Asymptotic cyclic-conditional freeness of random matrices

TL;DR

This paper introduces the Vortex model, a new random matrix framework demonstrating asymptotic cyclic-conditional freeness, and develops cyclic-conditional freeness as a unifying concept for second-order asymptotics in free probability.

Contribution

It proposes the Vortex model with conditioned unitary invariance and introduces cyclic-conditional freeness, unifying various notions of independence in free probability.

Findings

Vortex model exhibits asymptotic cyclic-conditional freeness.

Cyclic-conditional freeness unifies infinitesimal, cyclic-monotone, and cyclic-Boolean independence.

Infinitesimal distribution in the Vortex model can be explicitly computed.

Abstract

Voiculescu's freeness emerges in computing the asymptotic of spectra of polynomials on $N\times N$ random matrices with eigenspaces in generic positions: they are randomly rotated with a uniform unitary random matrix $U_N$. In this article we elaborate on the previous point by proposing a random matrix model, which we name the Vortex model, where $U_N$ has the law of a uniform unitary random matrix conditioned to leave invariant one deterministic vector $v_N$. In the limit $N \to +\infty$, we show that $N\times N$ matrices randomly rotated by the matrix $U_N$ are asymptotically conditionally free with respect to the normalized trace and the state vector $v_N$. To describe second order asymptotics, we define cyclic-conditional freeness, a new notion of independence unifying infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. The infinitesimal distribution in the Vortex model can be computed thanks to this new independence. Finally, we elaborate on the Vortex model in order to build random matrix models for ordered freeness and for indented independence.