Stochastic geometry and topology of non-Gaussian fields

Gaussian random fields pervade all areas of science. However, it is often the departures from Gaussianity that carry the crucial signature of the nonlinear mechanisms at the heart of diverse phenomena, ranging from structure formation in condensed matter and cosmology to biomedical imaging. The standard test of non-Gaussianity is to measure higher order correlation functions. In the present work, we take a different route. We show how geometric and topological properties of Gaussian fields, such as the statistics of extrema, are modified by the presence of a non-Gaussian perturbation. The resulting discrepancies give an independent way to detect and quantify non-Gaussianities. In our treatment, we consider both local and nonlocal mechanisms that generate non-Gaussian fields, both statically and dynamically through nonlinear diffusion.Comment: 8 pages, 4 figure

The distribution of extremal points of Gaussian scalar fields

We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without boundary). We apply the general results to random waves and random surfaces. Furthermore, we find a generating functional for the two-point function. Its Legendre transform is the integral over the scalar curvature of a 4-dimensional Riemannian manifold.Comment: 22 pages, 8 figures, corrected published versio

Quasi-exact-solution of the Generalized Exe Jahn-Teller Hamiltonian

We consider the solution of a generalized Exe Jahn-Teller Hamiltonian in the context of quasi-exactly solvable spectral problems. This Hamiltonian is expressed in terms of the generators of the osp(2,2) Lie algebra. Analytical expressions are obtained for eigenstates and eigenvalues. The solutions lead to a number of earlier results discussed in the literature. However, our approach renders a new understanding of ``exact isolated'' solutions

On the wave-induced difference in mean sea level between the two sides of a submerged breakwater

Very simple formulae are derived for the difference in mean level between the two sides of a submerged breakwater when waves are incident on it at an arbitrary angle. The formulae apply also to waves undergoing refraction due to changes in depth and to waves in open channel transitions

On the statistical distribution of the heights of sea waves

The statistical distribution of wave-heights is derived theoretically on the assumptions (a) that the wave spectrum contains a single narrow band of frequencies, and (b) that the wave energy is being received from a large number of different sources whose phases are random. Theoretical relations are found between the root-meansquare wave-height, the mean height of the highest one-third (or highest one-tenth) waves and the most probable height of the largest wave in a given interval of time. There is close agreement with observation

Signed zeros of Gaussian vector fields-density, correlation functions and curvature

We calculate correlation functions of the (signed) density of zeros of Gaussian distributed vector fields. We are able to express correlation functions of arbitrary order through the curvature tensor of a certain abstract Riemann-Cartan or Riemannian manifold. As an application, we discuss one- and two-point functions. The zeros of a two-dimensional Gaussian vector field model the distribution of topological defects in the high-temperature phase of two-dimensional systems with orientational degrees of freedom, such as superfluid films, thin superconductors and liquid crystals.Comment: 14 pages, 1 figure, uses iopart.cls, improved presentation, to appear in J. Phys.

Steep sharp-crested gravity waves on deep water

A new type of steady steep two-dimensional irrotational symmetric periodic gravity waves on inviscid incompressible fluid of infinite depth is revealed. We demonstrate that these waves have sharper crests in comparison with the Stokes waves of the same wavelength and steepness. The speed of a fluid particle at the crest of new waves is greater than their phase speed.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

A note on wave set-up

Seaward of the breaker zone, the observations of Saville are in good qualitative agreement with the prediction that the mean surface level is increasingly depressed towards the shoreline

Asymptotic behaviour of the Rayleigh--Taylor instability

We investigate long time numerical simulations of the inviscid Rayleigh-Taylor instability at Atwood number one using a boundary integral method. We are able to attain the asymptotic behavior for the spikes predicted by Clavin & Williams\cite{clavin} for which we give a simplified demonstration. In particular we observe that the spike's curvature evolves like $t^3$ while the overshoot in acceleration shows a good agreement with the suggested $1/t^5$ law. Moreover, we obtain consistent results for the prefactor coefficients of the asymptotic laws. Eventually we exhibit the self-similar behavior of the interface profile near the spike.Comment: 4 pages, 6 figure