Zeroes of random Reinhardt polynomials

Abstract

For a Reinhardt domain $\Omega$ with the smooth boundary in ${\mathbb{C}}^{m+1}$ and a positive smooth measure $\mu$ on the boundary of $\Omega$, we consider the ensemble $P_N$ of polynomials of degree $N$ with the Gaussian probability measure ${\gamma }_N$ which is induced by $L^2(\partial \Omega ,d\mu )$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\in \partial \Omega$. First of all we apply stationary phase method to the Boutet de Monvel-Sjöstrand theorem to get the asymptotic for the partial szegö kernel, $S_N(z,z)$, and then we compute the scaling limit partial szegö kernel in any direction in ${\mathbb{C}}^{m+1}$, then by using well-known Kac-Rice formula we compute scaling limit distribution function and scaling limit pair correlation function between zeros.

References in corpus (2)

• Universality and scaling of correlations between zeros on complex manifolds
• Correlations between zeros and supersymmetry