Explicit Multi-Matrix Topological Expansion for Quaternionic Random Matrices

TL;DR

This paper develops an explicit topological expansion formula for expected values of products of quaternionic random matrices, incorporating trace and real part functions, with algorithms for expressing these in terms of index contractions and matrix cumulants.

Contribution

It introduces a novel topological expansion with dual topologies for quaternionic matrices and provides algorithms and formulas for their expected values.

Findings

Derived explicit formulas for quaternionic matrix expectations.

Computed matrix cumulants for key quaternionic ensembles.

Provided algorithms for expressing complex expectations in terms of index contractions.

Abstract

We present an explicit formula for the expected value of a product of several independent symplectically invariant matrices in which the trace and real part function may be applied, possibly to different subexpressions. This takes the form of a topological expansion; however, each term has two topologies: one for the trace, and another for the real part. The traces and real parts can always be written in terms of index contraction, but in some cases, it is possible to write the expression as a product in which the two functions are applied to bracketed intervals in a legal bracket diagram. We present the conditions under which this may be done, and an algorithm to construct such an expression given the contracted indices when possible. The summands in the topological expansion are written in terms of matrix cumulants. We compute the matrix cumulants of quaternionic Ginibre, Gaussian symplectic, quaternionic Wishart, and Haar-distributed symplectic matrices, which allow direct computation of an expression constructed from several independent ensembles of any of these matrices.