# Freeness and The Partial Transposes of Wishart Random Matrices

## TL;DR

This paper demonstrates that partial transposes of complex Wishart matrices become asymptotically free, explores regimes with fixed block numbers, and examines real Wishart matrices, revealing new insights into their operator-level freeness.

## Contribution

It introduces the asymptotic freeness of partial transposes of Wishart matrices and analyzes different regimes, including real matrices, at the operator level.

## Key findings

- Partial transposes of complex Wishart matrices are asymptotically free.
- Freeness persists when the number of blocks is fixed and block size increases.
- Real Wishart matrices also exhibit similar freeness properties.

## Abstract

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.