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This work concerns superharmonic perturbations of a Gaussian measure given bya special class of positive weights in the complex plane of the form $w(z) =\exp(-|z|^2 + U^{\mu}(z))$, where $U^{\mu}(z)$ is the logarithmic potential ofa compactly supported positive measure $\mu$. The equilibrium measure of thecorresponding weighted energy problem is shown to be supported on subharmonicgeneralized quadrature domains for a large class of perturbing potentials$U^{\mu}(z)$. It is also shown that the $2\times 2$ matrix d-bar problem fororthogonal polynomials with respect to such weights is well-defined and has aunique solution given explicitly by Cauchy transforms. Numerical evidence ispresented supporting a conjectured relation between the asymptotic distributionof the zeroes of the orthogonal polynomials in a semi-classical scaling limitand the Schwarz function of the curve bounding the support of the equilibriummeasure, extending the previously studied case of harmonic polynomialperturbations with weights $w(z)$ supported on a compact domain.

Superharmonic Perturbations of a Gaussian Measure, Equilibrium Measures and Orthogonal Polynomials