Overcrowding Estimates for the Nodal Volume of Stationary Gaussian Processes

Abstract

Abstract For smooth centered stationary Gaussian processes on ${{\mathbb {R}}}^d$ ($d \geq 1$), we study the unlikely event of overcrowding of the nodal set; this is the event that the volume of the nodal set in a region is much larger than its expected value. Under some mild assumptions on the spectral measure of the process, we obtain estimates for the probability of the overcrowding event.