arXiv subjects

Mihai Ciucu

Publications and source records attributed to Mihai Ciucu.

At least 37 records · Page 2Linked to original sources

arXiv2014

Lozenge tilings of hexagons with arbitrary dents

Eisenk"olbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.

math.CO↗

arXiv2014

A triangular gap of size two in a sea of dimers on a $60^\circ$ angle

We consider a triangular gap of side two in a $60^\circ$ angle on the triangular lattice whose sides are zig-zag lines. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle. This provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work.

math-ph↗

arXiv2014

Macroscopically separated gaps in dimer coverings of Aztec rectangles

In this paper we determine the interaction of diagonal defect clusters in regions of an Aztec rectangle that scale to arbitrary points on its symmetry axis (in earlier work we treated the case when this point was the center of the scaled Aztec rectangle). We use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary. In several of the treated situations this seems not to be accomplishable by previous methods. Our applications include the case of two long neutral strings, which turn out to interact by an analog of the Casimir force, two families of neutral doublets that turn out to interact completely independently of one another, a neutral doublet and a very long neutral string, a general collection of macroscopically separated monomer and separation defects, and the case of long strings consisting of consecutive monomers.

math-ph↗

arXiv2014

A generalization of Kuo condensation

Kuo introduced his 4-point condensation in 2003 for bipartite planar graphs. In 2006 Kuo generalized this 4-point condensation to planar graphs that are not necessarily bipartite. His formula expressed the product between the number of perfect matching of the original graph $G$ and that of the subgraph obtained from $G$ by removing the four distinguished vertices as a Pfaffian of order 4, whose entries are numbers of perfect matchings of subgraphs of $G$ obtained by removing various pairs of vertices chosen from among the four distinguished ones. The compelling elegance of this formula is inviting of generalization. Kuo generalized it to $2k$ points under the special assumption that the subgraph obtained by removing some subset of the $2k$ vertices has precisely one perfect matching. In this paper we prove that the formula holds in the general case. We also present a couple of applications.

math.CO↗

arXiv2014

A triangular gap of size two in a sea of dimers in a $90^\circ$ angle with mixed boundary conditions, and a heat flow conjecture for the general case

We consider a triangular gap of side two in a $90^\circ$ angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. The image in the side with constrained boundary has the same orientation as the original gap, while the image in the side with free boundary has the opposite orientation. This, together with the parallel between the correlation of gaps in dimer packings and electrostatics we developed in previous work, provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we phrase as a conjecture. While the electrostatic interpretation is equivalent to a steady state heat flow interpretation in the bulk, it turns out that the latter view is more natural in the context of the interaction of the gaps with the boundary. The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).

math-ph↗

arXiv2014

Proof of Blum's conjecture on hexagonal dungeons

Matt Blum conjectured that the number of tilings of the Hexagonal Dungeon of sides $a,\ 2a,\ b,\ a,\ 2a,\ b$ (where $b\geq 2a$) is $13^{2a^2}14^{\lfloor\frac{a^2}{2}\rfloor}$ (J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present a proof for this conjecture using Kuo's Graphical Condensation Theorem (E. Kuo, Applications of Graphical Condensation for Enumerating Matchings and Tilings, Theoretical Computer Science, 2004).

math.CO↗

arXiv2013

Proof of two conjectures of Ciucu and Krattenthaler on the enumeration of lozenge tilings of hexagons with cut off corners

In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions from two non-adjacent corners. For the case when the staircases are removed from adjacent corners of the hexagon, they presented two conjectural formulas, whose proofs, as they remarked, seemed at the time "a formidable task". In this paper we prove those two conjectures. Our proofs proceed by first generalizing the conjectures, and then proving them by induction, using Kuo's graphical condensation method.

math.CO↗

arXiv2012

A dual of MacMahon's theorem on plane partitions

A classical theorem of MacMahon states that the number of lozenge tilings of any centrally symmetric hexagon drawn on the triangular lattice is given by a beautifully simple product formula. In this paper we present a counterpart of this formula, corresponding to the {\it exterior} of a concave hexagon obtained by turning 120 degrees after drawing each side (MacMahon's hexagon is obtained by turning 60 degrees after each step).

math.CO↗

arXiv2012

The interaction of collinear gaps of arbitrary charge in a two dimensional dimer system

The correlation of gaps in dimer systems was introduced in 1963 by Fisher and Stephenson, who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we considered the analogous problem on the hexagonal lattice, and we extended the set-up to include the correlation of any finite number of monomer clusters. For fairly general classes of monomer clusters we proved that the asymptotics of their correlation is given, for large separations between the clusters, by a multiplicative version of Coulomb's law for 2D electrostatics. However, our previous results required that the monomer clusters consist (with possibly one exception) of an even number of monomers. In this paper we determine the asymptotics of general defect clusters along a lattice diagonal in the square lattice (involving an arbitrary, even or odd number of monomers), and find that it is given by the same Coulomb law. We also obtain a conceptual interpretation for the multiplicative constant as the product of the correlations of the individual clusters.

math.CO↗

arXiv2009

The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4πr}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.

math.CO↗

arXiv2009

The emergence of the electrostatic field as a Feynman sum in random tilings with holes

We consider random lozenge tilings on the triangular lattice with holes $Q_1,...,Q_n$ in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge covering it. We show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole $Q_i$ as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside $Q_i$. This is then restated in terms of random surfaces, yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is, in the scaling limit, a sum of helicoids.

math-ph↗

arXiv2008

A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings

We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A similar factorization theorem is proven for the sum of two Schur functions of respective rectangular shapes $(M^n)$ and $(M^{n-1})$.

math.CO↗

arXiv2007

Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole

We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid $G_n$ of order $n$ is similar to the disjoint union of two copies of the quartered Aztec diamond $QAD_{n-1}$ of order $n-1$ with the path $P_n^{(2)}$ on $n$ vertices having edge weights equal to~2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. In particular, this allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described ``linear squarishness'' property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases--graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases.

math.CO↗

arXiv2007

The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions

We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of lattice-triangular holes of even sides satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of right-pointing and left-pointing unit triangles in each hole. We detail this parallel by indicating that, as a consequence of our result, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approaches, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. We give an equivalent phrasing of our result in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatics arises by averaging over all possible discrete geometries of the covering surfaces.

math-ph↗

arXiv2005

Perfect matchings and perfect powers

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers. In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version.

math.CO↗

arXiv2005

Rotational invariance of quadromer correlations on the hexagonal lattice

In 1963 Fisher and Stephenson \cite{FS} conjectured that the monomer-monomer correlation on the square lattice is rotationally invariant. In this paper we prove a closely related statement on the hexagonal lattice. Namely, we consider correlations of two quadromers (four-vertex subgraphs consisting of a monomer and its three neighbors) and show that they are rotationally invariant.

math.CO↗

arXiv2004

A random tiling model for two dimensional electrostatics

We consider triangular holes on the hexagonal lattice and we study their interaction when the rest of the lattice is covered by dimers. More precisely, we analyze the joint correlation of these triangular holes in a ``sea'' of dimers. We determine the asymptotics of the joint correlation (for large separations between the holes) in the case when one of the holes has side 1, all remaining holes have side 2, and the holes are distributed symmetrically with respect to a symmetry axis. Our result has a striking physical interpretation. If we regard the holes as electrical charges, with charge equal to the difference between the number of down-pointing and up-pointing unit triangles in a hole, the logarithm of the joint correlation behaves exactly like the electrostatic potential energy of this two-dimensional electrostatic system: it is obtained by a Superposition Principle from the interaction of all pairs, and the pair interactions are according to Coulomb's law.

math-ph↗

arXiv2004

Plane partitions I: a generalization of MacMahon's formula

The number of plane partitions contained in a given box was shown by MacMahon to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of side-lengths $a,b,c,a,b,c$ (in cyclic order) and angles of 120 degrees. We present a generalization in the case $b=c$ by giving simple product formulas enumerating lozenge tilings of regions obtained from a hexagon of side-lengths $a,b+k,b,a+k,b,b+k$ (where $k$ is an arbitrary non-negative integer) and angles of 120 degrees by removing certain triangular regions along its symmetry axis.

math.CO↗