1,820 research outputs found
Labelled tree graphs, Feynman diagrams and disk integrals
In this note, we introduce and study a new class of "half integrands" in
Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called
Parke-Taylor factors; these are dubbed Cayley functions as each of them
corresponds to a labelled tree graph. The CHY formula with a Cayley function
squared gives a sum of Feynman diagrams, and we represent it by a combinatoric
polytope whose vertices correspond to Feynman diagrams. We provide a simple
graphic rule to derive the polytope from a labelled tree graph, and classify
such polytopes ranging from the associahedron to the permutohedron.
Furthermore, we study the linear space of such half integrands and find (1) a
nice formula reducing any Cayley function to a sum of Parke-Taylor factors in
the Kleiss-Kuijf basis (2) a set of Cayley functions as a new basis of the
space; each element has the remarkable property that its CHY formula with a
given Parke-Taylor factor gives either a single Feynman diagram or zero. We
also briefly discuss applications of Cayley functions and the new basis in
certain disk integrals of superstring theory.Comment: 30+8 pages, many figures;typos fixe
The automorphism group of a graphon
We study the automorphism group of graphons (graph limits). We prove that
after an appropriate "standardization" of the graphon, the automorphism group
is compact. Furthermore, we characterize the orbits of the automorphism group
on -tuples of points. Among applications we study the graph algebras defined
by finite rank graphons and the space of node-transitive graphons.Comment: 29 pages, 2 figure
The diameter of random Cayley digraphs of given degree
We consider random Cayley digraphs of order with uniformly distributed
generating set of size . Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as and
. We find a sharp phase transition from 0 to 1 at around . In particular, if is asymptotically linear in , the
probability converges exponentially fast to 1.Comment: 11 page
Abelian Cayley digraphs with asymptotically large order for any given degree
Abelian Cayley digraphs can be constructed by using a generalization to Z(n) of the concept of congruence in Z. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known large dense results were all non-constructive.Peer ReviewedPostprint (author's final draft
Paths in quantum Cayley trees and L^2-cohomology
We study existence, uniqueness and triviality of path cocycles in the quantum
Cayley graph of universal discrete quantum groups. In the orthogonal case we
find that the unique path cocycle is trivial, in contrast with the case of free
groups where it is proper. In the unitary case it is neither bounded nor
proper. From this geometrical result we deduce the vanishing of the first
L^2-Betti number of A_o(I_n).Comment: 30 pages ; v2: major update with many improvements and new results
about the unitary case ; v3: accepted versio
- …