Asymptotic cyclic-conditional freeness of random matrices

Abstract

Voiculescu’s freeness emerges when computing the asymptotic 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 paper, we elaborate on the previous result 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 $v_N$. We define a new concept called cyclic-conditional freeness “unifying” three independences: infinitesimal freeness, cyclic-monotone independence and cyclic-Boolean independence. Infinitesimal distributions 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 $\alpha$-freeness and for $\beta \gamma$-freeness (formerly named indented independence and ordered freeness).