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

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