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- W2950271390 abstract "We consider ensembles of random polynomials of the form $p(z)=sum_{j = 1}^N a_j P_j$ where ${a_j}$ are independent complex normal random variables and where ${P_j}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $Omega subset C$ relative to an analytic weight $rho(z) |dz|$. In the simplest case where $Omega$ is the unit disk and $rho=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform measure on $S^1$. We show that for any analytic $(Omega, rho)$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $partialOmega$ as $Ntoinfty$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(Omega, rho)$." @default.
- W2950271390 created "2019-06-27" @default.
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- W2950271390 date "2002-06-17" @default.
- W2950271390 modified "2023-09-23" @default.
- W2950271390 title "Equilibrium distribution of zeros of random polynomials" @default.
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