Variance of the number of zeroes of shift-invariant Gaussian analytic functions (Q1617942)

A Gaussian analytic function (GAF) in the strip \(D_\Delta:=\{z\in\mathbb{C} : \vert\mathrm{Im}(z)\vert <\Delta \}\), where \(\Delta\in(0,\infty]\), is a random variable \(f\) taking values in the space of analytic functions on \(D_\Delta\) such that for every \(n\in\mathbb{N}\) and every \(z_1,\ldots,z_n\in D_\Delta\), the random vector \((f(z_1),\ldots,f(z_n))\) has a mean zero complex Gaussian distribution. A GAF \(f\) in \(D_\Delta\) is called stationary, if for any \(t\in\mathbb{R}\), any \(n\in\mathbb{N}\), and any \(z_1,\ldots,z_n\in D_\Delta\), the random vectors \((f(z_1),\ldots,f(z_n))\) and \((f(z_1+t),\ldots,f(z_n+t))\) have the same distribution. A stationary GAF is called degenerate if the spectral measure of its covariance kernel \(\mathbb{E}(f(z)\overline{f(w)})\), \(z,w\in D_\Delta\), consists of exactly one atom. The author shows that, given a non-degenerate stationary GAF in a strip \(D_\Delta\), the variance of the number of its zeroes in a long horizontal rectangle \([-T,T] \times [a,b]\) with \(-\Delta < a < b < \Delta\) is asymptotically between \(cT\) and \(CT^2\), where \(c\) and \(C\) are some positive constants. Especially, the results indicate that the zeroes are never ``super-concentrated'' around their mean. Conditions in terms of the spectral measure in question are also provided under which the variance grows asymptotically linearly with \(T\), as a quadratic function of \(T\), or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.