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arXiv · 1402.1302

Volume fluctuations of random analytic varieties in the unit ball

Xavier Massaneda·Bharti Pridhnani
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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 $Z(f_L)$. We study the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a pseudo-hyperbolic ball of radius $r$. We first express this variance as an integral of a positive function in the unit disk. Then we study its asymptotic behaviour as $L\to\infty$ and as $r\to 1^{-}$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.

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Xavier Massaneda, Bharti Pridhnani. 2014-02-06. Volume fluctuations of random analytic varieties in the unit ball. https://arxiv.org/abs/1402.1302

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