Zeros of Gaussian power series with dependent random variables

Abstract

Peres and Virág in 2005 studied the zeros of the random power series ${\sum }_{n=1}^{\infty }{a}_n{z}^{n-1}$ in the unit disc whose coefficients $\left(a_n\right)$ are independent and identically distributed symmetrical Gaussian complex random variables. They discovered that its zeros constitute a determinant point process and derived many other interesting related properties. In this paper, we study the general Gaussian power series where the independence requirement is relaxed by taking $f\left(z\right)={\sum }_{n=1}^{\infty }{\Delta }_n{z}^n$, where $\left({\Delta }_n\right)$ is the classical fractional Gaussian noise of index $0\le H<1$. The intensity of the point process of the zeros of $f$ is used to derive important properties. As for the case of independent variables (corresponding to $H=1/2$), the zeros cluster around the unit circle, but in any domain inside the open unit disc, the dependent case (that is, $H\ne 1/2$) yields fewer zeros than the classical case of independent random variables. This implies that the clustering around the unit circle is faster than in the independent case. Moreover, in contrast to independent random variables, the zeros of $f$ are generally asymmetrically distributed in the unit circle. However, asymptotically, the two point processes are equivalent except near the sole point $z=1$. An interesting open problem is whether this point process is also a determinantal process.