Classical and free infinitely divisible distributions and random matrices Annals of Probability
Classical and free infinitely divisible distributions and random matrices Annals of Probability
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经典和自由的无限可分分布和随机矩阵概率年鉴
DOI：
10.1214/009117904000000982
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
Florent Benaych
中科院分区：
文献类型：
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作者：
Florent Benaych
我们构造了经典分布和自由无限可分分布之间的双射Ψ的随机矩阵模型：对于任意d≥1，我们在d×d Hermite矩阵空间上相当自然地将每个*-无穷可分分布μa分布Pμd联系在一起，使得Pμd*Pνd=Pμ*νd.分布为Pμd的随机矩阵的谱分布在d趋于+Ψ(μ时以概率收敛于∞。此外，它还给出了GUE矩阵的谱分布几乎必然收敛的新证明和Marchenko-Pastur分布的投影模型。类似地，对于每一个d≥1，我们都与每个*-无穷可分分布μ相关联，即复(非埃尔米特)d×d随机矩阵空间上的一个分布Lμd。如果μ是对称的，则当Md是Lμd分布时，|MD|的谱分布的对称化按概率收敛到Ψ(μ)。
We construct a random matrix model for the bijection Ψ between classical and free infinitely divisible distributions: for every d ≥ 1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution P μ d on the space of d × d Hermitian matrices such that P μ d * P ν d = P μ*ν d . The spectral distribution of a random matrix with distribution P μ d converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko-Pastur distribution. In an analogous way, for every d ≥ 1, we associate to each *-infinitely divisible distribution μ, a distribution L μ d on the space of complex (non-Hermitian) d x d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |M d |, when M d is L μ d -distributed, converges in probability to Ψ(μ).