One-cut solution of the $\beta$-ensembles in the Zhukovsky variable

摘要

In this article, we study in detail the modified topological recursion of theone matrix model for arbitrary $\beta$ in the one cut case. We show that forpolynomial potentials, the recursion can be computed as a sum of residues.However the main difference with the hermitian matrix model is that theresidues cannot be set at the branchpoints of the spectral curve but requirethe knowledge of the whole curve. In order to establish non-ambiguous formulas,we place ourselves in the context of the globalizing parametrization which isspecific to the one cut case (also known as Zhukovsky parametrization). Thissituation is particularly interesting for applications since in most cases thepotentials of the matrix models only have one cut in string theory. Finally,the article exhibits some numeric simulations of histograms of limiting densityof eigenvalues for different values of the parameter $\beta$.