通信量分布和独立性：置换不变随机矩阵和独立性的三个概念
Traffic distributions and independence: permutation invariant random matrices and the three notions of independence
摘要：
Voiculescu的渐近自由独立的概念是已知的一个大类的随机矩阵，包括独立的酉不变矩阵。通过置换矩阵的共轭，将这一概念推广到规律不变的独立随机矩阵。这一事实自然导致了自由概率的扩展，在流量概率的概念下形式化。我们首先建立随机矩阵的这种结构。我们定义了随机矩阵的流量分布，它比自由概率的 *-分布更丰富。独立置换不变矩阵族的个体流量分布的知识足以计算连接族的极限分布。在因子分解假设下，我们称流量独立性为流量分布独立的渐近规则。Wigner矩阵、Haar酉矩阵和均匀置换矩阵在流量分布中收敛，这一事实对我们可以构造的几个矩阵的极限 *-分布产生了新的结果。然后定义抽象交通空间为具有更强结构的非交换概率空间。我们证明，在代数水平上，交通的独立性在某种意义上统一了三个典型的概念张量，自由和布尔独立。中心极限定理是在这种情况下，张量之间的插值，自由和布尔中心极限定理。
英文摘要：
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the *-distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the join family. Under a factorization assumption, we call traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting *-distributions of several matrices we can construct from them. Then we define the abstract traffic spaces as non commutative probability spaces with more structure. We prove that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.
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