We consider a general class of symmetric or Hermitian random band matrices H=(hxy)x,y∈〚1,N〛d\documentclass[12pt]{minimal}
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\begin{document}$$H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket ^d}$$\end{document} in any dimension d≥1\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 1$$\end{document}, where the entries are independent, centered random variables with variances sxy=E|hxy|2\documentclass[12pt]{minimal}
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\begin{document}$$s_{xy}=\mathbb {E}|h_{xy}|^2$$\end{document}. We assume that sxy\documentclass[12pt]{minimal}
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\begin{document}$$s_{xy}$$\end{document} vanishes if |x-y|\documentclass[12pt]{minimal}
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\begin{document}$$|x-y|$$\end{document} exceeds the band width W, and we are interested in the mesoscopic scale with 1≪W≪N\documentclass[12pt]{minimal}
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\begin{document}$$1\ll W\ll N$$\end{document}. Define the generalized resolvent of H as G(H,Z):=(H-Z)-1\documentclass[12pt]{minimal}
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\begin{document}$$G(H,Z):=(H - Z)^{-1}$$\end{document}, where Z is a deterministic diagonal matrix with entries Zxx∈C+\documentclass[12pt]{minimal}
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\begin{document}$$Z_{xx}\in \mathbb {C}_+$$\end{document} for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions d≥2\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 2$$\end{document}. More precisely, for any fixed d≥2\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 2$$\end{document}, we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if N≤W1+d2\documentclass[12pt]{minimal}
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\begin{document}$$N\le W^{1+\frac{d}{2}}$$\end{document}. This improves the corresponding results in He and Marcozzi (Diffusion profile for random band matrices: a short proof, 2018. arXiv:1804.09446) that imposed the assumption N≪W1+dd+1\documentclass[12pt]{minimal}
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\begin{document}$$N\ll W^{1+\frac{d}{d+1}}$$\end{document}, and the results in Erdős and Knowles (Ann Henri Poincaré12(7):1227–1319, 2011; Commun Math Phys 303(2): 509–554, 2011) that imposed the assumption N≪W1+d6\documentclass[12pt]{minimal}
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\begin{document}$$N\ll W^{1+\frac{d}{6}}$$\end{document}. For 1D random band matrices, our fluctuation averaging result was used in Bourgade et al. (J Stat Phys 174:1189–1221, 2019; Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality, 2018. arXiv:1807.01559) to prove the delocalization conjecture and bulk universality for random band matrices with N≪W4/3\documentclass[12pt]{minimal}
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\begin{document}$$N\ll W^{4/3}$$\end{document}.
