163 research outputs found
Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
We analyze combinatorial structures which play a central role in determining
spectral properties of the volume operator in loop quantum gravity (LQG). These
structures encode geometrical information of the embedding of arbitrary valence
vertices of a graph in 3-dimensional Riemannian space, and can be represented
by sign strings containing relative orientations of embedded edges. We
demonstrate that these signature factors are a special representation of the
general mathematical concept of an oriented matroid. Moreover, we show that
oriented matroids can also be used to describe the topology (connectedness) of
directed graphs. Hence the mathematical methods developed for oriented matroids
can be applied to the difficult combinatorics of embedded graphs underlying the
construction of LQG. As a first application we revisit the analysis of [4-5],
and find that enumeration of all possible sign configurations used there is
equivalent to enumerating all realizable oriented matroids of rank 3, and thus
can be greatly simplified. We find that for 7-valent vertices having no
coplanar triples of edge tangents, the smallest non-zero eigenvalue of the
volume spectrum does not grow as one increases the maximum spin \jmax at the
vertex, for any orientation of the edge tangents. This indicates that, in
contrast to the area operator, considering large \jmax does not necessarily
imply large volume eigenvalues. In addition we give an outlook to possible
starting points for rewriting the combinatorics of LQG in terms of oriented
matroids.Comment: 43 pages, 26 figures, LaTeX. Version published in CQG. Typos
corrected, presentation slightly extende
General duality for abelian-group-valued statistical-mechanics models
We introduce a general class of statistical-mechanics models, taking values
in an abelian group, which includes examples of both spin and gauge models,
both ordered and disordered. The model is described by a set of ``variables''
and a set of ``interactions''. A Gibbs factor is associated to each variable
and to each interaction. We introduce a duality transformation for systems in
this class. The duality exchanges the abelian group with its dual, the Gibbs
factors with their Fourier transforms, and the interactions with the variables.
High (low) couplings in the interaction terms are mapped into low (high)
couplings in the one-body terms. The idea is that our class of systems extends
the one for which the classical procedure 'a la Kramers and Wannier holds, up
to include randomness into the pattern of interaction. We introduce and study
some physical examples: a random Gaussian Model, a random Potts-like model, and
a random variant of discrete scalar QED. We shortly describe the consequence of
duality for each example.Comment: 26 pages, 2 Postscript figure
Hopf algebras for matroids over hyperfields
Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
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