Characteristic Polynomials for 1D Random Band Matrices from the Localization Side

Abstract

We study the special case of n × n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by J = (−W 2 △ + 1) −1 . Assuming that the band width W ≪ √ n, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as W, n → ∞) is equal to one, and so it does not coincides with those for GUE. This complements the result of [18] and proves the expected crossover for 1D Hermitian random band matrices at W ∼ √ n on the level of characteristic polynomials.

Evidence of the Poisson/Gaudin–Mehta phase transition for band matrices on global scales

“…if α > 1 2 [10]. These results, although consistent with Poisson/Gaudin-Mehta local statistics, do not imply it.…”

Universality for 1d Random Band Matrices: Sigma-Model Approximation

“…The sigma-model approximation for RBM was introduced by Efetov (see [7]), and the spins there are 4 × 4 matrices with both complex and Grassmann entries. As it was shown in [20], the mechanism of the crossover for the sigma-model is essentially the same as for the correlation functions of characteristic polynomials (see [19]), but the structure of the transfer operator for the sigma-model is more complicated: it is a 6 × 6 matrix kernel whose entries are kernels depending on two unitary 2 × 2 matrices U, U ′ and two hyperbolic 2 × 2 matrices S, S ′ . As it will be shown below, in the case of the second correlation function of (1.4) -(1.5) which is the main point of interest in this paper, the transfer operator K becomes 70 × 70 matrix whose elements are kernels defined on L 2 (U (2)) ⊗ L 2 (H + 2 L), where U (2) is 2 × 2 unitary group, H + 2 is a space of 2 × 2 positive hermitian matrices, and L = diag{1, −1}, and so the spectral analysis of K provides serious structural problems.…”

Random Band Matrices

“…Theorem 2.2 (Transition for characteristic polynomials [86,89]). For any E ∈ (−2, 2) and ε > 0, we have lim N →∞…”