2017/11/14 by Rivera, Alejandro, Vanneuville, Hugo · 2 citations
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1711.05009
We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance (x,y)↦κ(x-y). As a first application, we study percolation for nodal lines in the spirit of [BG16]. In the said article, Beffara and Gayet rely on Tassion's method ([Tas16]) to prove that, under some assumptions on κ, most notably that κ≥ 0 and κ(x)=O(|x|-325), the nodal set satisfies a box-crossing property. The decay exponent was then lowered to 16+ε by Beliaev and Muirhead in [BM17]. In the present work we lower this exponent to 4+ε 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 [NS15].