Equilibrium distribution of zeros of random polynomials

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 $\mathrm{\Omega }\subset C$ relative to an analytic weight $\rho (z)\mathrm{\mid }dz\mathrm{\mid }$. In the simplest case where $\mathrm{\Omega }$ 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 $(\mathrm{\Omega },\rho )$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $\mathrm{\partial }\mathrm{\Omega }$ as $N\to \mathrm{\infty }$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(\mathrm{\Omega },\rho )$.