Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic   Feynman Graphs
  We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality between quaternionic self-adjoint and real symmetric matrix integrals. The asymptotic expansions of these integrals are given in terms of summations over topologies of compact surfaces, both orientable and non-orientable, for all genera and an arbitrary positive number of marked points on them. We show that the Gaussian Orthogonal Ensemble (GOE) and Gaussian Symplectic Ensemble (GSE) have exactly the same graphical expansion term by term (when appropriately normalized),except that the contributions from non-orientable surfaces with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials. Indeed, we show that this duality is equivalent to Poincare duality of graphs drawn on a compact surface. Another application of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble. 
Motohico Mulase and Andrew Waldron