Fluctuations in the Zero Set of the Hyperbolic Gaussian Analytic Function

Abstract

The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disc whose distribution is invariant under automorphisms of the disc. We study the variance of the number of points in a disc of increasing radius. Somewhat surprisingly, we find a change of behaviour at a certain value of the 'intensity' of the process, which appears to be novel.

Hole probabilities for $\beta $-ensembles and determinantal point processes in the complex plane

“…In this chapter we calculate V[X L (r D)] and V(X L (ϕ)), when ϕ(z) = (1 − |z| 2 r 2 ) p 2 + for p > 0, as r → 1 − . The problem is inspired by the work of Buckley [Buc15]. He consider the point processes X f L in the unit disk for L > 0, the zeros set of hyperbolic Gaussian analytic functions…”

Fluctuations for zeros of Gaussian Taylor series

“…For the family of hyperbolic GAFs, whose zero sets are invariant with respect to the isometries of the unit disk, Buckley [4] found the asymptotics of the variance of the number of zeros (see also Section 2.4). Buckley and Sodin [5] studied fluctuations in the increment of the argument along curves for the planar GAF (GEF).…”

The hyperbolic-type point process

“…In particular, it was proved there that (5) V 1 0 (N r ) = r 2 1 − r 4 , and another proof of this result is given in [4]. However, beware that the Gaussian analytic series in the disc studied in [4] are in general not determinantal and that asymptotic formulas for the variance of N r are derived there from a formula quite similar to (2). More generally, the adaptation of our previous proof of Shirai's result yields the following formula for V ν m (N r ): Proposition 2.…”

“…In particular, P 1 0 is nothing else but the hyperbolic Gaussian determinantal process defined and studied in [14], and is realized as the zeros of a Gaussian analytic series. In particular, it was proved there that (5) V 1 0 (N r ) = r 2 1 − r 4 , and another proof of this result is given in [4]. However, beware that the Gaussian analytic series in the disc studied in [4] are in general not determinantal and that asymptotic formulas for the variance of N r are derived there from a formula quite similar to (2).…”