Zeroes of random Reinhardt polynomials

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

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