Paper detail

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

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