32,884 research outputs found
The component sizes of a critical random graph with given degree sequence
Consider a critical random multigraph with vertices
constructed by the configuration model such that its vertex degrees are
independent random variables with the same distribution (criticality
means that the second moment of is finite and equals twice its first
moment). We specify the scaling limits of the ordered sequence of component
sizes of as tends to infinity in different cases. When
has finite third moment, the components sizes rescaled by
converge to the excursion lengths of a Brownian motion with parabolic drift
above past minima, whereas when is a power law distribution with exponent
, the components sizes rescaled by converge to the excursion lengths of a certain nontrivial
drifted process with independent increments above past minima. We deduce the
asymptotic behavior of the component sizes of a critical random simple graph
when has finite third moment.Comment: Published in at http://dx.doi.org/10.1214/13-AAP985 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Average Characteristic Polynomials of Determinantal Point Processes
We investigate the average characteristic polynomial where the 's are real random variables
which form a determinantal point process associated to a bounded projection
operator. For a subclass of point processes, which contains Orthogonal
Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a
sufficient condition for its limiting zero distribution to match with the
limiting distribution of the random variables, almost surely, as goes to
infinity. Moreover, such a condition turns out to be sufficient to strengthen
the mean convergence to the almost sure one for the moments of the empirical
measure associated to the determinantal point process, a fact of independent
interest. As an application, we obtain from a theorem of Kuijlaars and Van
Assche a unified way to describe the almost sure convergence for classical
Orthogonal Polynomial Ensembles. As another application, we obtain from
Voiculescu's theorems the limiting zero distribution for multiple Hermite and
multiple Laguerre polynomials, expressed in terms of free convolutions of
classical distributions with atomic measures.Comment: 26 page
Complements of hyperplane sub-bundles in projective space bundles over the projective line
We establish that the isomorphy type as an abstract algebraic variety of the
complement of an ample hyperplane sub-bundle H of a projective space bundle of
rank r-1 over the projective line depends only on the the r-fold
self-intersection of H . In particular it depends neither on the ambient bundle
nor on a particular ample hyperplane sub-bundle with given r-fold
self-intersection. Our proof exploits the unexpected property that every such
complement comes equipped with the structure of a non trivial torsor under a
vector bundle on the affine line with a double origin
Semisimple Lie groups satisfy property RD, a short proof
We give a short elementary proof of the fact that connected semisimple real
Lie groups satisfy property RD. The proof is based on a process of
linearization
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