2,470 research outputs found
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
A Note on Dimer Models and D-brane Gauge Theories
The connection between quiver gauge theories and dimer models has been well
studied. It is known that the matter fields of the quiver gauge theories can be
represented using the perfect matchings of the corresponding dimer model.We
conjecture that a subset of perfect matchings associated with an internal point
in the toric diagram is sufficient to give information about the charge matrix
of the quiver gauge theory. Further, we perform explicit computations on some
aspects of partial resolutions of toric singularities using dimer models. We
analyse these with graph theory techniques, using the perfect matchings of
orbifolds of the form \BC^3/\Gamma, where the orbifolding group may
be noncyclic. Using these, we study the construction of the superpotential of
gauge theories living on D-branes which probe these singularities, including
the case where one or more adjoint fields are present upon partial resolution.
Applying a combination of open and closed string techniques to dimer models, we
also study some aspects of their symmetries.Comment: Discussions expanded, clarifications added, typos fixed. 1+49 page
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Graded Quivers, Generalized Dimer Models and Toric Geometry
The open string sector of the topological B-model model on CY -folds
is described by -graded quivers with superpotentials. This correspondence
extends to general the well known connection between CY -folds and
gauge theories on the worldvolume of D-branes for . We
introduce -dimers, which fully encode the -graded quivers and their
superpotentials, in the case in which the CY -folds are toric.
Generalizing the well known cases, -dimers significantly simplify
the connection between geometry and -graded quivers. A key result of this
paper is the generalization of the concept of perfect matching, which plays a
central role in this map, to arbitrary . We also introduce a simplified
algorithm for the computation of perfect matchings, which generalizes the
Kasteleyn matrix approach to any . We illustrate these new tools with a few
infinite families of CY singularities.Comment: 54 pages, 6 figure
Stable Marriage with Multi-Modal Preferences
We introduce a generalized version of the famous Stable Marriage problem, now
based on multi-modal preference lists. The central twist herein is to allow
each agent to rank its potentially matching counterparts based on more than one
"evaluation mode" (e.g., more than one criterion); thus, each agent is equipped
with multiple preference lists, each ranking the counterparts in a possibly
different way. We introduce and study three natural concepts of stability,
investigate their mutual relations and focus on computational complexity
aspects with respect to computing stable matchings in these new scenarios.
Mostly encountering computational hardness (NP-hardness), we can also spot few
islands of tractability and make a surprising connection to the \textsc{Graph
Isomorphism} problem
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