Zero distribution of complex orthogonal polynomials with respect to exponential weights

We study the limiting zero distribution of orthogonal polynomials with respect to some particular exponential weights e−nV(z)e−nV(z) along contours in the complex plane. We are especially interested in the question under which circumstances the zeros of the orthogonal polynomials accumulate on a single analytic arc (one cut case), and in which cases they do not. In a family of cubic polynomial potentials V(z)=−iz33+iKz, we determine the precise values of KK for which we have the one cut case. We also prove the one cut case for a monomial quintic V(z)=−iz55 on a contour that is symmetric in the imaginary axis.