Random Matrix Pencils, Branching Points, and Monodromy

We define complex projective n-space CP as C \ {0} under the equivalence relation (z0 : z1 : ... : zn) ∼ (cz0 : cz1 : ... : czn) where c ∈ C\{0}. A complex projective curve is then the set of points (a0 : a1 : ... : an) ∈ CP such that p(a0, a1, ..., an) = 0 for a fixed complex homogeneous polynomial p(z0, z1, ..., zn). Such a curve is a 2-dimensional, real, orientable Riemannian manifold, or Riemann surface. For a non-constant holomorphic map f : X → Y between compact, connected Riemann surfaces X and Y , f is locally a covering map. The preimage f−1(p) of a point p ∈ Y generally consists of d distinct points, where d is the degree of the map f . If the preimage f−1(p) consists of fewer than d distinct points, p is called a branching point of the map f . In appropriate local coordinates at each point x ∈ f−1(p), the map f looks like z 7→ z for some positive integer k ∈ Z called the ramification index of x. A branching point p is called simple if some point x ∈ f−1(p) has ramification index 2 and the rest have index 1. Let br(f) be the set of branching points of f and fix a base point y ∈ Y which is not a branching point. Each loop in Y \ br(f) with base point y lifts to d distinct paths in X with endpoints in f−1(y), and thus we get a group homomorphism from the fundamental group π1(Y \ br(f)) to the symmetric group Sf−1(y) ' Sd called the monodromy. The image of a simple branching point is a transposition, and the image of br(f) is called the monodromy group. For complex n × n matrices A and B, define the characteristic polynomial χ(t, λ) = det(A + tB + λI). By considering the homogenized polynomial χ̃(t, λ, z) = zχ(t/z, λ/z), let R be the zero set of χ in CP. Let f : R→ CP be the projection onto the t-coordinate. The branching points of this map are precisely the values of t such that A+ tB has repeated eigenvalues. In this project, we studied the distribution of the branching points of this map where A and B are random matrices sampled from the Gaussian Orthogonal Ensemble GOEn or from the Gaussian Unitary Ensemble GUEn and investigate the monodromy of each.