Non-crossing Annular Pairings and The Infinitesimal Distribution of the GOE

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number of typographical errors; I am grateful to the readers who brought these to my attention.