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摘要
Consider real symmetric, complex Hermitian Toeplitz, and real symmetric Hankel band matrix models where the bandwidth b (N) -> a but b (N) /N -> ba[0,1] as N -> a. We prove that the distributions of eigenvalues converge weakly to universal symmetric distributions gamma (T) (b) and gamma (H) (b). In the case b > 0 or b=0 but with the addition of b(N) >= CN(1/2+epsilon 0) for some positive constants epsilon(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, gamma (T) (0) is the standard Gaussian distribution, and gamma (H) (0) is the distribution |x|exp (-x (2)). In addition, from the fourth moments, we know that gamma (T) (b) are different for different b, gamma (H) (b) different for different b is an element of [0, 1/2], and gamma(H)(b) different for different b is an element of [1/2, 1].
- 出版日期2011-12
- 单位北京大学
