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Random Matrices: Theory and Applications cover
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About the Journal

Volume 03, Issue 02 (April 2014)

Research Articles
No Access
Universality of local eigenvalue statistics in random matrices with external source
  • 1450005

https://doi.org/10.1142/S2010326314500051

Preview Abstract

Consider a random matrix of the form formula, where Mn is a Wigner matrix and Dn is a real deterministic diagonal matrix (Dn is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of Wn for a general class of Wigner matrices Mn and diagonal matrices Dn. Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones. The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

Research Articles
No Access
Bridges and random truncations of random matrices
  • 1450006

https://doi.org/10.1142/S2010326314500063

Preview Abstract

Let U be a Haar distributed matrix in 𝕌(n) or 𝕆(n). In a previous paper, we proved that after centering, the two-parameter process

formula
converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of U by a random one, in which each row (respectively, column) is chosen with probability s (respectively, t) independently. We prove that the corresponding two-parameter process, after centering and normalization by n-1/2 converges to a Gaussian process. On the way we meet other interesting convergences.

Research Articles
No Access
Bulk behavior of Schur–Hadamard products of symmetric random matrices
  • 1450007

https://doi.org/10.1142/S2010326314500075

Preview Abstract

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur–Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSD of Schur–Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur–Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.

Research Articles
No Access
On the largest eigenvalue of products from the β-Laguerre ensemble
  • 1450008

https://doi.org/10.1142/S2010326314500087

Preview Abstract

We determine the limiting distribution of the largest eigenvalue of products from the β-Laguerre ensemble. This limiting distribution is given by a Tracy–Widom law with parameter β0 > 0 depending on the ratio of the parameters of the two matrices involved.

Research Articles
No Access
Eigenvalue distributions of beta-Wishart matrices
  • 1450009

https://doi.org/10.1142/S2010326314500099

Preview Abstract

We derive explicit expressions for the distributions of the extreme eigenvalues of the beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0.

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