Abstract
We consider a general class of symmetric or Hermitian random band matrices \(H=(h_{xy})_{x,y \in \llbracket 1,N\rrbracket ^d}\) in any dimension \(d\ge 1\), where the entries are independent, centered random variables with variances \(s_{xy}=\mathbb {E}|h_{xy}|^2\). We assume that \(s_{xy}\) vanishes if \(|x-y|\) exceeds the band width W, and we are interested in the mesoscopic scale with \(1\ll W\ll N\). Define the generalized resolvent of H as \(G(H,Z):=(H - Z)^{-1}\), where Z is a deterministic diagonal matrix with entries \(Z_{xx}\in \mathbb {C}_+\) 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 \(d\ge 2\). More precisely, for any fixed \(d\ge 2\), we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if \(N\le W^{1+\frac{d}{2}}\). 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 \(N\ll W^{1+\frac{d}{d+1}}\), 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 \(N\ll W^{1+\frac{d}{6}}\). 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 \(N\ll W^{4/3}\).
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Acknowledgements
The second author would like to thank Paul Bourgade, L. Fu, Benedek Valkó and Horng-Tzer Yau for fruitful discussions and valuable suggestions. We also want to thank the editors and three anonymous referees for their helpful comments, which have improved the paper significantly.
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The work of Jun Yin is partially supported by the NSF grant DMS-1552192.
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The work of Jun Yin is partially supported by the NSF Grant DMS-1552192.
Appendices
A Proof of (2.35)
With Taylor expansion, we can write
Since \(\Vert S\Vert _{l^\infty \rightarrow l^\infty } = 1\) and \(|m|\le 1 - c\eta \) for some constant \(c>0\) by (1.4), it is easy to see that by taking \(K=\eta ^{-1} \) in (A.1), we have
Since S is a doubly stochastic matrix, \((S^{k})_{xy}\) can be understood through a k-step random walk on the torus \(\mathbb {Z}_N^d\). We first prove the following lemma. Here different from the previous proof, for any vector \(v\in \mathbb {R}^d\) we denote \(| v|\equiv \Vert v\Vert _2\).
Lemma 30
Let \(B_n = \sum _{i=1}^n X_i\) be a random walk on \(\mathbb {Z}^d\) with i.i.d. steps \(X_i\) that satisfy the following conditions: (i) \(X_1\) is symmetric; (ii) \(|X_1| \le L\) almost surely; (iii) there exists constants \(C^*,c_*>0\) such that
Let \(\varSigma \) be the covariance matrix of \(X_1\) with \(\varSigma _{ij}=\mathbb {E} [ (X_1)_i (X_1)_j ]\). Assume that \(n\in \mathbb {N}\) satisfies
for some constant \(c_0>0\). Then for any fixed (large) \(D>0\), we have
for large enough L (and n).
Proof of Lemma 30
Note that (A.5) is in accordance with the central limit theorem. Our proof below is in fact a variant of the proof of CLT with characteristic functions.
Combining condition (ii), i.e., \(|X_i|\le L\), with a large deviation estimate, with (A.4), we get that
for any fixed (small) \(\tau >0\) and (large) \(D>0\). Thus to prove (A.5), we only need to focus on the case
for some small enough constant \(\tau _0>0 \). In the following proof, we always make this assumption.
For \(p\in \mathbb {R}^d \) with \(0<|p|\le L^{-1}n^{-1/2+\tau _0}\), we have
where \(\hat{p}= p/|p|\) and \(\kappa _k(\hat{p})\) is the kth cumulant of \(\hat{p}\cdot X_1\). It gives that
By the condition (A.3), it is easy to verify that
in the sense of operators, and
for some constant \(C>0\). Then for \(|p|\le L^{-1}n^{-1/2+\tau _0}\), we have
where \(\alpha _k\in \mathbb {C}\) are coefficients (independent of p) satisfying
and \(K_D={{\,\mathrm{O}\,}}(1)\) is a fixed integer depending only on D and the constant \(c_0\) in (A.4).
Now we estimate \(\mathbb {E}e^{\mathrm {i}p\cdot B_n}\) for large p. Because of the existence of the core in (A.3), it is easy to see that for some constant \(c>0\),
which implies that for some \(c>0\),
Together with (A.7), with
and \(H_n\) being the Hermite polynomials, we have
where in the third step we used \(C^{-1/2}|q|\le |L\varSigma ^{-1/2}q|\le C^{1/2}|q|\) by (A.6) and approximated the \(\int _{|L\varSigma ^{-1/2}q|\le n^{\tau _0}}\) with \(\int _{q\in \mathbb {R}}\) up to an error \({{\,\mathrm{O}\,}}(L^{-D})\) due to the factor \(e^{-q^2/2}\), and in the last step we used (A.8), \(|y|={{\,\mathrm{O}\,}}(n^{\tau _0})\) and stationary approximation to bound the integrals. This proves (A.5). \(\square \)
Now we can give a proof of (2.35).
Proof of (2.35)
Fix any small constant \(\tau >0\). We now bound the sum in (A.2). Let \(B_n = \sum _{i=1}^n X_i\) be a random walk on \(\mathbb {Z}_N^d\) with i.i.d. steps \(X_i\), with distribution \(\mathbb {P}(X_1 = y-x)=s_{xy}\). Then it is easy to see that
For \(1\le k \le N^{\tau }\), with (2.4) we can bound
For \(N^{\tau } \le k \le N^{2-\tau }/W^2\), we have a large deviation estimate
for some constant \(c>0\). In particular, with high probability, \(b_x\) can be regarded as a random walk on the full lattice \(\mathbb {Z}^d\) if \(k \le N^{2-\tau }/W^2\), and we can apply (A.5) to get that
for some constant \(c>0\) and for any large constant \(D>0\). Finally, for \(N^{2-\tau }/W^2 \le k \le \eta ^{-1}\), using \(\Vert S\Vert _{l^\infty \rightarrow l^\infty } \le 1\) we get that
where we used (A.10) in the last step. Applying (A.9)–(A.11) to (A.2), we obtain that
where it is easy to verify that
This finishes the proof of (2.35) since \(\tau \) can be arbitrarily small and D can be arbitrarily large. \(\square \)
B Proof of Lemma 29
We fix \(x,y, y'\) in the proof. For simplicity, we ignore “x” from the coefficients \(c_{xw}\) and \(\widetilde{c}_{xw}\). With (6.3), we can write
Similarly for \(x\notin \{y,y'\}\cup \dot{I}\cup \ddot{I}\), together with (5.5), we get that
Now we define the “error” part as
and
With the above definition and (B.1), we have for any \(x\in \mathbb {Z}_N^d\),
It implies (with \(\{y,y'\}\cup \dot{I}\cup \ddot{I}\subset J\))
Then using (2.15), we obtain that for any fixed \(D>0\),
for some coefficients satisfying
Furthermore, by the definition of \({\mathcal B}_w\), we have
for some coefficients
Therefore, up to the error term \({{\,\mathrm{O}\,}}_\prec (N^{-D})\), \(\dot{G}_{xy}\ddot{G}_{xy'}-Q_x\left( \dot{G}_{xy}\ddot{G}_{xy'}\right) \) is equal to (see the explanation below)

where \(c_{1,w}\) and \(c_{2,w}\) are some coefficients that also satisfy (B.5). Here we have only drawn the dashed lines and ignored the \(\times \)-dashed lines. Moreover, the y and \(y'\) can be the same atom, but we did not draw this case. The first graph in the first row represents the first term on the right-hand side of (B.4). The second and third graphs in the first row represent the two terms in the second line of (B.4). The second row of (B.6) represents the the first row of (B.2), and the third row of (B.6) represents the the second row of (B.2). The graphs of \(\sum _{w\notin J}c_w {\mathcal B}_w \) have the same structures as the graphs in the second and third rows of (B.6), and we used “\(\ldots \)” to represent them in the fourth row.
Now the first graph in (B.6) gives the second term on the right-hand side of (6.54). All the other graphs in the first and second rows of (B.6) can be included into the third term on the right-hand side of (6.54) by relabelling w, \(w'\) as \(\alpha \) atoms. It is easy to check that these graphs satisfy the conditions for \({\mathcal G}_\kappa \) below (6.56). Therefore to finish the proof of (6.54), it remains to write the graphs in the third line of (B.6) into the form of the third term on the right-hand side of (6.54).
Following the idea in the proof for Lemma 21, we can write the graph with \(P_x\) color into a sum of colorless graphs. More precisely, using
and taking partial expectation \(\mathbb {E}_x\), we can write the graphs in the third row of (B.6) as
where
and \({\mathcal G}_\kappa (\mathbf {\alpha }, x, y,y')\) are colorless graphs which look like the graphs in (6.56). In fact, it is easy to check that \({\mathcal G}_\kappa \) either has a light weight (i.e. \(f_4\) light weight on the atom x or \(f_6\) weight on some \(\alpha \) atom) or there exists a solid line between \(\alpha \) atoms. Hence (B.7) can be written into the form of the third term on the right-hand side of (6.54) and satisfies the conditions below (6.56). This completes the proof of (6.54) in Lemma 29.
For (6.55), we use (B.3) and see that it suffices to write \({\mathcal B}_x\) into the form of the third term on the right-hand side of (6.55), which have been done above. Thus we finish the proof of (6.55).
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Yang, F., Yin, J. Random band matrices in the delocalized phase, III: averaging fluctuations. Probab. Theory Relat. Fields 179, 451–540 (2021). https://doi.org/10.1007/s00440-020-01013-5
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DOI: https://doi.org/10.1007/s00440-020-01013-5
Keywords
- Random band matrices
- Delocalization
- Averaging fluctuations
- Generalized resolvent
Mathematics Subject Classification
- 60B20
- 15B52
- 82B44