Density of critical points for a Gaussian random function

Details Density of critical points for a Gaussian random function Density of critical points for a Gaussian random function

2007-07-03

arxiv.org/abs/0707.0457v2

Physics

Fluid Dynamics

11 pages 1 figure, changes in list of references, corrected typos

Scientific paper

10.1088/1751-8113/41/2/025210

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words, the density of the saddle points and extrema agree. This argument breaks down in three dimensions. All ratios of the densities of saddle points and extrema larger than one are possible. For a homogeneous Gaussian random field one finds no longer an equidistribution of signs, saddle points are slightly more frequent.

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