On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

Abstract

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of 1D Gaussian band matrices, i.e. of the Hermitian N × N matrices H N with independent Gaussian entries such that H ij H lk = δ ik δ jl J ij , where J = (−W 2 △ + 1) −1 . Assuming that W 2 = N 1+θ , 0 < θ ≤ 1, we show that the moment's asymptotic behavior (as N → ∞) in the bulk of the spectrum coincides with that for the Gaussian Unitary Ensemble.

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

“…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. …”

“…Moreover, correlation functions of characteristic polynomials are expected to exhibit a crossover which is similar to that of local eigenvalue statistics. In particular, for 1D RBM they are expected to have the same local behaviour as for GUE for W ≫ √ n, and the different behaviour for W ≪ √ n. The first part of this conjecture was proved in [18]. The main result of [18] is Theorem 1.1 ( [18]) For the 1D RBM of (1.1) -(1.2) with W 2 = n 1+θ , where 0 < θ ≤ 1, we have 6) i.e.…”

On Fluctuations of Eigenvalues of Random Band Matrices

“…Till now this result is not proven rigorously, but the problem is one of the most challenging in the random matrix theory (see, e.g. [19], [3], [4], [18] and references therein).…”

Delocalization for a class of random block band matrices

“…Localization has been shown for M N 1/8 in [23], while delocalization in a certain weak sense for the most eigenvectors was proven for M N 4/5 in [11]. Interestingly, for a special Gaussian model even the sine kernel behavior of the 2-point correlation function of the characteristic polynomials could be proven down to the optimal band width M N 1/2 , see [19,21]. Note that the sine kernel is consistent with the delocalization but does not imply it.…”

“…Second, we assume that the distribution of the matrix elements matches a Gaussian up to four moments in the spirit of [28]. Supersymmetry heavily uses Gaussian integrations, in fact all mathematically rigorous works on random band matrices with supersymmetric method assume that the matrix elements are Gaussian, see [4][5][6][19][20][21]26,27]. The Green's function comparison method [14] allows one to compare Green's functions of two matrix ensembles provided that the distributions match up to four moments and provided that G ii are bounded.…”