13,010 research outputs found
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 nested arches is polynomial in if is large enough,
thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11
(2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio
Loops, matchings and alternating-sign matrices
The appearance of numbers enumerating alternating sign matrices in stationary
states of certain stochastic processes is reviewed. New conjectures concerning
nest distribution functions are presented as well as a bijection between
certain classes of alternating sign matrices and lozenge tilings of hexagons
with cut off corners.Comment: LaTeX, 26 pages, 44 figures, extended version of a talk given at the
14th International Conference on Formal Power Series and Algebraic
Combinatorics (Melbourne 2002); Version2: Changed title, expanded some
sections and included more picture
On some polynomials enumerating Fully Packed Loop configurations
We are interested in the enumeration of Fully Packed Loop configurations on a
grid with a given noncrossing matching. By the recently proved
Razumov--Stroganov conjecture, these quantities also appear as groundstate
components in the Completely Packed Loop model. When considering matchings with
p nested arches, these numbers are known to be polynomials in p. In this
article, we present several conjectures about these polynomials: in particular,
we describe all real roots, certain values of these polynomials, and conjecture
that the coefficients are positive. The conjectures, which are of a
combinatorial nature, are supported by strong numerical evidence and the proofs
of several special cases. We also give a version of the conjectures when an
extra parameter tau is added to the equations defining the groundstate of the
Completely Packed Loop model.Comment: 27 pages. Modifications reflecting the recent proof of the
Razumov--Stroganov conjecture; also added some references and a more detailed
conclusio
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
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
A Bijection between classes of Fully Packed Loops and Plane Partitions
It has recently been observed empirically that the number of FPL
configurations with 3 sets of a, b and c nested arches equals the number of
plane partitions in a box of size a x b x c. In this note, this result is
proved by constructing explicitly the bijection between these FPL and plane
partitions
Configurational statistics of densely and fully packed loops in the negative-weight percolation model
By means of numerical simulations we investigate the configurational
properties of densely and fully packed configurations of loops in the
negative-weight percolation (NWP) model. In the presented study we consider 2d
square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where
edge weights are drawn from a Gaussian distribution. For a given realization of
the disorder we then compute a configuration of loops, such that the
configurational energy, given by the sum of all individual loop weights, is
minimized. For this purpose, we employ a mapping of the NWP model to the
"minimum-weight perfect matching problem" that can be solved exactly by using
sophisticated polynomial-time matching algorithms. We characterize the loops
via observables similar to those used in percolation studies and perform
finite-size scaling analyses, up to side length L=256 in 2d, L=48 in 3d and
L=20 in 4d (for which we study only some observables), in order to estimate
geometric exponents that characterize the configurations of densely and fully
packed loops. One major result is that the loops behave like uncorrelated
random walks from dimension d=3 on, in contrast to the previously studied
behavior at the percolation threshold, where random-walk behavior is obtained
for d>=6.Comment: 11 pages, 7 figure
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