Variance of the Number of Zeroes of Shift-Invariant Gaussian Analytic Functions

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle $[0,T]\times [a,b]$ is asymptotically between $cT$ and $CT^2$, with positive constants $c$ and $C$. We also supply with conditions (in terms of the spectral measure) under which the variance asymptotically grows linearly, as a quadratic function of $T$, or has intermediate growth.Comment: 31 pages (22 pages without appendices

The Brownian web is a two-dimensional black noise

The Brownian web is a random variable consisting of a Brownian motion starting from each space-time point on the plane. These are independent until they hit each other, at which point they coalesce. Tsirelson mentions this model in his paper "Scaling limit, Noise, Stability", along with planar percolation, in suggesting the existence of a two-dimensional black noise. A two-dimensional noise is, roughly speaking, a random object on the plane whose distribution is translation invariant and whose behavior on disjoint subsets is independent. Black means sensitive to the resampling of sets of arbitrarily small total area. Tsirelson implicitly asks: "Is the Brownian web a two-dimensional black noise?". We give a positive answer to this question, providing the second known example of such after the scaling limit of critical planar percolation.Comment: 16 Pages, 3 Figure