High local maxima of stationary smooth Gaussian fields

Abstract

Consider the point process (in $\mathbb{R}^d$) of local maxima of smooth Gaussian fields, with sufficient decay of correlation at infinity, above a level $u$. We show that this point process, rescaled appropriately, converges weakly to a Poisson point process in the limit $u \to \infty$. Our proof relies on the classical observation that simple point processes are characterised by avoidance probabilities (i.e. $\mathbb{P}(η(B)=0)$ for a point process $η$ and Borel set $B$). Then we approximate avoidance probability with the excursion probability, where the latter is well studied. Second main result is a quantified version of the Poisson convergence of high local maxima of the Bargmann-Fock field in $\mathbb{R}^2$. We prove that, for Bargmann-Fock field in two dimensions, the total variation distance between a Poisson random variable and the number of local maxima of the field above a threshold $u$ in an $R \times R$ box in $\mathbb{R}^2$ decays like $\exp(- βu^2)$, for some fixed $β>0$. As an immediate consequence, when the level $u$ is a function of $R$ such that $u(R) \to \infty$ and $u(R)/ \sqrt{\log R} \to 0$ as $R \to \infty$, we have a quantitative central limit theorem for the number of high local maxima. The proof is based on the Chen-Stein method for quantitative Poisson approximation. We produce a close coupling of a stationary smooth field and its Palm version, which might be of independent interest.

Figures (8)

• Figure 1: (Left) Bargmann-Fock field in a 20 by 20 box. (Right) Palm coupling of the field with maxima at origin with height at least 20.
• Figure 3: (Left) Bargmann-Fock critical points in dimension 2. (Right) Poisson point process with the same intensity.
• Figure 4: (Left) A sample of RPW in a box of sidelength around 200 wavelengths. (Right) The same with around 2000 wavelengths. Picture by Alex Barnett (Link: https://users.flatironinstitute.org/ ahb/rpws/)
• Figure 5: The entire shaded area is defined to be $\lambda_{a,u}$
• Figure : (a) RPW: All maxima in a 20 by 20 box.

Theorems & Definitions (28)

• Theorem 2.2
• Theorem 2.3
• Theorem 2.4: Quantitative central limit theorem
• Theorem 2.6
• Definition 3.1: DC-ring
• Theorem 3.2: c.f. Theorem 4.18 of kallenberg_random_2017
• Theorem 3.3: Theorem 7.1,piterbarg_asymptotic_1996
• Lemma 3.4
• proof
• Lemma 3.5