Realisability conditions for second order marginals of biphased media

16 pagesInternational audienceThis paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper

Diophantine Gaussian excursions and random walks

We investigate the asymptotic variance of Gaussian nodal excursions in the Euclidean space, focusing on the case where the spectral measure has incommensurable atoms. This study requires to establish fine recurrence properties in 0 for the associated irrational random walk on the torus. We show in particular that the recurrence magnitude depends strongly on the diophantine properties of the atoms, and the same goes for the variance asymptotics of nodal excursions. More specifically, if the spectral measures contains atoms which ratios are well approximable by rationals, the variance is likely to have large fluctuations as the observation window grows, whereas the variance is bounded by the (d -- 1)-dimensional measure of the window boundary if these ratio are badly approximable. We also show that, given any reasonable function, there are uncountably many sets of parameters for which the variance is asymptotically equivalent to this function.Comment: Added a proof of the variance cancellation phenomenon for planar Gaussian waves nodal excursion

Bounds to the normal for proximity region graphs

In a proximity region graph ${\cal G}$ in $\mathbb{R}^d$, two distinct points $x,y$ of a point process $\mu$ are connected when the 'forbidden region' $S(x,y)$ these points determine has empty intersection with $\mu$. The Gabriel graph, where $S(x,y)$ is the open disc with diameter the line segment connecting $x$ and $y$, is one canonical example. When $\mu$ is a Poisson or binomial process, under broad conditions on the regions $S(x,y)$, bounds on the Kolmogorov and Wasserstein distances to the normal are produced for functionals of ${\cal G}$, including the total number of edges and the total length. Variance lower bounds, not requiring strong stabilization, are also proven to hold for a class of such functionals.Comment: 33 pages; changes in response to referees' comment