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arXiv · math-ph/0301041

The distribution of extremal points of Gaussian scalar fields

Georg Foltin
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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.

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Georg Foltin. 2003-04-14. The distribution of extremal points of Gaussian scalar fields. https://doi.org/10.1088/0305-4470%2F36%2F16%2F307

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