Duality of real and quaternionic random matrices

TL;DR

This paper explores the duality between real and quaternionic Gaussian random matrices, extending Wick's formula through graphical enumeration, and reveals a deep connection between their moments.

Contribution

It introduces a generalized Wick formula for quaternionic Gaussian variables and demonstrates a duality in moments between quaternionic and real random matrix families.

Findings

Quaternionic Gaussian variables satisfy a generalized Wick formula.

Duality between moments of quaternionic and real Wigner and Wishart matrices.

Graphical enumeration characterizes the expected values of products.

Abstract

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.