The distribution of extremal points of Gaussian scalar fields

Abstract

We consider the signed density of the extremal points of (two-dimensional) scalar fields with a Gaussian distribution. We assign a positive unit charge to the maxima and minima of the function and a negative one to its saddles. At first, we compute the average density for a field in half-space with Dirichlet boundary conditions. Then we calculate the charge-charge correlation function (without boundary). We apply the general results to random waves and random surfaces. Furthermore, we find a generating functional for the two-point function. Its Legendre transform is the integral over the scalar curvature of a 4-dimensional Riemannian manifold.

22 pages, 8 figures, corrected published version

References in corpus (3)

• The Geometry of Soft Materials: A Primer
• Current and vortex statistics in microwave billiards
• Signed zeros of Gaussian vector fields-density, correlation functions and curvature

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