36,123 research outputs found
Duality and fluctuation relations for statistics of currents on cyclic graphs
We consider stochastic motion of a particle on a cyclic graph with
arbitrarily periodic time dependent kinetic rates. We demonstrate duality
relations for statistics of currents in this model and in its continuous
version of a diffusion in one dimension. Our duality relations are valid beyond
detailed balance constraints and lead to exact expressions that relate
statistics of currents induced by dual driving protocols. We also show that
previously known no-pumping theorems and some of the fluctuation relations,
when they are applied to cyclic graphs or to one dimensional diffusion, are
special consequences of our duality.Comment: 2 figure, 6 pages (In twocolumn). Accepted by JSTA
On the refined counting of graphs on surfaces
Ribbon graphs embedded on a Riemann surface provide a useful way to describe
the double line Feynman diagrams of large N computations and a variety of other
QFT correlator and scattering amplitude calculations, e.g in MHV rules for
scattering amplitudes, as well as in ordinary QED. Their counting is a special
case of the counting of bi-partite embedded graphs. We review and extend
relevant mathematical literature and present results on the counting of some
infinite classes of bi-partite graphs. Permutation groups and representations
as well as double cosets and quotients of graphs are useful mathematical tools.
The counting results are refined according to data of physical relevance, such
as the structure of the vertices, faces and genus of the embedded graph. These
counting problems can be expressed in terms of observables in three-dimensional
topological field theory with S_d gauge group which gives them a topological
membrane interpretation.Comment: 57 pages, 12 figures; v2: Typos corrected; references adde
Enumerating Subgraph Instances Using Map-Reduce
The theme of this paper is how to find all instances of a given "sample"
graph in a larger "data graph," using a single round of map-reduce. For the
simplest sample graph, the triangle, we improve upon the best known such
algorithm. We then examine the general case, considering both the communication
cost between mappers and reducers and the total computation cost at the
reducers. To minimize communication cost, we exploit the techniques of (Afrati
and Ullman, TKDE 2011)for computing multiway joins (evaluating conjunctive
queries) in a single map-reduce round. Several methods are shown for
translating sample graphs into a union of conjunctive queries with as few
queries as possible. We also address the matter of optimizing computation cost.
Many serial algorithms are shown to be "convertible," in the sense that it is
possible to partition the data graph, explore each partition in a separate
reducer, and have the total computation cost at the reducers be of the same
order as the computation cost of the serial algorithm.Comment: 37 page
On Perturbative Gravity and Gauge Theory
We review some applications of tree-level (classical) relations between
gravity and gauge theory that follow from string theory. Together with
-dimensional unitarity, these relations can be used to perturbatively
quantize gravity theories, i.e. they contain the necessary information for
obtaining loop contributions. We also review recent applications of these ideas
showing that N=1 D=11 supergravity diverges, and review arguments that N=8 D=4
supergravity is less divergent than previously thought, though it does appear
to diverge at five loops. Finally, we describe field variables for the
Einstein-Hilbert Lagrangian that help clarify the perturbative relationship
between gravity and gauge theory.Comment: Talk presented at Third Meeting on Constrained Dynamics and Quantum
Gravity, Villasimius (Sardinia, Italy) September 13-17, 1999 and at the
Workshop on Light-Cone QCD and Nonperturbative Hadron Physics, University of
Adelaide (Australia) December 13-22, 1999. Latex, 9 page
Monodromy--like Relations for Finite Loop Amplitudes
We investigate the existence of relations for finite one-loop amplitudes in
Yang-Mills theory. Using a diagrammatic formalism and a remarkable connection
between tree and loop level, we deduce sequences of amplitude relations for any
number of external legs.Comment: 24 pages, 6 figures, v2 typos corrected, reference adde
On the Relationship between Yang-Mills Theory and Gravity and its Implication for Ultraviolet Divergences
String theory implies that field theories containing gravity are in a certain
sense `products' of gauge theories. We make this product structure explicit up
to two loops for the relatively simple case of N=8 supergravity four-point
amplitudes, demonstrating that they are `squares' of N=4 super-Yang-Mills
amplitudes. This is accomplished by obtaining an explicit expression for the
-dimensional two-loop contribution to the four-particle S-matrix for N=8
supergravity, which we compare to the corresponding N=4 Yang-Mills result. From
these expressions we also obtain the two-loop ultraviolet divergences in
dimensions D=7 through D=11. The analysis relies on the unitarity cuts of the
two theories, many of which can be recycled from a one-loop computation. The
two-particle cuts, which may be iterated to all loop orders, suggest that
squaring relations between the two theories exist at any loop order. The
loop-momentum power-counting implied by our two-particle cut analysis indicates
that in four dimensions the first four-point divergence in N=8 supergravity
should appear at five loops, contrary to the earlier expectation, based on
superspace arguments, of a three-loop counterterm.Comment: Latex, 52 pages, discussion of 2 loop divergences in D=8,10 adde
MHV, CSW and BCFW: field theory structures in string theory amplitudes
Motivated by recent progress in calculating field theory amplitudes, we study
applications of the basic ideas in these developments to the calculation of
amplitudes in string theory. We consider in particular both non-Abelian and
Abelian open superstring disk amplitudes in a flat space background, focusing
mainly on the four-dimensional case. The basic field theory ideas under
consideration split into three separate categories. In the first, we argue that
the calculation of alpha'-corrections to MHV open string disk amplitudes
reduces to the determination of certain classes of polynomials. This line of
reasoning is then used to determine the alpha'^3-correction to the MHV
amplitude for all multiplicities. A second line of attack concerns the
existence of an analog of CSW rules derived from the Abelian Dirac-Born-Infeld
action in four dimensions. We show explicitly that the CSW-like perturbation
series of this action is surprisingly trivial: only helicity conserving
amplitudes are non-zero. Last but not least, we initiate the study of BCFW
on-shell recursion relations in string theory. These should appear very
naturally as the UV properties of the string theory are excellent. We show that
all open four-point string amplitudes in a flat background at the disk level
obey BCFW recursion relations. Based on the naturalness of the proof and some
explicit results for the five-point gluon amplitude, it is expected that this
pattern persists for all higher point amplitudes and for the closed string.Comment: v3: corrected erroneous statement about Virasoro-Shapiro amplitude
and added referenc
Toric ideals of normalized graph algebras
A graph-theoretic method, simpler than existing ones, is used to characterize
the minimal set of monomial generators for the integral closure of any algebra
of polynomials generated by quadratic monomials. The toric ideal of relations
between these generators is generated by a set of binomials, defined
graphically. The spectra of the original algebra and of its integral closure
turn out to be canonically homeomorphic.Comment: 9 pages, no figures. v2: major rewrite. v3: minor improvements. v4:
new title, old reference located, other small change
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