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

TL;DR

This paper analyzes the modified topological recursion for the one matrix model with arbitrary beta in the one cut case, providing residue-based formulas and numerical eigenvalue density simulations relevant to string theory applications.

Contribution

It introduces residue computations for the modified topological recursion in the one cut case using Zhukovsky parametrization, applicable to arbitrary beta.

Findings

Residue formulas depend on the entire spectral curve, not just branchpoints.

Numerical simulations illustrate eigenvalue density distributions for various beta values.

The approach is relevant for string theory models with single-cut potentials.

Abstract

In this article, we study in detail the modified topological recursion of the one matrix model for arbitrary $\beta$ 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 branchpoints of the spectral curve but require the knowledge of the whole curve. In order to establish non-ambiguous formulas, we place ourselves in the context of the globalizing parametrization which is specific to the one cut case (also known as Zhukovsky parametrization). 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 article exhibits some numeric simulations of histograms of limiting density of eigenvalues for different values of the parameter $\beta$.