Planar orthogonal polynomials and two dimensional Coulomb gases

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Abstract

In this thesis we focus on the relation between random matrix theory and orthogonal polynomial theory in the complex plane. It is well known that even if the entries of a random matrix are independent, the eigenvalues will be highly correlated. This corre- lation, which is a pairwise logarithmic repulsion between the eigenvalues, leads one to think that the eigenvalues of a random matrix behave like particles in a Coulomb gas, since the logarithmic repulsion is the Coulomb interaction in two dimensions.<br />
We consider the case when the particles are confined to an ellipse in the plane. At inverse temperature beta equal 2, we introduce new families of exactly solvable two-dimensional Coulomb gases for a fixed and finite number of particles N. We find, in the analysis of local fluctuations in the weak non-Hermiticity limit of the correlation functions, old and new universality classes. This is achieved by showing that certain subfamilies of Jacobi polynomials extend to orthogonality relations over a weighted ellipse in the plane.

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