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Complexity of random smooth functions on the high-dimensional sphere. (English) Zbl 1288.15045

The paper deals with the number of critical points of Gaussian smooth functions on the \(N\) dimensional sphere, and more especially, it tries to characterize a Morse function on a high-dimensional sphere, and to determine the number of critical values of a given index, or below a given index. The main result is based on an identity which relates the mean number of critical points of index \(k\) with the \(k\)th smallest eigenvalue of the Gaussian orthogonal ensemble, and shows that there is an exponentially large number of critical points of given index. The asymptotic complexity of the mean number of critical points is carefully investigated and an explicit formula is derived.

MSC:

15B52 Random matrices (algebraic aspects)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
60G60 Random fields
82B27 Critical phenomena in equilibrium statistical mechanics
60B20 Random matrices (probabilistic aspects)
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