IUMJ

Title: Volume fluctuations of random analytic varieties in the unit ball

Authors: Xavier Massaneda and Bharti Pridhnani

Issue: Volume 64 (2015), Issue 6, 1667-1695

Abstract:

Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb{C}^n$, $n\geq2$, consider its (random) zero variety $Z(f_L)$. We reduce the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ 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 $r\to1^{-}$ and as $L\to\infty$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.