Diffusion Profile for Random Band Matrices: A Short Proof

Abstract

Let H be a Hermitian random matrix whose entries $H_{xy}$ are independent, centred random variables with variances $S_{xy} = {\mathbb {E}}|H_{xy}|^2$, where $x, y \in ({\mathbb {Z}}/L{\mathbb {Z}})^d$ and $d \geqslant 1$. The variance $S_{xy}$ is negligible if $|x - y|$ is bigger than the band width W. For $d = 1$ we prove that if $L \ll W^{1 + \frac{2}{7}}$ then the eigenvectors of H are delocalized and that an averaged version of $|G_{xy}(z)|^2$ exhibits a diffusive behaviour, where $G(z) = (H-z)^{-1}$ is the resolvent of H. This improves the previous assumption $L \ll W^{1 + \frac{1}{4}}$ of Erdős et al. (Commun Math Phys 323:367–416, 2013). In higher dimensions $d \geqslant 2$, we obtain similar results that improve the corresponding ones from Erdős et al. (Commun Math Phys 323:367–416, 2013). Our results hold for general variance profiles $S_{xy}$ and distributions of the entries $H_{xy}$. The proof is considerably simpler and shorter than that of Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013), Erdős et al. (Commun Math Phys 323:367–416, 2013). It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It is completely self-contained and avoids the intricate fluctuation averaging machinery from Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013).

Proof of Lemma 5.2

We recall that Lemma 5.2 basically coincides with Corollary 5.4 in [9]. Here we give a proof which does not rely on the averaging fluctuations estimate in [7].

We will assume that H is Hermitian with Gaussian entries and such that ${\mathbb {E}}H_{ij}^2 =0$, but the result holds also in the general complex case and the additional terms are treated as we saw in Sect. 8.

To get the desired bounds, we consider the expectation ${\mathcal {F}}_{ab} := {\mathbb {E}}|F_{ab}|^{2p}$ where $F_{ab}:=G_{ab} - \mathfrak m \, \delta _{ab}$ and p is an arbitrary strictly positive integer. The cumulant expansion yields

$$\begin{aligned} z {\mathcal {F}}_{ab}&= {\mathbb {E}}(zG_{ab}-z \mathfrak m \, \delta _{ab})F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} = {\mathbb {E}}\bigg (\sum _i H_{ai}G_{ib}-(1+z \mathfrak m) \, \delta _{ab}\bigg )F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} \\&= \mathfrak m^2 \delta _{ab}{\mathbb {E}}F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} - {\mathbb {E}}\sum _i S_{ai} \partial _{ia} G_{ib} F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} \\&= \mathfrak m^2 \delta _{ab} {\mathbb {E}}F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} - {\mathbb {E}}\sum _i S_{ai} G_{ii}G_{ab} F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} + {\mathbb {E}}\sum _i S_{ai} G_{ib} \partial _{ia}F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} \end{aligned}$$

where we used (2.10). Using the trivial identities $G_{ii} = G_{ii} - \mathfrak m + \mathfrak m$ and $G_{ab} = F_{ab} + \mathfrak m \delta _{ab}$, we get

$$\begin{aligned} (z + \mathfrak m) {\mathcal {F}}_{ab} =&- {\mathbb {E}}\sum _i S_{ai} (G_{ii}-\mathfrak m)F_{ab} F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} - \mathfrak m \delta _{ab} {\mathbb {E}}\sum _i S_{ai} (G_{ii}-\mathfrak m) F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} \\&+ {\mathbb {E}}\sum _i S_{ai} G_{ib} \partial _{ia}F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p}\,. \end{aligned}$$

Using (6.5), (2.10) and that $|F_{ab}| \leqslant \Lambda \prec \Psi$, one obtains

$$\begin{aligned} {\mathcal {F}}_{ab} = O_{\prec }(\Psi ^2 {\mathbb {E}}| F_{ab}|^{2p-1}) + O_{\prec }\bigg (\bigg \vert {\mathbb {E}}\sum _i S_{ai} G_{ib} \partial _{ia}F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p} \bigg \vert \bigg ). \end{aligned}$$

(A.1)

Let us examine the second term on the right hand side of (A.1): we note that

$$\begin{aligned} \partial _{ia} F_{ab} = - G_{ai}G_{ab}, \ \ \ \partial _{ia} \overline{F} \!\,_{ab} = - \overline{G} \!\,_{aa} \overline{G} \!\,_{ib}, \end{aligned}$$

therefore (A.1) becomes

$$\begin{aligned} {\mathcal {F}}_{ab} =&\, O_{\prec }(\Psi ^2 {\mathbb {E}}| F_{ab}|^{2p-1}) + O_{\prec }\bigg (\bigg \vert {\mathbb {E}}\sum _i S_{ai} G_{ib} G_{ai}G_{ab}F_{ab}^{p-2} \overline{F} \!\,_{ab}^{p} \bigg \vert \bigg ) \nonumber \\&+ O_{\prec }\bigg (\bigg \vert {\mathbb {E}}\sum _i S_{ai} | G_{ib}|^2 \overline{G} \!\,_{aa} F_{ab}^{p-1} \overline{F} \!\,_{ab}^{p-1} \bigg \vert \bigg ). \end{aligned}$$

(A.2)

In the second term on the right hand side of (A.2), when $a \ne b$ we can use (7.11) so that

$$\begin{aligned} {\mathcal {F}}_{ab} \prec \Psi ^2 {\mathbb {E}}| F_{ab}|^{2p-1} + {\mathbb {E}}(\Psi ^4 + T_{ab})| F_{ab}|^{2p-2}, \end{aligned}$$

while, when $a = b$, the trivial inequality $|xy| \leqslant (|x|^2 + |y|^2)/2$ yields

$$\begin{aligned} \sum _i S_{ai}G_{ib} G_{ai} \prec \sum _i S_{ai}(|G_{ai}|^2 + |G_{ia}|^2) = T_{aa} + T_{aa}', \end{aligned}$$

so that

$$\begin{aligned} {\mathcal {F}}_{aa} \prec \Psi ^2 {\mathbb {E}}| F_{aa}|^{2p-1} + {\mathbb {E}}(T_{aa} + T'_{aa})| F_{aa}|^{2p-2}. \end{aligned}$$

Thus, we finally have

$$\begin{aligned} {\mathcal {F}}_{ab}&\prec \Psi ^2 {\mathbb {E}}| F_{ab}|^{2p-1} + (\Psi ^4 + \Omega _{ab}){\mathbb {E}}| F_{ab}|^{2p-2} \prec (\Psi ^2 + \Omega _{ab}){\mathbb {E}}| F_{ab}|^{2p-1} \\&\quad + (\Psi ^4 + \Omega ^2_{ab}) {\mathbb {E}}| F_{ab}|^{2p-2}. \end{aligned}$$

Lemma 6.3 completes the proof.

Remark A.1

The estimate in Lemma 5.2 cannot be improved in the sense that $F_{ab}$ will always be bounded at least by $T_{ab} = \sum _i S_{ai}|G_{ib}|^2$ because $T_{ab}$ appears explicitly in the third term on the right hand side of (A.2).