Duality of real and quaternionic random matrices

Electronic Journal Probab. Vol. 14 (2009), Paper no. 17, pages 452--476 We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families.