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Register a new userDelocalization and quantum diffusion of random band matrices in high dimensions II: $T$-expansion
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We consider Greens functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(mathbb Z/Lmathbb Z)^d$. The entries $h_{xy}=overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=mathbb E|h_{xy}|^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any $nin mathbb N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 sum_{alpha}s_{xalpha}|G_{alpha y}|^2$, with an error $O(W^{-nd/2})$, and use it to prove a local law on the Greens function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $dge 8$ in part I of this series.
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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.
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.
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.
This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $Ntimes N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1ll Wll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=left(z 1_{1leq i leq W}+widetilde z 1_{ i > W} right) delta_{ij}$, with two distinct spectral parameters $zin mathbb C_+:={zin mathbb C:{rm Im} z>0}$ and $widetilde zin mathbb C_+cup mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $Wgg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in cite{PartIII}.
Consider $Ntimes N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W geq N^{3/4+varepsilon}$ for any $varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+varepsilon}$ for any $varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${rm L}^infty$ norms are all simultaneously bounded by $N^{-frac{1}{2}+varepsilon}$ (after normalization in ${rm L}^2$) with overwhelming probability, for any $varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method cite{BouErdYauYin2017}, which required $W=Omega(N)$, to the current setting $W ge N^{3/4+varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W geq N^{3/4+varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation cite{BouYau2017} under the matrix Brownian motions.
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