Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second version mainly resolves minor LaTeX problems.

Evolution of Random Wave Fields in the Water of Finite Depth

The evolution of random wave fields on the free surface is a complex process which is not completely understood nowadays. For the sake of simplicity in this study we will restrict our attention to the 2D physical problems only (i.e. 1D wave propagation). However, the full Euler equations are solved numerically in order to predict the wave field dynamics. We will consider the most studied deep water case along with several finite depths (from deep to shallow waters) to make a comparison. For each depth we will perform a series of Monte--Carlo runs of random initial conditions in order to deduce some statistical properties of an average sea state.Comment: 12 pages, 5 figures, 28 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Non Gaussian Minkowski functionals and extrema counts for 2D sky maps

In the conference presentation we have reviewed the theory of non-Gaussian geometrical measures for the 3D Cosmic Web of the matter distribution in the Universe and 2D sky data, such as Cosmic Microwave Background (CMB) maps that was developed in a series of our papers. The theory leverages symmetry of isotropic statistics such as Minkowski functionals and extrema counts to develop post- Gaussian expansion of the statistics in orthogonal polynomials of invariant descriptors of the field, its first and second derivatives. The application of the approach to 2D fields defined on a spherical sky was suggested, but never rigorously developed. In this paper we present such development treating effects of the curvature and finiteness of the spherical space $S_2$ exactly, without relying on the flat-sky approximation. We present Minkowski functionals, including Euler characteristic and extrema counts to the first non-Gaussian correction, suitable for weakly non-Gaussian fields on a sphere, of which CMB is the prime example.Comment: 6 pages, to appear as proceedings of the IAU Symposium No. 308, 2014 The Zeldovich Universe, Genesis and Growth of the Cosmic Web Rien van de Weygaert, Sergei Shandarin, Enn Saar and Jaan Einast

The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D

The full moments expansion of the joint probability distribution of an isotropic random field, its gradient and invariants of the Hessian is presented in 2 and 3D. It allows for explicit expression for the Euler characteristic in ND and computation of extrema counts as functions of the excursion set threshold and the spectral parameter, as illustrated on model examples.Comment: 4 pages, 2 figures. Corrected expansion coefficients for orders n>=5. Relation between Gram-Charlier and Edgeworth expansions is clarified

Non-Gaussian Minkowski functionals & extrema counts in redshift space

In the context of upcoming large-scale structure surveys such as Euclid, it is of prime importance to quantify the effect of peculiar velocities on geometric probes. Hence the formalism to compute in redshift space the geometrical and topological one-point statistics of mildly non-Gaussian 2D and 3D cosmic fields is developed. Leveraging the partial isotropy of the target statistics, the Gram-Charlier expansion of the joint probability distribution of the field and its derivatives is reformulated in terms of the corresponding anisotropic variables. In particular, the cosmic non-linear evolution of the Minkowski functionals, together with the statistics of extrema are investigated in turn for 3D catalogues and 2D slabs. The amplitude of the non-Gaussian redshift distortion correction is estimated for these geometric probes. In 3D, gravitational perturbation theory is implemented in redshift space to predict the cosmic evolution of all relevant Gram-Charlier coefficients. Applications to the estimation of the cosmic parameters sigma(z) and beta=f/b1 from upcoming surveys is discussed. Such statistics are of interest for anisotropic fields beyond cosmology.Comment: 35 pages, 15 figures, matches version published in MNRAS with a typo corrected in eq A1