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4. S-curves

    1. S-curves for complex potentials in the scalar case


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      Problem 4.1.

      [A. Kuijlaars] Analysis of existence and properties of S-curves in the scalar case when the potential is \phi(z)=\textrm{Re}\, z^p, for p\geq 3, and we take \tau as the set of continua that join neighboring valleys of \phi(z) (where \phi(z)\to-\infty) in the complex plane. In particular, prove that the support of the equilibrium measure on a curve with the S-property is one-cut case, or give bounds on the number of cuts depending on p.

        • Numerical computation of S-curves


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          Problem 4.2.

          [M. Bertola] A general question about numerical methods for vector equilibrium problems and computation of corresponding S-curves. More in particular, how to produce critical trajectories in a reliable way, given the spectral curve of the problem?

            • S-curves with piecewise harmonic external field


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              Problem 4.3.

              [E. Rakhmanov] Consider the external potential \phi(z)=\begin{cases} -k_1\, \textrm{Re}\, z, & \textrm{Re}\, z>0,\\ k_2\, \textrm{Re}\, z, & \textrm{Re}\, z<0, \end{cases}
              where k_1,k_2>0. Is there a continuum with the S-property connecting the points z_1=-\varepsilon_1+i and z_2=\varepsilon_2-i in the complex plane, for small positive \varepsilon_1 and \varepsilon_2? Also, consider the max-min problem in this setting. Does the curve obtained have a portion along the imaginary axis?

                • Green’s potential


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                  Problem 4.4.

                  [A. Martínez-Finkelshtein] Existence theory for S-curves in a general sense, if the logarithmic potential is replaced by Green’s potential.

                      Cite this as: AimPL: Vector equilibrium problems and random matrix models, available at http://aimpl.org/vectorequilib.