Asymptotic normality of linear statistics of zeros of random polynomials

Abstract

ABSTRACT. In this note, we prove a central limit theorem for smooth linear statistics of zeros of random polynomials which are linear combinations of orthogonal polynomials with iid standard complex Gaussian coefficients. Along the way, we obtain Bergman kernel asymptotics for weighted L 2 -space of polynomials endowed with varying measures of the form e −2nϕn(z) dz under suitable assumptions on the weight functions ϕ n .

Universality results for zeros of random holomorphic sections

“…In this paper we study the asymptotic distribution of zeros of sequences of random holomorphic sections of singular Hermitian holomorphic line bundles. We generalize our previous results from [CM1,CM2,CM3,CMM,Ba1,Ba3,Ba2] in several directions. We consider sequences (L p , h p ), p ≥ 1, of singular Hermitian holomorphic line bundles over Kähler spaces instead of the sequence of powers (L p , h p ) = (L ⊗p , h ⊗p ) of a fixed line bundle (L, h).…”

“…It is by now well established that the off-diagonal decay of the Bergman/Szegő kernel for powers L p of a line bundle L implies the asymptotics of the variance current and variance number for zeros of random holomorphic sections of L p , cf. [Ba2,ST,SZ2]. Note also that the Bergman kernel provides the 2-point correlation function for the determinantal random point process defined by the Bergman projection [Ber,§6.1].…”

“…also [Be]). In [Ba2,Theorem 2.4] the exponential decay was obtained for a family of weights having super logarithmic growth at infinity.…”

Equidistribution of zeros of random holomorphic sections

“…On the other hand, Shiffman and Zelditch [SZ4] pursued the idea of Sodin and Tsirelson and generalized their result to the setting of random holomorphic sections for a positive line bundle L → X defined over a projective manifold. Building upon ideas from [SoT,SZ4] and using the near and off diagonal Bergman kernel asymptotics (see [Ba2,§2]) we proved a CLT for linear statistics:…”

“…Theorem 2.5 ([Ba2, Theorem 1.2]). Let ϕ : C n → R be a C 2 weight function satisfying (2.1) and Φ be a real (n − 1, n − 1) test form with C 3 coefficients such that ∂∂Φ ≡ 0 and ∂∂Φ is supported in the interior of the bulk of ϕ (see [Ba2,(2.3)]). Then the linear statistics…”