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Register a new userDelocalization for a class of random block band matrices
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We consider $Ntimes N$ Hermitian random matrices $H$ consisting of blocks of size $Mgeq N^{6/7}$. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width $M$. We show that the entries of the Greens function $G(z)=(H-z)^{-1}$ satisfy the local semicircle law with spectral parameter $z=E+mathbf{i}eta$ down to the real axis for any $eta gg N^{-1}$, using a combination of the supersymmetry method inspired by cite{Sh2014} and the Greens function comparison strategy. Previous estimates were valid only for $etagg M^{-1}$. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
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We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, $Wsim N$. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $s_{xy}=mathbb E|h_{xy}|^2$. The variance matrix $S=(s_{xy})$ has a banded structure so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $dge 8$, we prove that, as long as $Wge L^epsilon$ for a small constant $epsilon>0$, with high probability most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Denote by $G(z)=(H-z)^{-1}$ the Greens function of the band matrix. For ${mathrm Im}, zgg W^2/L^2$, we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of $mathbb E|G_{xy}(z)|^2$ with respect to $x-y$ is of the form $({mathrm Im}, z + a(p))^{-1}$ for some $a(p)$ quadratic in $p$, where $p$ is the Fourier variable. Our method is based on an expansion of $T_{xy}=|m|^2 sum_{alpha}s_{xalpha}|G_{alpha y}|^2$ and it requires a self-energy renormalization up to error $W^{-K}$ for any large constant $K$ independent of $W$ and $L$. We expect that this method can be extended to non-Gaussian band matrices.
Recently, T. and M. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of band matrices by Constantinescu. Oth
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.
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