Stationary Gaussian Field Research Articles

Upper Bound for the Conjunction Probability of Smooth Stationary Two-dimensional Gaussian Fields

Upper Bound for the Conjunction Probability of Smooth Stationary Two-dimensional Gaussian Fields

Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Limit theorems for p-domain functionals of stationary Gaussian fields

Limit theorems for p-domain functionals of stationary Gaussian fields

Uniqueness of Unbounded Component for Level Sets of Smooth Gaussian Fields

Abstract For a large family of stationary continuous Gaussian fields $f$ on ${\mathbb {R}}^{d}$, including the Bargmann–Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell \}$ (as well as in the excursion set $\{f\geq \ell \}$) almost surely for every level $\ell \in {\mathbb {R}}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez, and Vanneuville. As the fields considered are typically very rigid (e.g., analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron–Martin theorem.

Local repulsion of planar Gaussian critical points

We study the local repulsion between critical points of a stationary isotropic smooth planar Gaussian field. We show that the critical points can experience a soft repulsion which is maximal in the case of the random planar wave model, or a soft attraction of arbitrary high order. If the type of critical points is specified (extremum, saddle point), the points experience a hard local repulsion, that we quantify with the precise magnitude of the second factorial moment of the number of points in a small ball.

On convergence of volume of level sets of stationary smooth Gaussian fields

On convergence of volume of level sets of stationary smooth Gaussian fields

Boundedness of the nodal domains of additive Gaussian fields

We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets { f ≤ ℓ } \{f \le \ell \} of additive planar Gaussian fields are bounded almost surely at the critical level ℓ c = 0 \ell _c = 0 . Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension d ≥ 3 d \ge 3 the excursion sets have unbounded components at all levels.

Effective Generation of Compressed Stationary Gaussian Fields

Effective Generation of Compressed Stationary Gaussian Fields

Quantitative estimates in stochastic homogenization for correlated coefficient fields

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations. It deduces optimal estimates of the homogenization error from optimal growth estimates of the (extended) corrector. In line with the heuristics, there are transitions at dimension $d=2$, and for a correlation-decay exponent $\beta=2$; we capture the correct power of logarithms coming from these two sources of criticality. The decay of correlations is sharply encoded in terms of a multiscale logarithmic Sobolev inequality (LSI) for the ensemble under consideration --- the results would fail if correlation decay were encoded in terms of an $\alpha$-mixing condition. Among other ensembles popular in modelling of random media, this class includes coefficient fields that are local transformations of stationary Gaussian fields. The optimal growth of the corrector $\phi$ is derived from bounding the size of spatial averages $F=\int g\cdot\nabla\phi $ of its gradient. This in turn is done by a (deterministic) sensitivity estimate of $F$, that is, by estimating the functional derivative $\frac{\partial F}{\partial a}$ of $F$ w.~r.~t.~the coefficient field $a$. Appealing to the LSI in form of concentration of measure yields a stochastic estimate on $F$. The sensitivity argument relies on a large-scale Schauder theory for the heterogeneous elliptic operator $-\nabla\cdot a\nabla$. The treatment allows for non-symmetric $a$ and for systems like linear elasticity.

On the monofractality of many stationary continuous Gaussian fields

In this article we focus on a general real-valued continuous stationary Gaussian field X characterized by its spectral density |g|2, where g is any even real-valued deterministic square integrable function. Our starting point consists in drawing a close connection between critical Besov regularity of the inverse Fourier transform of g and αX the random pointwise Hölder exponent function of X, which measures local roughness/smoothness of its sample paths at each point. Then, thanks to Littlewood-Paley methods and Hausdorff-Young inequalities, under weak conditions on g, we show that the random function αX is actually a deterministic constant which does not change from point to point. This result means that the field X is of monofractal nature. Also, it is worth mentioning that such a result can easily be extended to the case where X is no longer stationary but has stationary increments.

Mean Euler characteristic of stationary random closed sets

The translative intersection formula of integral geometry yields an expression for the mean Euler characteristic of a stationary random closed set intersected with a fixed observation window. We formulate this result in the setting of sets with positive reach and using flag measures which yield curvature measures as marginals. As an application, we consider excursion sets of stationary random fields with C1,1 realizations, in particular, stationary Gaussian fields, and obtain results which extend those known from the literature.

Talagrand’s inequality in planar Gaussian field percolation

Let f be a stationary isotropic non-degenerate Gaussian field on R2. Assume that f=q∗W where q∈L2(R2)∩C2(R2) and W is the L2 white noise on R2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that q∗q is pointwise non-negative and has fast enough decay, the set {f≥−l} percolates with probability one when l>0 and with probability zero if l≤0. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold T(g) which is the minimal value such that {g≥−T(g)} contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.

Statistics for Gaussian random fields with unknown location and scale using Lipschitz‐Killing curvatures

AbstractIn the present article we study the average of Lipschitz‐Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on . The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters of X, we introduce novel fundamental quantities called effective level and effective spectral moment. We propose unbiased and asymptotically normal estimators of these parameters. From these asymptotic results, we build a test to determine if two images of excursion sets can be compared. This test is applied on both synthesized and real mammograms. Meanwhile, we establish the consistency of the empirical variance estimators of the third LK curvature under a weak condition on the correlation function of X.

Extremes of locally stationary Gaussian and chi fields on manifolds

Depending on a parameter h∈(0,1], let {Xh(t),t∈Mh} be a class of centered Gaussian fields indexed by compact manifolds Mh with positive reach. For locally stationary Gaussian fields Xh, we study the asymptotic excursion probabilities of Xh on Mh. Two cases are considered: (i) h is fixed and (ii) h→0. These results are also extended to obtain the limit behaviors of the extremes of locally stationary χ-fields on manifolds.

A covariance formula for topological events of smooth Gaussian fields

We derive a covariance formula for the class of ‘topological events’ of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example, (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g., the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 1679–1711), in which a correlation inequality was derived for certain topological events on the plane, as well as by (Asymptotic Methods in the Theory of Gaussian Processes and Fields (1996) Amer. Math. Soc.), in which a similar covariance formula was established for finite-dimensional Gaussian vectors.

A second moment bound for critical points of planar Gaussian fields in shrinking height windows

We consider the number of critical points of a stationary planar Gaussian field, restricted to a large domain, whose heights lie in a certain interval. Asymptotics for the mean of this quantity are simple to establish via the Kac–Rice formula, and recently Estrade and Fournier proved a second moment bound that is optimal in the case that the height interval does not depend on the size of the domain. We establish an improved bound in the more delicate case of height windows that are shrinking with the size of the domain.

No repulsion between critical points for planar Gaussian random fields

We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a ‘generic’ field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.

On the absolute continuity of random nodal volumes

We study the absolute continuity with respect to the Lebesgue measure of the distribution of the nodal volume associated with a smooth, nondegenerate and stationary Gaussian field $(f(x),{x\in\mathbb{R}^{d}})$. Under mild conditions, we prove that in dimension $d\geq3$, the distribution of the nodal volume has an absolutely continuous component plus a possible singular part. This singular part is actually unavoidable bearing in mind that some Gaussian processes have a positive probability to keep a constant sign on some compact domain. Our strategy mainly consists in proving closed Kac–Rice type formulas allowing one to express the volume of the set $\{f=0\}$ as integrals of explicit functionals of $(f,\nabla f,\operatorname{Hess}(f))$ and next to deduce that the random nodal volume belongs to the domain of a suitable Malliavin gradient. The celebrated Bouleau–Hirsch criterion then gives conditions ensuring the absolute continuity.

Quasi-independence for nodal lines

\abstract{We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance $(x,y)\mapsto\kappa(x-y)$. As a first application, we study percolation for nodal lines in the spirit of~\cite{bg_16}. In the said article, Beffara and Gayet rely on Tassion's method (\cite{tassion2014crossing}) to prove that, under some assumptions on $\kappa$, most notably that $\kappa \geq 0$ and $\kappa(x)=O(|x|^{-325})$, the nodal set satisfies a box-crossing property. The decay exponent was then lowered to $16+\varepsilon$ by Beliaev and Muirhead in \cite{bm_17}. In the present work we lower this exponent to $4+\varepsilon$ thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from~\cite{nazarov2015asymptotic}.

An Improved Second-Order Poincaré Inequality for Functionals of Gaussian Fields

We present an improved version of the second-order Gaussian Poincare inequality, first introduced in Chatterjee (Probab Theory Relat Fields 143(1):1–40, 2009) and Nourdin et al. (J Funct Anal 257(2):593–609, 2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative central limit theorems for nonlinear functionals of stationary Gaussian fields related to the Breuer–Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.