Density of Complex Critical Points of a Real Random SO(m+1) Polynomials

Details Density of Complex Critical Points of a Real Random SO(m+1) Polynomials Density of Complex Critical Points of a Real Random SO(m+1) Polynomials

2010-10-28

arxiv.org/abs/1010.6104v1

Journal of Statistical Physics, 2010, Volume 141, Number 3, Pages 517-531

Physics

Mathematical Physics

15 pages

Scientific paper

10.1007/s10955-010-0057-y

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper [Mac09], the author used the Poincare- Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac- Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.

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