We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furthermore, we obtain a variance estimate for those random zero currents, which yields the almost sure convergence under some geometric condition.
Abstract. In this paper, we study hole probabilities P 0,m (r, N ) of SU (m + 1) Gaussian random polynomials of degree N over a polydisc (D(0, r)) m . When r ≥ 1, we find asymptotic formulas and decay rate of log P 0,m (r, N ). In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities P k,1 (r, N ) over a disc D(0, r). IntroductionHole probability is the probability that some random field never vanishes over some set. The case of Gaussian random entire functions was studied by Sodin and Tsirelson:In [9], the authors considered the case of Gaussian random sections: let M be a compact Kähler manifold with complex dimension m and (L, h) → M be a positive holomorphic line bundle. γ N denotes the Gaussian probability measure on H Therefore, it is natural to ask: can we find sharp constants C 1 , C 2 in the above two theorems and furthermore, is it possible to obtain an asymptotic formula and a decay rate for the hole probability? Using Cauchy's integral estimates, Nishry answered this question in the random entire function case:Gaussian random variables. ThenThis inspires us that for those line bundles with polynomial sections, maybe it is possible to find an asymptotic formula for the hole probability.If P 0,m (r, N ) denotes the hole probability of SU (m+1) Gaussian random polynomials over the polydisc D(0, r) m , d m x is the Lebesgue measure on R m andis a continuous function defined over the standard simplex Σ m ∶= {x = (x 1 , . . . ,x i ≤ 1}(here we adopt the convention that 0 log 0 = 0), we have the following results:where,Here when m = 1, we takeRemark 0.3. Theorem 0.1 can be derived from Theorem 0.2 as when r ≥ 1, {x ∈ Σ m ∶ E r (x) ≥ 0} = Σ m and α 0 (r, m) = 1. In fact we could have proved this general case directly. But the idea of the proof would turn out to be extremely difficult to follow.Corollary 0.4. In the case of m = 1, the asymptotic formula for the logarithm of the hole probability over a disc exists for all r > 0:α 0 (2 log r + 1 − log α 0 ),Because of the simplicity of one dimensional case, we can obtain more about the hole probability of SU (2) Gaussian random polynomials:Theorem 0.5. If U ⊂ C is a bounded simply connected domain containing 0 and ∂U is a Jordan curve. Let φ ∶ D(0, 1) → U be a biholomorphism given by the Riemmann mapping theorem such that φ(0) = 0(thus φ is unique up to the composition of a unitary transformation of C). Then the hole probability P 0,1 (U, N ) of SU (2) Gaussian random polynomials of degree N over U satisfiesAlso in dimension one, it makes sense to study the number of zeros in some set. So let a generalized hole probability P k,1 (r, N ) be the probability that an SU (2) Gaussian random polynomial of degree N has no more than k zeros in D(0, r), then the following theorem shows that asymptotic formula of log P k,1 (r, N ) exists: Theorem 0.6. For all k ≥ 0 and r > 0:where α 0 = α 0 (r, 1) ∈ (0, 1] is given in Theorem 0.2.We should remark here that in all the cases w...
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