| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

4. $S$-curves

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

      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

          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

              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

                  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.