2,875 research outputs found
The spatial structure of networks
We study networks that connect points in geographic space, such as
transportation networks and the Internet. We find that there are strong
signatures in these networks of topography and use patterns, giving the
networks shapes that are quite distinct from one another and from
non-geographic networks. We offer an explanation of these differences in terms
of the costs and benefits of transportation and communication, and give a
simple model based on the Monte Carlo optimization of these costs and benefits
that reproduces well the qualitative features of the networks studied.Comment: 5 pages, 3 figure
Planar Drawings of Fixed-Mobile Bigraphs
A fixed-mobile bigraph G is a bipartite graph such that the vertices of one
partition set are given with fixed positions in the plane and the mobile
vertices of the other part, together with the edges, must be added to the
drawing. We assume that G is planar and study the problem of finding, for a
given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In
the most general case, we show NP-hardness. For k=0 and under additional
constraints on the positions of the fixed or mobile vertices, we either prove
that the problem is polynomial-time solvable or prove that it belongs to NP.
Finally, we present a polynomial-time testing algorithm for a certain type of
"layered" 1-bend drawings
Pfaffian Correlation Functions of Planar Dimer Covers
The Pfaffian structure of the boundary monomer correlation functions in the
dimer-covering planar graph models is rederived through a combinatorial /
topological argument. These functions are then extended into a larger family of
order-disorder correlation functions which are shown to exhibit Pfaffian
structure throughout the bulk. Key tools involve combinatorial switching
symmetries which are identified through the loop-gas representation of the
double dimer model, and topological implications of planarity.Comment: Revised figures; corrected misprint
On emergence in gauge theories at the 't Hooft limit
The aim of this paper is to contribute to a better conceptual understanding
of gauge quantum field theories, such as quantum chromodynamics, by discussing
a famous physical limit, the 't Hooft limit, in which the theory concerned
often simplifies.
The idea of the limit is that the number of colours (or charges) goes to
infinity. The simplifications that can happen in this limit, and that we will
consider, are: (i) the theory's Feynman diagrams can be drawn on a plane
without lines intersecting (called `planarity'); and (ii) the theory, or a
sector of it, becomes integrable, and indeed corresponds to a well-studied
system, viz. a spin chain. Planarity is important because it shows how a
quantum field theory can exhibit extended, in particular string-like,
structures; in some cases, this gives a connection with string theory, and thus
with gravity.
Previous philosophical literature about how one theory (or a sector, or
regime, of a theory) might be emergent from, and-or reduced to, another one has
tended to emphasize cases, such as occur in statistical mechanics, where the
system before the limit has finitely many degrees of freedom. But here, our
quantum field theories, including those on the way to the 't Hooft limit, will
have infinitely many degrees of freedom.
Nevertheless, we will show how a recent schema by Butterfield and taxonomy by
Norton apply to the quantum field theories we consider; and we will classify
three physical properties of our theories in these terms. These properties are
planarity and integrability, as in (i) and (ii) above; and the behaviour of the
beta-function reflecting, for example, asymptotic freedom.
Our discussion of these properties, especially the beta-function, will also
relate to recent philosophical debate about the propriety of assessing quantum
field theories, whose rigorous existence is not yet proven.Comment: 44 pp. arXiv admin note: text overlap with arXiv:1012.3983,
arXiv:hep-ph/9802419, arXiv:1012.3997 by other author
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