Density of Eigenvalues of Random Normal Matrices with an Arbitrary Potential, and of Generalized Normal Matrices

Following the works by Wiegmann–Zabrodin, Elbau–Felder, Hedenmalm–Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic eigenvalue density of such matrices (when the size of the matrices goes to infinity) is related to the problem of Hele–Shaw flows on curved surfaces, considered by Entov and the first author in 1990-s. In the case when the potential function is the sum of a rotationally invariant function and the real part of a polynomial of the complex coordinate, we use this relation and the conformal mapping method developed by Entov and the first author to find the shape of the support domain explicitly (up to finitely many undetermined parameters, which are to be found from a finite system of equations). In the case when the rotationally invariant function is $\beta |z|^2$, this is done by Wiegmann–Zabrodin and Elbau–Felder. We apply our results to the generalized normal matrix model, which deals with random block matrices that give rise to $*$-representations of the deformed preprojective algebra of the affine quiver of type $\hat A_{m-1}$. We show that this model is equivalent to the usual normal matrix model in the large $N$ limit. Thus the conformal mapping method can be applied to find explicitly the support domain for the generalized normal matrix model.