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-Sj\"{o}strand theorem to get the asymptotic for the partial szeg\"{o} kernel, $S_{N}(z,z)$, and then we compute the scaling limit partial szeg\"{o} 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.