We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

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.

a b s t r a c tWe study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

This paper studies the winding of a continuously differentiable Gaussian stationary process f : R → C in the interval [0, T ]. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with T , and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in L 2 (R), then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Malevich, Cuzick, Slud and others.

ABSTRACT. The Gaussian entire function is a random entire function, characterised by a certain invariance with respect to isometries of the plane. We study the fluctuations of the increment of the argument of the Gaussian entire function along planar curves. We introduce an inner product on finite formal linear combinations of curves (with real coefficients), that we call the signed length, which describes the limiting covariance of the increment. We also establish asymptotic normality of fluctuations.Let (ζ n ) ∞ n=0 be a sequence of iid standard complex Gaussian random variables (that is, each ζ n has density 1 π e −|z| 2 with respect to the Lebesgue measure on the plane), and define the Gaussian entire function byA remarkable feature of this random entire function is the invariance of the distribution of its zero set with respect to isometries of the plane. The invariance of the distribution of f under rotations is obvious, by the invariance of the distribution of each ζ n . The translation invariance arises from the fact that, for any w ∈ C, the Gaussian processes f (z + w) and e zw+ 1 2 |w| 2 f (z) have the same distribution; this follows, for instance, by inspecting the covariancesFurther, by Calabi's rigidity, f is (essentially) the only Gaussian entire function whose zeroes satisfy such an invariance (see [HKPV09, Chapter 2] for details and further references).Given a large parameter R > 0, the function log f (Rz) gives rise to multi-valued fields with a high intensity of logarithmic branch points, which is somewhat reminiscent of chiral bosonic fields as described by Kang and Makarov [KM13, Lecture 12]. One way to understand asymptotic fluctuations of these fields as R → ∞ is to study asymptotic fluctuations of the increment of the argument of f (Rz) along a given curve, which will be our concern in this paper. Note that, by the argument principle, if the curve bounds a domain G then this observable coincides with the number of zeroes of f in RG (the dilation of the set G), up to a factor 2π (and a sign change if the curve is negatively oriented with respect to the domain it bounds).We begin with the following definition. Definition 1.In what follows a curve Γ is always a C 1 -smooth regular oriented simple curve in the plane, of finite length 1 . An R-chain is a finite formal sum Γ = i a i Γ i , where Γ i are curves and the coefficients a i are real numbers.Note that if the coefficients a i are integer valued, then we can assign an obvious geometric meaning to the formal sum Γ = i a i Γ i .Definition 2. Given a curve Γ and R > 0 we define ∆ R (Γ) to be the random variable given by the increment of the argument of f (Rz) along Γ. Given an R-chain Γ = i a i Γ i we defineIn order for this definition to make sense, we need to see that almost surely f does not vanish on a fixed curve. Note that the mean number of zeroes in a (measurable) subset of the plane is proportional to the Lebesgue measure of the set. Since the number of zeroes on a fixed curve is a non-negative random variable, whose mean is zero, t...