On the Distribution of Roots of Random Polynomials on the Complex Plane

Two special cases

Henryk Zawadzki

The Karol Adamiecki University of Economics in Katowice, Poland

Abstract

When the coefficients (Subscript[a, j]) (j = 0, 1, ..., n - 1) of the polynomial

(Subscript[p, n])(z)=(∑_(j=0)^(n-1))(Subscript[a, j])(z^j)

are random variables, the number (Subscript[υ, n])(Ω) of its roots in any measurable set Ω ⊂ C ( C - the complex plane) is also a random variable. The parameters of (Subscript[υ, n])(Ω) and the distribution (location) of roots on the complex plane C depend on distribution of (Subscript[a, j]). 
In the presentation we show how are the roots of (1) distributed on the complex plane when:
1) coefficients (Subscript[a, j]) are independent identically distributed standard Gaussian random variables,
2) coefficients (Subscript[a, j]) are independent Gaussian variables of zero mean and variance (D^2)((Subscript[a, j])) = (e^(-(j^( α))))indexed by a real parameter α ≥ 0.
In the first case, using the formula

E[(Subscript[υ, n])(Ω)]=(^)∫(∫_Ω)(Subscript[h, n])(x,y)dxdy+(∫_(Ω⋂R))(Subscript[g, n])(x)dx

(Shepp, Vanderbei (1995)) we calculate the expected value of (Subscript[υ, n])(Ω) for some n and Ω. We also show the evolution of graphs of intensity functions (Subscript[h, n]) when n is increasing. Even for small values of n the shape of the graphs explains the phenomenon of concentration (or condensation) of roots of (1) in a neighborhood of the circle |z| = 1. In the second case we show how with an increasing of parameter α, the concentration of roots around the unit circle changes into the concentration of roots on the real axis (according to the result obtained recently by Schehr, Majumdar (2009)).

References

1. Schehr G., Majumdar S. N. : Condensation of the roots of real random polynomials on the real axis. Journal of Statistical Physics, Vol. 135, Number 4, May 2009, pp. 587-598.
2. Shepp L. A., Vanderbei R. J. : The complex zeros of random polynomials. Transactions of the American Mathematical Society, Vol. 347, Number 11, November 1995, pp. 4365-4394.