Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Dang-Zheng Liu () and Zheng-Dong Wang
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Dang-Zheng Liu: Peking University
Zheng-Dong Wang: Peking University

Journal of Theoretical Probability, 2011, vol. 24, issue 4, 988-1001

Abstract: Abstract Consider real symmetric, complex Hermitian Toeplitz, and real symmetric Hankel band matrix models where the bandwidth b N →∞ but b N /N→b∈[0,1] as N→∞. We prove that the distributions of eigenvalues converge weakly to universal symmetric distributions γ T (b) and γ H (b). In the case b>0 or b=0 but with the addition of $b_{N}\geq CN^{\frac{1}{2}+\epsilon_{0}}$ for some positive constants ε 0 and C, we prove the almost sure convergence. The even moments of these distributions are the sums of some integrals related to certain pair partitions. In particular, when the bandwidth grows slowly, i.e., b=0, γ T (0) is the standard Gaussian distribution, and γ H (0) is the distribution |x|exp (−x 2). In addition, from the fourth moments, we know that γ T (b) are different for different b, γ H (b) different for different $b\in[0,\frac{1}{2}]$ , and γ H (b) different for different $b\in [\frac{1}{2},1]$ .

Keywords: Random matrix theory; Distribution of eigenvalues; Toeplitz band matrices; Hankel band matrices; 15A52 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-009-0260-4

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