Quaternionic Second-Order Freeness and the Fluctuations of Large Symplectically Invariant Random Matrices

TL;DR

This paper introduces a new concept of second-order freeness for quaternionic matrices, demonstrating its validity for symplectically invariant matrices and highlighting unique asymmetries due to quaternionic trace properties.

Contribution

It defines second-order freeness in the quaternionic setting and shows its asymptotic validity for symplectically invariant matrices, differing from complex and real cases.

Findings

Quaternionic second-order freeness is asymptotically satisfied by symplectically invariant matrices.

The quaternionic trace's non-cyclic property causes asymmetries in multi-matrix interactions.

Distinct behavior from complex and real second-order freeness due to quaternionic trace properties.

Abstract

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behaviour of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.