Zeroes of random Reinhardt polynomials

For a Reinhardt domain $\mathrm{\Omega }$ with the smooth boundary in ${\mathbb{C}}^{m+1}$ and a positive smooth measure $\mu$ on the boundary of $\mathrm{\Omega }$, we consider the ensemble $P_N$ of polynomials of degree $N$ with the Gaussian probability measure ${\gamma }_N$ which is induced by $L^2(\mathrm{\partial }\mathrm{\Omega },d\mu )$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\in \mathrm{\partial }\mathrm{\Omega }$. 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.