32,927 research outputs found
Distribution of levels in high-dimensional random landscapes
We prove empirical central limit theorems for the distribution of levels of
various random fields defined on high-dimensional discrete structures as the
dimension of the structure goes to . The random fields considered
include costs of assignments, weights of Hamiltonian cycles and spanning trees,
energies of directed polymers, locations of particles in the branching random
walk, as well as energies in the Sherrington--Kirkpatrick and Edwards--Anderson
models. The distribution of levels in all models listed above is shown to be
essentially the same as in a stationary Gaussian process with regularly varying
nonsummable covariance function. This type of behavior is different from the
Brownian bridge-type limit known for independent or stationary weakly dependent
sequences of random variables.Comment: Published in at http://dx.doi.org/10.1214/11-AAP772 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Globally and Locally Minimal Weight Spanning Tree Networks
The competition between local and global driving forces is significant in a
wide variety of naturally occurring branched networks. We have investigated the
impact of a global minimization criterion versus a local one on the structure
of spanning trees. To do so, we consider two spanning tree structures - the
generalized minimal spanning tree (GMST) defined by Dror et al. [1] and an
analogous structure based on the invasion percolation network, which we term
the generalized invasive spanning tree or GIST. In general, these two
structures represent extremes of global and local optimality, respectively.
Structural characteristics are compared between the GMST and GIST for a fixed
lattice. In addition, we demonstrate a method for creating a series of
structures which enable one to span the range between these two extremes. Two
structural characterizations, the occupied edge density (i.e., the fraction of
edges in the graph that are included in the tree) and the tortuosity of the
arcs in the trees, are shown to correlate well with the degree to which an
intermediate structure resembles the GMST or GIST. Both characterizations are
straightforward to determine from an image and are potentially useful tools in
the analysis of the formation of network structures.Comment: 23 pages, 5 figures, 2 tables, typographical error correcte
Factorization of Spanning Trees on Feynman Graphs
In order to use the Gaussian representation for propagators in Feynman
amplitudes, a representation which is useful to relate string theory and field
theory, one has to prove first that each - parameter (where is
the parameter associated to each propagator in the -representation of
the Feynman amplitudes) can be replaced by a constant instead of being
integrated over and second, prove that this constant can be taken equal for all
propagators of a given graph. The first proposition has been proven in one
recent letter when the number of propagators is infinite. Here we prove the
second one. In order to achieve this, we demonstrate that the sum over the
weighted spanning trees of a Feynman graph can be factorized for disjoint
parts of . The same can also be done for cuts on , resulting in a
rigorous derivation of the Gaussian representation for super-renormalizable
scalar field theories. As a by-product spanning trees on Feynman graphs can be
used to define a discretized functional space.Comment: 47 pages, Plain Tex, 3 PostScript figure
How to Resum Feynman Graphs
In this paper we reformulate in a simpler way the combinatoric core of
constructive quantum field theory We define universal rational combinatoric
weights for pairs made of a graph and one of its spanning trees. These weights
are nothing but the percentage of Hepp's sectors in which the tree is leading
the ultraviolet analysis. We explain how they allow to reshuffle the divergent
series formulated in terms of Feynman graphs into convergent series indexed by
the trees that these graphs contain. The Feynman graphs to be used are not the
ordinary ones but those of the intermediate field representation, and the
result of the reshuffling is called the Loop Vertex Expansion.Comment: 18 pages, 6 figures; minor revisions; improves and replaces
arXiv:1006.461
Setting Parameters by Example
We introduce a class of "inverse parametric optimization" problems, in which
one is given both a parametric optimization problem and a desired optimal
solution; the task is to determine parameter values that lead to the given
solution. We describe algorithms for solving such problems for minimum spanning
trees, shortest paths, and other "optimal subgraph" problems, and discuss
applications in multicast routing, vehicle path planning, resource allocation,
and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations
of Computer Science (FOCS '99
- …