Random complex zeroes, I. Asymptotic normality

By M. Sodin, Boris Tsirelson. 2002

135 citations recorded by the source corpus.

We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of zeroes.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/31d7e019a738671d26c94f705813f85ae3c75be3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Fluctuations of eigenvalues of random normal matrices (2008, result): “ of linear statistics ofeigenvaluesfromsomeclassicalcompactgroups. Themethodhasalsobeenusedintheparallel work on linear statistics of zeros of Gaussian analytic functions initiated by Sodin–Tsirelson [34] and generalized by Shiﬀman–Zelditch [33]. A brief comparison of these results to those of the p”
Asymptotic distribution of complex zeros of random analytic functions (2014, background): “alues in C, are nondegenerate, and satisfy the condition Elog(1 +|ξ0|) < ∞. Fix a parameter α > 0. We start by considering the following three ensembles of random analytic functions (see, e.g., [13, 33]): Fn(z)= Xn k=0 ξk n(n−1)···(n−k +1) k! α zk (elliptic, n∈N, z ∈C), X∞ k=0 ξk”
Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions (2005, background): “GAF that is conformally invariant in the unit disk and has this density. See Sodin and Tsirelson [ 9 ]”
EXPECTED TOPOLOGY OF RANDOM REAL ALGEBRAIC SUBMANIFOLDS (2013, background): “ds and in [16], we gave a lower bound for them. A similar probabilistic study of complex projective manifolds has been performed by Shiﬀman and Zelditch, see [30], [33], [2] for example, or also [3], [36]. In particular, the asymptotic equidistribution of critical points of random sections over a ﬁx”
The distribution of the zeros of random trigonometric polynomials (2008, background): “uess, but have not proved, that the same result may be true for Dunnage’s ensemble (3). The zeros of random complex analytic functions were examined in a series of papers by Sodin-Tsirelson (see e.g. [ST]), and Shiﬀman-Zelditch (see e.g. [SZ]). 1.2. Statement of results. Let XN : [0,2π] → Rbe Qualls”
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process (2003, background): “Close to the topic of this paper are Shiffman and Zelditch (2003), (with a brief review on the history of Gaussian random polynomials) and Sodin and Tsirelson (2003), which includes an extensive bibliography.”
Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on S^d (2014, unknown): “≡ − the weight function w ( t ) = (1 − t ) (cid:12) In order to formulate our main result, let us introduce the usual Wasserstein distance d between random variables Z , N : where Lip(1) is the set of Lipschitz functions whose Lipschitz constant equals 1.”
Quantitative Breuer-Major Theorems (2010, methodology): “recently the only available techniques for proving results such asTheorem 1.1were those based oncombinatorialcumulants/diagrams computations. These techniques are quite eﬀective and ﬂexible (see e.g. [24, 34] for further instances of their applicability), but suﬀer of a fundamental drawback, namely ”