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 →∞ 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 $b\in [\frac{1}{2},1]$ .