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We extend our recent study of winding number density statistics in Gaussianrandom matrix ensembles of the chiral unitary (AIII) and chiral symplectic(CII) classes. Here, we consider the chiral orthogonal (BDI) case which is themathematically most demanding one. The key observation is that we can map thetopological problem on a spectral one, rendering the toolbox of random matrixtheory applicable. In particular, we employ a technique that exploitssupersymmetry structures without reformulating the problem in superspace.

Winding Number Statistics for Chiral Random Matrices: Averaging Ratios of Parametric Determinants in the Orthogonal Case