Abstract
We present a simple and versatile method for deriving (an)isotropic local laws for general random matrices constructed from independent random variables. Our method is applicable to mean-field random matrices, where all independent variables have comparable variances. It is entirely insensitive to the expectation of the matrix. In this paper we focus on the probabilistic part of the proof—the derivation of the self-consistent equations. As a concrete application, we settle in complete generality the local law for Wigner matrices with arbitrary expectation.
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Acknowledgements
We thank Torben KrĂĽger for drawing our attention to the importance of the equation (1.3) for general mean-field matrix models. In a private communication, Torben KrĂĽger also informed us that the idea of organizing proofs of local laws by bootstrapping is being developed independently for random matrices with general correlated entries in [17].
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Appendices
Appendix 1: Proof of Lemma 2.4
In this section we prove the cumulant expansion formula with the remainder term (2.3). We start with an elementary inequality.
Lemma 5.1
Let X be a nonnegative random variable with finite moments. Then for any \(a,b,t \geqslant 0\), we have
Proof
It suffices to assume \(a>0\). Let us abbreviate \(\Vert X\Vert _{a}:=(\mathbb {E}X^a)^{1/a}\). For \(t \geqslant \Vert X\Vert _a\), we have
which is the desired result. For \(t < \Vert X\Vert _a\), we have
Jensen’s (or Hölder’s) inequality yields
and the claim follows from (5.2) and (5.3), using \(1 = \mathbf {1}_{X \leqslant t} + \mathbf {1}_{X > t}\).\(\square \)
Let \(\chi (t):=\log \mathbb E e^{\mathrm {i}t h}\). For \(n \geqslant 1\), we have
hence
For \(g(h)=h^\ell \), we have
and by linearity the same relation holds for any polynomial P of degree \(\leqslant \ell \):
Next, let f be as in the statement of Lemma 2.4, and fix \(\ell \in \mathbb {N}\). By Taylor expansion we can find a polynomial P of degree at most \(\ell \), such that for any \(0 \leqslant k \leqslant \ell \),
where \(\xi _k\equiv \xi _k(h)\) is a random variable taking values between 0 and h.
By (5.4), (5.5), homogeneity of the cumulants, and Jensen’s inequality we find that the error term in (2.2) satisfies
The desired result then follows from estimating the last term of (5.6) by Cauchy-Schwarz inequality and Lemma 5.1.
Appendix 2: The complex case
In this appendix we explain how to generalize our results to the complex case. We briefly explain how to conclude the proof of Theorem 1.5 (i); Theorem 1.5 (ii) follows in a similar fashion.
If H is complex Hermitian, in general \(\mathbb E H^2_{ij}\) and \(\mathbb E |H_{ij}|^2\) are different. We define, in addition to (3.1),
and, in addition to (3.2), for a vector \({\mathbf{x}}=(x_i)_{i \in \mathbb N} \in \mathbb C^N\),
Now (3.11) generalizes to
and \(\mathcal {D}_{{\mathbf{v}}{\mathbf{w}}}\) remains the same as in (3.12).
Following a similar argument as in Sect. 3.1, we see that Proposition 3.1 is enough to conclude Theorem 1.5 (i). As in (3.21), we fix a constant \(p \in \mathbb N\) and write
We compute the second term on the right-hand side of (6.4) using the complex cumulant expansion, given in [27, Lemma 7.1], which replaces the real cumulant expansion from Lemma 2.4. The result is
where \(\tilde{X_k}\) and \(\tilde{\mathcal {R}}^{ij}_{\ell +1}\) are defined analogously to (3.24) and (3.25), respectively. Using the same proof, one can easily extend Lemma 3.4 (ii)–(iii) with \(X_k\) and \(\mathcal {R}^{ij}_{\ell +1}\) replaced by \( \tilde{X}_k\) and \(\tilde{\mathcal {R}}^{ij}_{\ell +1}\), so that it remains to estimate \(\mathbb {E}[\mathcal {K}_{{\mathbf{v}}{\mathbf{w}}}\mathcal {D}_{{\mathbf{v}}{\mathbf{w}}}^{p-1}\overline{\mathcal {D}}_{{\mathbf{v}}{\mathbf{w}}}^p]+\tilde{X}_1\). Using the complex cumulant expansion, we find
where in the second step we used (6.3) and
(Here we take the conventions of [27, Section 7] for the derivatives in the complex entries of H.) Thus we have
By estimating the right-hand side of (6.7) in a similar way as we dealt with the right-hand side of (3.27), we obtain
which completes the proof.
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He, Y., Knowles, A. & Rosenthal, R. Isotropic self-consistent equations for mean-field random matrices. Probab. Theory Relat. Fields 171, 203–249 (2018). https://doi.org/10.1007/s00440-017-0776-y
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DOI: https://doi.org/10.1007/s00440-017-0776-y
Mathematics Subject Classification
- 15B52
- 82B44
- 82C44