The limiting distributions of large heavy Wigner and arbitrary random matrices

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• 作者：Camille Male ^{camille.male@math.u-bordeaux.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Random matrices ; Free probability ; Asymptotic freeness ; Wigner matrices
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：272
• 期：1
• 页码：1-46
• 全文大小：976 K

文摘

A heavy Wigner matrix X_N$X_N$ is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of $\sqrt{N}{X}_N$ tend to infinity with N  , as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family X_N${\mathbf{X}}_N$ of independent heavy Wigner matrices and an independent family Y_N${\mathbf{Y}}_N$ of arbitrary random matrices with a bound condition and converging in ^⁎-distribution in the sense of free probability. We characterize the possible limiting joint ^⁎-distributions of (X_N,Y_N)$({\mathbf{X}}_N,{\mathbf{Y}}_N)$, giving explicit formulas for joint ^⁎-moments. We find that they depend on more than the ^⁎-distribution of Y_N${\mathbf{Y}}_N$ and that in general X_N${\mathbf{X}}_N$ and Y_N${\mathbf{Y}}_N$ are not asymptotically ^⁎-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on Y_N${\mathbf{Y}}_N$ and describe the limiting ^⁎-distribution of (X_N,Y_N)$({\mathbf{X}}_N,{\mathbf{Y}}_N)$. We develop this approach for related models and give recurrence relations for the limiting ^⁎-distribution of heavy Wigner and independent diagonal matrices.