Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Consider real symmetric, complex Hermitian Toeplitz and real symmetric Hankel band matrix models, where the bandwidth $b_{N}\ra \iy$ but $b_{N}/N \to b$, $b\in [0,1]$ as $N\to \infty$. 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 $b_{N}\geq C N^{{1/2}+\epsilon_{0}}$ for some positive constants $\epsilon_{0}$ and $C$, we prove almost sure convergence. The even moments of these distributions are the sum 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 the $\gamma_{_{T}}(b)$'s are different for different $b$'s, the $\gamma_{_{H}}(b)$'s different for different $b\in [0,{1/2}]$ and the $\gamma_{_{H}}(b)$'s different for different $b\in [{1/2},1]$.