptotics of the correlators are found to be proportional to the squared numbers of boxed plane partitions.

The Five-Vertex Model and Enumerations of Plane Partitions

We consider the five-vertex model on an M ×2N lattice with fixed boundary conditions of special type. We discuss a determinantal formula for the partition function in application to description of various enumerations of N × N × (M − N) boxed plane partitions. It is shown that at the free-fermion point of the model, this formula reproduces the MacMahon formula for the number of boxed plane partitions, while for generic weights (outside the free-fermion point), it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horizontal) rows. Various representations for the partition function which describes such enumerations are obtained.

Lozenge tilings with free boundary

We study tilings with lozenges of a domain with free boundary conditions on one side. These correspondto boxed symmetric plane partitions. We show that the positions of the horizontal lozenges near the left flatboundary, in the limit, have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (theGUE-corners/minors process). We also prove the existence of a limit shape of the height function (the symmetricplane partition). We also consider domains where the sides converge to $\infty$ at different rates and recover again theGUE-corners process. Nous étudions les pavages par losanges d’un domaine dont le bord vertical est “libre”. Nous montrons queles positions des losanges horizontaux proches du bord gauche ont la même distribution que les valeurs propres del’ensemble gaussien unitaire. Nous montrons aussi l’existence d’une limite de la forme de la fonction de hauteur (unepartition plane symétrique). Nous considérons aussi des domaines ou des bords différents convergent vers $\infty$ destaux différents et nous retrouvons nouveau les processus EGU au bord.

A Combinatorial Interpretation of the Scalar Products of State Vectors of Integrable Models

The representation of the Bethe wave functions of certain integrable models via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The algebraic relations arising in the calculation of the scalar products and the correlation functions are based on the Binet-Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of the Bethe state-vectors in terms of nests of the self-avoiding lattice paths constituting the so-called watermelon configurations. The interpretation proposed is, in its turn, related to the enumeration of the boxed plane partitions.

Lozenge tiling constrained codes

While the field of one-dimensional constrained codes is mature, with theoretical as well as practical aspects of code- and decoder-design being well-established, such a theoretical treatment of its two-dimensional (2D) counterpart is still unavailable. Research has been conducted on a few exemplar 2D constraints, e.g., the hard triangle model, run-length limited constraints on the square lattice, and 2D checkerboard constraints. Excluding these results, 2D constrained systems remain largely uncharacterized mathematically, with only loose bounds of capacities present. In this paper we present a lozenge constraint on a regular triangular lattice and derive Shannon noiseless capacity bounds. To estimate capacity of lozenge tiling we make use of the bijection between the counting of lozenge tiling and the counting of boxed plane partitions.

Correlation functions of XX0 Heisenberg chain, q-binomial determinants, and random walks

The XX0 Heisenberg model on a cyclic chain is considered. The representation of the Bethe wave functions via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The determinantal expressions of the form-factors and of the thermal correlation functions are obtained. The q-binomial determinants enable the connection of the form-factors with the generating functions both of boxed plane partitions and of self-avoiding lattice paths. The asymptotical behavior of the thermal correlation functions is studied in the limit of low temperature provided that the characteristic parameters of the system are large enough.

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

A Gelfand–Tsetlin scheme of depth $$N$$ is a triangular array with $$m$$ integers at level $$m$$ , $$m=1,\ldots ,N$$ , subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand–Tsetlin schemes with arbitrary fixed $$N$$ th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its $$q$$ -deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).

Weighted enumerations of boxed plane partitions and the inhomogeneous five-vertex model

We consider the five-vertex model on a square lattice with fixed boundary conditions which corresponds to weighted (with weight q per elementary cube) enumerations of boxed plane partitions. We calculate the one-point correlation function of the model which describes the probability of a given state on an edge (polarization). This generalizes an analogous result obtained previously by the authors for unweighted (weighted with weight q = 1) enumerations of plane partitions. Bibliography: 14 titles.

Q-Distributions on boxed plane partitions

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon’s product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of these weights that are related to orthogonal polynomials; they form three 2-D families. For distributions from these families, we prove two types of results. First, we construct explicit Markov chains that preserve these distributions. In particular, this leads to a relatively simple exact sampling algorithm. Second, we consider a limit when all dimensions of the box grow and plane partitions become large and prove that the local correlations converge to those of ergodic translation invariant Gibbs measures. For fixed proportions of the box, the slopes of the limiting Gibbs measures (that can also be viewed as slopes of tangent planes to the hypothetical limit shape) are encoded by a single quadratic polynomial.

Random Sampling of Plane Partitions

This article presents uniform random generators of plane partitions according to size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity isO(n(lnn)3) in approximate-size sampling andO(n4/3) in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a×b)-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

The five-vertex model and boxed plane partitions

Boxed plane partitions are considered in terms of the five-vertex model on a finite lattice with fired boundary conditions. Assuming that all weights of the model have the same value, the one-point correlation function. describing the probability of having a given state on an arbitrary horizontal edge of the lattice is calculated. This is equivalent to the enumeration of boxed plane partitions that correspond to rhombus tilings of a hexagon with one fired rhombus of a particular type. The solution of the problem is given for the case of a boar of generic size. Bibliography: 24 titles.

Form factors, plane partitions, and random walks

An exactly solvable boson model, the so-called “phase model,” is considered. A relation between certain transition matrix elements of this model and boxed plane partitions, three-dimensional Young diagrams placed into a box of finite size, is established, It is shown that the natural model describing the behavior of friendly walkers, ones that can share the same lattice sites, is the “phase model.” An expression for the number of all admissible nests of lattice paths made by a fixed number of friendly walkers for a certain number of steps is obtained. Bibliography 35 titles.