that although we assume the discriminant $\Delta$ in Theorem \[thm:main1\] is negative, which means that the Riemann surface (\[eq:curve20\]) has 2 real and 2 complex branch points, it is unclear which Riemann sheet do these branch points belong to and these different situations will result in different sheet structures of the Riemann surface as indicated in Figure \[fig:struc\].

\[\]\[\]\[1\]\[0.0\][$\xi_1$]{} \[\]\[\]\[1\]\[0.0\][$\xi_2$]{} \[\]\[\]\[1\]\[0.0\][$\xi_3$]{} ![Possible sheet structures of the Riemann surface. For the implementation of the Riemann-Hilbert analysis, we need to show that the Riemann surface has the sheet structure shown on the right hand side.[]{data-label="fig:struc"}](struc2 "fig:")

In order to determine the sheet structure of the Riemann surface, we need to analyze the analyticity of the solutions $\xi_j$ in the vicinity of the points $\lambda_k$ in Theorem \[thm:side1\]. This requires the type of analysis used in [@Mc], which is very difficult to carry out in our case. As in [@Mo], we overcome these difficulties by showing that the Stieltjes transform of the limiting eigenvalue distribution satisfies (\[eq:curve20\]). Then by using properties of the Stieltjes transform obtained in [@BS95], [@BS98], [@BS99], [@CS], [@S95], we were able to determine the sheet structure of the Riemann surface (\[eq:curve20\]).

The determination of the topology of the set 