One-cut solution of the β ensembles in the Zhukovsky variable

In this paper, we study in detail the modified topological recursion of the one-matrix model for arbitrary β in the one-cut case. We show that, for polynomial potentials, the recursion can be computed as a sum of residues. However, the main difference with the Hermitian matrix model is that the residues cannot be set at the branch points of the spectral curve but require the knowledge of the whole curve. In order to establish non-ambiguous formulae, we place ourselves in the context of the globalizing parameterization which is specific to the one-cut case (also known as Zhukovsky parameterization). This situation is particularly interesting for applications since in most cases the potentials of the matrix models only have one cut in string theory. Finally, the paper exhibits some numerical simulations of histograms of the limiting density of eigenvalues for different values of the parameter β