5,210 research outputs found
Gaussian processes, kinematic formulae and Poincar\'e's limit
We consider vector valued, unit variance Gaussian processes defined over
stratified manifolds and the geometry of their excursion sets. In particular,
we develop an explicit formula for the expectation of all the
Lipschitz--Killing curvatures of these sets. Whereas our motivation is
primarily probabilistic, with statistical applications in the background, this
formula has also an interpretation as a version of the classic kinematic
fundamental formula of integral geometry. All of these aspects are developed in
the paper. Particularly novel is the method of proof, which is based on a an
approximation to the canonical Gaussian process on the -sphere. The
limit, which gives the final result, is handled via recent
extensions of the classic Poincar\'e limit theorem.Comment: Published in at http://dx.doi.org/10.1214/08-AOP439 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Expected volume and Euler characteristic of random submanifolds
In a closed manifold of positive dimension , we estimate the expected
volume and Euler characteristic for random submanifolds of codimension in two different settings. On one hand, we consider a closed
Riemannian manifold and some positive . Then we take independent
random functions in the direct sum of the eigenspaces of the Laplace-Beltrami
operator associated to eigenvalues less than and consider the random
submanifold defined as the common zero set of these functions. We compute
asymptotics for the mean volume and Euler characteristic of this random
submanifold as goes to infinity. On the other hand, we consider a
complex projective manifold defined over the reals, equipped with an ample line
bundle and a rank holomorphic vector bundle
that are also defined over the reals. Then we get asymptotics for the expected
volume and Euler characteristic of the real vanishing locus of a random real
holomorphic section of as goes to
infinity. The same techniques apply to both settings.Comment: Final version, accepted for publication in J. Funct. Anal., 50
pages.A change in notational convention impacts the statement of the main
theorems and most formula
Counting arcs in negative curvature
Let M be a complete Riemannian manifold with negative curvature, and let C_-,
C_+ be two properly immersed closed convex subsets of M. We survey the
asymptotic behaviour of the number of common perpendiculars of length at most s
from C_- to C_+, giving error terms and counting with weights, starting from
the work of Huber, Herrmann, Margulis and ending with the works of the authors.
We describe the relationship with counting problems in circle packings of
Kontorovich, Oh, Shah. We survey the tools used to obtain the precise
asymptotics (Bowen-Margulis and Gibbs measures, skinning measures). We describe
several arithmetic applications, in particular the ones by the authors on the
asymptotics of the number of representations of integers by binary quadratic,
Hermitian or Hamiltonian forms.Comment: Revised version, 44 page
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