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).

Voiculescu 的自由性在计算多项式的渐近谱时出现$氮\times 氮$特征空间位于通用位置的随机矩阵：它们使用统一的酉随机矩阵随机旋转$U_{氮}$。在本文中，我们通过提出一个随机矩阵模型来详细阐述之前的结果，我们将其命名为Vortex 模型，其中$U_{氮}$具有均匀酉随机矩阵定律，条件是留下不变的一个确定性向量$v_{氮}$。在极限内$氮\to +\mathrm{无穷大}$，我们证明$氮\times 氮$由矩阵随机旋转的矩阵$U_{氮}$对于归一化迹和状态向量是渐近条件自由的$v_{氮}$。我们定义了一个称为循环条件自由度的新概念，“统一”了三个独立性：无穷小自由度、循环单调独立性和循环布尔独立性。由于这种新的独立性，可以计算涡模型中的无穷小分布。最后，我们详细阐述了 Vortex 模型，以便构建随机矩阵模型$\alpha$- 自由和为了$\beta \gamma$-自由度（以前称为缩进独立性和有序自由度）。