Some combinatorics of rhomboid-shaped fully packed loop configurations

Abstract. The study of rhomboid-shaped fully packed loop configurations (RFPLs) is inspired by the work of Fischer and Nadeau on triangular fully packed loop configurations (TFPLs). By using the same techniques as they did some nice combinatorics for RFPLs arise. To each RFPL and to each oriented RFPL a quadruple of binary words (α, β; γ, δ) – its so-called boundary – is assigned. There are necessary conditions for the boundary of an RFPL respectively an oriented RFPL. For instance, it has to fulfill the inequality d(γ) + d(δ) ≥ d(α) + d(β) + |α|0|β|1, where |α|i denotes the number of occurrences of i = 0, 1 in α and d(α) denotes the number of inversions of α. Furthermore, the number of ordinary RFPLs with boundary (α, β; γ, δ) can be expressed in terms of oriented RFPLs with the same boundary. Finally, oriented RFPLs with boundary (α, β; γ, δ) such that d(γ) + d(δ) = d(α) + d(β) + |α|0|β|1 are considered. They are in bijection with rhomboid-shaped Knutson-Tao puzzles. Also, Littlewood-Richardson tableaux of defect d are defined. They can be understood as a generalization of Littlewood-Richardson tableaux. Those tableaux are in bijection with rhomboid-shaped Knutson-Tao puzzles. Résumé. L’étude des configurations de boucles compactes dans un rhomboïde (”rhomboid-shaped fully packed loop configurations”, RFPLs) est inspirée des travaux de Fischer et Nadeau sur les configurations de boucles compactes dans un triangle (TFPLs). En utilisant les mêmes techniques, des résultats combinatoires sont obtenus pour les RPFLs. À chaque RPFL et à chaque RPFL orienté nous associons un quadruplet de mots binaires (α, β; γ, δ)

Fully Packed Loops in a triangle: matchings, paths and puzzles

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. Firstly, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Secondly, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d(u)+d(v)<=d(w) where u,v,w are 01-words that encode the boundary conditions of ordinary TFPLs and d(u) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d(w)- d(u)-d(v)=0,1; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.Comment: 40 pages, 31 figure

Hamiltonian Cycles on Random Eulerian Triangulations

A random Eulerian triangulation is a random triangulation where an even number of triangles meet at any given vertex. We argue that the central charge increases by one if the fully packed O(n) model is defined on a random Eulerian triangulation instead of an ordinary random triangulation. Considering the case n -> 0, this implies that the system of random Eulerian triangulations equipped with Hamiltonian cycles describes a c=-1 matter field coupled to 2D quantum gravity as opposed to the system of usual random triangulations equipped with Hamiltonian cycles which has c=-2. Hence, in this case one should see a change in the entropy exponent from the value gamma=-1 to the irrational value gamma=(-1-\sqrt{13})/6=-0.76759... when going from a usual random triangulation to an Eulerian one. A direct enumeration of configurations confirms this change in gamma.Comment: 22 pages, 9 figures, references and a comment adde

Proof of two conjectures of Zuber on fully packed loop configurations

Two conjectures of Zuber [``On the counting of fully packed loops configurations. Some new conjectures,'' preprint] on the enumeration of configurations in the fully packed loop model on the square grid with periodic boundary conditions, which have a prescribed linkage pattern, are proved. Following an idea of de Gier [``Loops, matchings and alternating-sign matrices,'' Discrete Math., to appear], the proofs are based on bijections between such fully packed loop configurations and rhombus tilings, and the hook-content formula for semistandard tableaux.Comment: 20 pages; AmS-LaTe

Fully Packed O(n=1) Model on Random Eulerian Triangulations

We introduce a matrix model describing the fully-packed O(n) model on random Eulerian triangulations (i.e. triangulations with all vertices of even valency). For n=1 the model is mapped onto a particular gravitational 6-vertex model with central charge c=1, hence displaying the expected shift c -> c+1 when going from ordinary random triangulations to Eulerian ones. The case of arbitrary n is also discussed.Comment: 12 pages, 3 figures, tex, harvmac, eps

Operator Spectrum and Exact Exponents of the Fully Packed Loop Model

We develop a Coulomb gas description of the critical fluctuations in the fully packed loop model on the honeycomb lattice. We identify the complete operator spectrum of this model in terms of electric and magnetic {\em vector}-charges, and we calculate the scaling dimensions of these operators exactly. We also study the geometrical properties of loops in this model, and we derive exact results for the fractal dimension and the loop size distribution function. A review of the many different representations of this model that have recently appeared in the literature, is given.Comment: 17 pages latex, 3 postscript figures, IOP style files include

On the number of fully packed loop configurations with a fixed associated matching

We show that the number of fully packed loop configurations corresponding to a matching with $m$ nested arches is polynomial in $m$ if $m$ is large enough, thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11 (2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio

Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n^2 Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully-packed, we analyze in details the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q \neq 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.Comment: 44 pages, 13 figure