5,190 research outputs found
From Free Fields to AdS -- II
We continue with the program of hep-th/0308184 to implement open-closed
string duality on free gauge field theory (in the large limit). In this
paper we consider correlators such as \la \prod_{i=1}^n
\Tr\Phi^{J_i}(x_i)\ra. The Schwinger parametrisation of this -point
function exhibits a partial gluing up into a set of basic skeleton graphs. We
argue that the moduli space of the planar skeleton graphs is exactly the same
as the moduli space of genus zero Riemann surfaces with holes. In other
words, we can explicitly rewrite the -point (planar) free field correlator
as an integral over the moduli space of a sphere with holes. A preliminary
study of the integrand also indicates compatibility with a string theory on
. The details of our argument are quite insensitive to the specific form
of the operators and generalise to diagrams of higher genus as well. We take
this as evidence of the field theory's ability to reorganise itself into a
string theory.Comment: 26 pages, 2 figures; v2. some additional comments, references adde
Convergent expansions for Random Cluster Model with q>0 on infinite graphs
In this paper we extend our previous results on the connectivity functions
and pressure of the Random Cluster Model in the highly subcritical phase and in
the highly supercritical phase, originally proved only on the cubic lattice
, to a much wider class of infinite graphs. In particular, concerning the
subcritical regime, we show that the connectivity functions are analytic and
decay exponentially in any bounded degree graph. In the supercritical phase, we
are able to prove the analyticity of finite connectivity functions in a smaller
class of graphs, namely, bounded degree graphs with the so called minimal
cut-set property and satisfying a (very mild) isoperimetric inequality. On the
other hand we show that the large distances decay of finite connectivity in the
supercritical regime can be polynomially slow depending on the topological
structure of the graph. Analogous analyticity results are obtained for the
pressure of the Random Cluster Model on an infinite graph, but with the further
assumptions of amenability and quasi-transitivity of the graph.Comment: In this new version the introduction has been revised, some
references have been added, and many typos have been corrected. 37 pages, to
appear in Communications on Pure and Applied Analysi
On some intriguing problems in Hamiltonian graph theory -- A survey
We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, -tough graphs, and claw-free graphs
Spatially embedded random networks
Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples
Subgraphs in random networks
Understanding the subgraph distribution in random networks is important for
modelling complex systems. In classic Erdos networks, which exhibit a
Poissonian degree distribution, the number of appearances of a subgraph G with
n nodes and g edges scales with network size as \mean{G} ~ N^{n-g}. However,
many natural networks have a non-Poissonian degree distribution. Here we
present approximate equations for the average number of subgraphs in an
ensemble of random sparse directed networks, characterized by an arbitrary
degree sequence. We find new scaling rules for the commonly occurring case of
directed scale-free networks, in which the outgoing degree distribution scales
as P(k) ~ k^{-\gamma}. Considering the power exponent of the degree
distribution, \gamma, as a control parameter, we show that random networks
exhibit transitions between three regimes. In each regime the subgraph number
of appearances follows a different scaling law, \mean{G} ~ N^{\alpha}, where
\alpha=n-g+s-1 for \gamma<2, \alpha=n-g+s+1-\gamma for 2<\gamma<\gamma_c, and
\alpha=n-g for \gamma>\gamma_c, s is the maximal outdegree in the subgraph, and
\gamma_c=s+1. We find that certain subgraphs appear much more frequently than
in Erdos networks. These results are in very good agreement with numerical
simulations. This has implications for detecting network motifs, subgraphs that
occur in natural networks significantly more than in their randomized
counterparts.Comment: 8 pages, 5 figure
Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II
Let be a connected graph with and with the spanning
forest . Let be a real number and let be a real function. In this paper, we show that if for all
, , then has a spanning tree
containing such that for each vertex , , where
denotes the number of components of and denotes the
number of edges of with both ends in . This is an improvement of several
results and the condition is best possible. Next, we also investigate an
extension for this result and deduce that every -edge-connected graph
has a spanning subgraph containing edge-disjoint spanning trees such
that for each vertex , , where ; also if contains
edge-disjoint spanning trees, then can be found such that for each vertex
, , where .
Finally, we show that strongly -tough graphs, including -tough
graphs of order at least three, have spanning Eulerian subgraphs whose degrees
lie in the set . In addition, we show that every -tough graph has
spanning closed walk meeting each vertex at most times and prove a
long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed
walk; connected factor; toughness; total exces
The random K-satisfiability problem: from an analytic solution to an efficient algorithm
We study the problem of satisfiability of randomly chosen clauses, each with
K Boolean variables. Using the cavity method at zero temperature, we find the
phase diagram for the K=3 case. We show the existence of an intermediate phase
in the satisfiable region, where the proliferation of metastable states is at
the origin of the slowdown of search algorithms. The fundamental order
parameter introduced in the cavity method, which consists of surveys of local
magnetic fields in the various possible states of the system, can be computed
for one given sample. These surveys can be used to invent new types of
algorithms for solving hard combinatorial optimizations problems. One such
algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo
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