149 research outputs found
Lower estimates for the expected Betti numbers of random real hypersurfaces
We estimate from below the expected Betti numbers of real hypersurfaces taken
at random in a smooth real projective n-dimensional manifold. These random
hypersurfaces are chosen in the linear system of a large d-th power of a real
ample line bundle equipped with a Hermitian metric of positive curvature. As
for the upper bounds that we recently established, these lower bounds read as a
product of a constant which only depends on the dimension n of the manifold
with the K\"ahlerian volume of its real locus RX and d^{n/2}. Actually, any
closed affine real algebraic hypersurface appears with positive probability as
part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).Comment: 19 page
Topology of random real hypersurfaces
These are notes of the mini-course I gave during the CIMPA summer school at
Villa de Leyva, Colombia, in July . The subject was my joint work with
Damien Gayet on the topology of random real hypersurfaces, restricting myself
to the case of projective spaces and focusing on our lower estimates. Namely,
we estimate from (above and) below the mathematical expectation of all Betti
numbers of degree random real projective hypersurfaces. For any closed
connected hypersurface of , we actually estimate from
below the mathematical expectation of the number of connected components of
these degree random real projective hypersurfaces which are diffeomorphic
to .Comment: 18 pages, notes of the course I gave during the CIMPA summer school
at Villa de Leyva, Colombia, in July 201
Topologies of nodal sets of random band limited functions
It is shown that the topologies and nestings of the zero and nodal sets of
random (Gaussian) band limited functions have universal laws of distribution.
Qualitative features of the supports of these distributions are determined. In
particular the results apply to random monochromatic waves and to random real
algebraic hyper-surfaces in projective space.Comment: An announcement of recent results. Includes an announcement of the
resolution of some open questions from the older version. 11 pages, 6 figure
On the expected Betti numbers of the nodal set of random fields
This note concerns the asymptotics of the expected total Betti numbers of the
nodal set for an important class of Gaussian ensembles of random fields on
Riemannian manifolds. By working with the limit random field defined on the
Euclidean space we were able to obtain a locally precise asymptotic result,
though due to the possible positive contribution of large percolating
components this does not allow to infer a global result. As a by-product of our
analysis, we refine the lower bound of Gayet-Welschinger for the important
Kostlan ensemble of random polynomials and its generalisation to K\"{a}hler
manifolds.Comment: 18 pages, 1 figur
Betti Numbers of Random Hypersurface Arrangements
We study the expected behavior of the Betti numbers of arrangements of the
zeros of random (distributed according to the Kostlan distribution) polynomials
in . Using a random spectral sequence, we prove an
asymptotically exact estimate on the expected number of connected components in
the complement of such hypersurfaces in . We also
investigate the same problem in the case where the hypersurfaces are defined by
random quadratic polynomials. In this case, we establish a connection between
the Betti numbers of such arrangements with the expected behavior of a certain
model of a randomly defined geometric graph. While our general result implies
that the average zeroth Betti number of the union of random hypersurface
arrangements is bounded from above by a function that grows linearly in the
number of polynomials in the arrangement, using the connection with random
graphs, we show an upper bound on the expected zeroth Betti number of random
quadrics arrangements that is sublinear in the number of polynomials in the
arrangement. This bound is a consequence of a general result on the expected
number of connected components in our random graph model which could be of
independent interest
Low-Degree Approximation of Random Polynomials
We prove that with âhigh probabilityâ a random Kostlan polynomial in n+1 many variables and of degree d can be approximated by a polynomial of âlow degreeâ without changing the topology of its zero set on the sphere Sn. The dependence between the âlow degreeâ of the approximation and the âhigh probabilityâ is quantitative: for example, with overwhelming probability, the zero set of a Kostlan polynomial of degree d is isotopic to the zero set of a polynomial of degree O(dlogdââââââ). The proof is based on a probabilistic study of the size of C1-stable neighborhoods of Kostlan polynomials. As a corollary, we prove that certain topological types (e.g., curves with deep nests of ovals or hypersurfaces with rich topology) have exponentially small probability of appearing as zero sets of random Kostlan polynomials
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