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Delocalization and quantum diffusion of random band matrices in high dimensions II: $T$-expansion

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 نشر من قبل Fan Yang
 تاريخ النشر 2021
  مجال البحث
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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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