On some high-dimensional limits of matricial stochastic processes seen from a quantum probability perspective

TL;DR

This paper extends previous results on the convergence of Brownian motions on unitary groups to a quantum Lévy process, now including orthogonal and symplectic groups in high-dimensional limits from a quantum probability perspective.

Contribution

It generalizes the convergence results of Brownian motions on various classical groups to a unified quantum Lévy process, covering orthogonal and symplectic groups.

Findings

Brownian motions on $O(nm)$ and $Sp(nm)$ converge to the same quantum Lévy process as on $U(nm)$

Block-wise convergence holds for orthogonal and symplectic groups in high dimensions

The results unify the behavior of different classical groups under quantum probability limits.

Abstract

We generalize the result of block-wise convergence of the Brownian motion on the unitary group $U(nm)$ towards a quantum L\'evy process on the unitary dual group $U\langle n\rangle$ when $m\rightarrow\infty$, obtained by the author in a previous paper, by showing that the Brownian motions on the orthogonal group $O(nm)$ and the symplectic group $Sp(nm)$ also converge block-wise to this same quantum L\'evy process.