3,082 research outputs found
Many 2-level polytopes from matroids
The family of 2-level matroids, that is, matroids whose base polytope is
2-level, has been recently studied and characterized by means of combinatorial
properties. 2-level matroids generalize series-parallel graphs, which have been
already successfully analyzed from the enumerative perspective.
We bring to light some structural properties of 2-level matroids and exploit
them for enumerative purposes. Moreover, the counting results are used to show
that the number of combinatorially non-equivalent (n-1)-dimensional 2-level
polytopes is bounded from below by , where
and .Comment: revised version, 19 pages, 7 figure
Revlex-Initial 0/1-Polytopes
We introduce revlex-initial 0/1-polytopes as the convex hulls of
reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are
special knapsack-polytopes. It turns out that they have remarkable extremal
properties. In particular, we use these polytopes in order to prove that the
minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the
graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and
a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their
graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and
Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at
least one.Comment: Accepted for publication in J. Comb. Theory Ser. A; 24 pages;
simplified proof of Theorem 1; corrected and improved version of Theorem 4
(the average degree is now bounded by d+4 instead of d+8); several minor
corrections suggested by the referee
Group field theories for all loop quantum gravity
Group field theories represent a 2nd quantized reformulation of the loop
quantum gravity state space and a completion of the spin foam formalism. States
of the canonical theory, in the traditional continuum setting, have support on
graphs of arbitrary valence. On the other hand, group field theories have
usually been defined in a simplicial context, thus dealing with a restricted
set of graphs. In this paper, we generalize the combinatorics of group field
theories to cover all the loop quantum gravity state space. As an explicit
example, we describe the GFT formulation of the KKL spin foam model, as well as
a particular modified version. We show that the use of tensor model tools
allows for the most effective construction. In order to clarify the
mathematical basis of our construction and of the formalisms with which we
deal, we also give an exhaustive description of the combinatorial structures
entering spin foam models and group field theories, both at the level of the
boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic
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