Volume fluctuations of random analytic varieties in the unit ball

Abstract

Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb{C}^n, n \geq 2$, consider its (random) zero variety $Z\left(f_L\right)$. We reduce the variance of the $(n-1)$-dimensional volume of $Z\left(f_L\right)$ inside a pseudo-hyperbolic ball of radius $r$ to an integral of a positive function in the unit disk. We illustrate the usefulness of this expression by describing the asymptotic behaviour of the variance as $r \rightarrow 1^{-}$and as $L \rightarrow \infty$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.