High local maxima of stationary smooth Gaussian fields

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. $\mathbb{P}(\eta(B)=0)$ for a point process $\eta$ 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 $\exp(- \beta u^2)$, for some fixed $\beta >0$. 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.