The spectral edge of some random band matrices

Abstract

We study the asymptotic distribution of the eigenvalues of random Hermitian periodic band matrices, focusing on the spectral edges. The eigenvalues close to the edges converge in distribution to the Airy point process if (and only if) the band is sufficiently wide (W N 5=6 ). Otherwise, a different limiting distribution appears.

Random matrix theory in statistics: A review

“…Ramírez et al (2011) described such limit laws in terms of the eigenstates of a stochastic Airy operator introduced by Edelman and Sutton (2007) and Sutton (2005). Sodin (2010) eigenvalues can be expressed in terms of Fredholm determinants of integral operators (for relevant operator theory one may refer to Gohberg andKrein, 1969 or Reed andSimon, 1972) (2003) to extend the domain of Theorem 3.6 to the setting when p=n-0 as p; n-1. Paul (2011) used the technique of El Karoui (2003) to prove that under the latter setting, the scaling limits of the normalized smallest eigenvalues are reflected TracyWidom laws, which paralleled a result of Baker et al (1998) that showed that the reflection of F 2 about the origin is the scaling limit for the largest eigenvalue of the LUE.…”

Structured Random Matrices

“…Such models have interesting applications in combinatorics and computer science (see, for example, [1]), and specific examples such as random band matrices are of significant interest in mathematical physics (cf. [22]). The "structure" of the matrix is determined by its sparsity pattern.…”

Delocalization for a class of random block band matrices

“…is not difficult to check the leading order terms of R k−1 (ı, j) are of the form 24) with some p, q , q ∈ N such that 25) and the leading order terms of R k (ı, j) possess the same form (with R replaced by S). Every other term has at least 6 factors of h ab or h g ab or their conjugates, thus their sizes are typically controlled by M −3 (N η) −n , i.e.…”

“…In addition, p(ρ, τ , u) is the expansion of 25) where p (ρ, τ , u) is a polynomial of the components of ρ and τ with degree 2 , regarding u as fixed parameters. Now, keeping the leading order term of p(ρ, τ , u), and discarding the remainder terms, one can get the final estimate of the integral by taking the Gaussian integral over u, ρ and τ .…”

“…We remark that our discussion concerns the bulk of the spectrum; the transition at the spectral edge is much better understood. In [25] it was shown with moment method that the edge spectrum follows the Tracy-Widom distribution, characteristic to mean field model, for M N 5/6 , but it yields a different distribution for narrow bands, M N 5/6 . Delocalization is closely related to estimates on the diagonal elements of the resolvent G(z) = (H − z) −1 at spectral parameters with small imaginary part η = Imz.…”