Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

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Consider real symmetric, complex Hermitian Toeplitz and real symmetric Hankel band matrix models, where the bandwidth bₙ but bₙ/N → b, b∈ [0,1] as N→ ∞. We prove that the distributions of eigenvalues converge weakly to universal, symmetric distributions γₜ(b) and γₕ(b). In the case b>0 or b=0 but with the addition of bₙ≥ C N1/2+ε₀ for some positive constants ε₀ 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, γₜ(0) is the standard Gaussian distribution and γₕ(0) is the distribution |x| (-x²). In addition, from the fourth moments we know that the γₜ(b)'s are different for different b's, the γₕ(b)'s different for different b∈ [0,1/2] and the γₕ(b)'s different for different b∈ [1/2,1].