Number Variance of Random Zeros
نویسندگان
چکیده
The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree N in an open set U ⊂ C as the degree N → ∞, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher dimensions, we give similar formulas for the variance of the volume inside a domain U of the zero hypersurface of a random holomorphic section of a high power of a positive line bundle over any compact Kähler manifold. These results generalize the variance asymptotics of Sodin and Tsirelson for special model ensembles of chaotic analytic functions in one variable to any ample line bundle and Riemann surface. We also combine our methods with those of Sodin-Tsirelson to generalize their asymptotic normality results for smoothed number statistics.
منابع مشابه
Domain of attraction of normal law and zeros of random polynomials
Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...
متن کاملNumber Variance of Random Zeros on Complex Manifolds
We show that the variance of the number of simultaneous zeros ofm i.i.d. Gaussian random polynomials of degree N in an open set U ⊂ C with smooth boundary is asymptotic to N νmm Vol(∂U), where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on C. Our res...
متن کاملMeta-analysis of incidence rate data in the presence of zero events
BACKGROUND When summary results from studies of counts of events in time contain zeros, the study-specific incidence rate ratio (IRR) and its standard error cannot be calculated because the log of zero is undefined. This poses problems for the widely used inverse-variance method that weights the study-specific IRRs to generate a pooled estimate. METHODS We conducted a simulation study to comp...
متن کامل