Matrix valued orthogonality and S-contours on a Riemann surface

I will discuss matrix valued orthogonal polynomials as they appear in certain random tiling problems with periodic weights. The matrix valued orthogonality can be viewed as scalar orthogonality for meromorphic functions on a Riemann surface. The limiting location of the zeros, as the degree tends to infinity, is given by a contour with the S-property, which is a notion that has to be properly defined on a higher genus Riemann surface.