Abstract
We provide a perturbative expansion for the empirical spectral distribution of a Hermitian matrix with large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first matrix are independent with a variance profile. We prove that, depending on the order of magnitude of the perturbation, several regimes can appear, called perturbative and semi-perturbative regimes. Depending on the regime, the leading terms of the expansion are related either to the one-dimensional Gaussian free field or to free probability theory.
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Notes
If the perturbing matrix belongs to the GOE or GUE, then its law is invariant under this change in basis, hence our results in fact apply to any self-adjoint matrix \(D_n\).
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Acknowledgements
We thank Jean-Philippe Bouchaud, Guy David and Vincent Vargas for some fruitful discussions. We are also glad to thank the GDR MEGA for partial support.
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Appendix
Appendix
The reader can find here the results we use along the paper, namely the Helffer–Sjöstrand formula, the CLT extension lemma of Shcherbina and Tirozzi and a functional density lemma with its proof.
1.1 Helffer–Sjöstrand Formula
The proof of the following formula can be found, e.g., in [7].
Proposition 9
(Helffer–Sjöstrand formula) Let \(n \in \mathbb {N}\) and \(\phi \in {\mathcal {C}}^{p+1}(\mathbb {R})\). We define the almost analytic extension of \(\phi \) of degree p through
Let \(\chi \in {\mathcal {C}}^\infty _c(\mathbb {C};[0,1])\) be a smooth cutoff function. Then for any \(\lambda \in \mathbb {R}\) satisfying \(\chi (\lambda ) = 1\) we have
where \(\mathrm {d}^2 z\) denotes the Lebesgue measure on \(\mathbb {C}\) and is the antiholomorphic derivative.
1.2 CLT Extension Lemma
The following CLT extension lemma is borrowed from the paper of Shcherbina and Tirozzi [20]. We state here the version that can be found in Appendix of [6].
Lemma 10
Let \((\mathcal {L}, \Vert \,\cdot \,\Vert )\) be a normed space with a dense subspace \(\mathcal {L}_1\) and, for each \(n\ge 1\), \((N_n(\phi ))_{\phi \in \mathcal {L}}\) a collection of real random variables such that:
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for each n , \(\phi \longmapsto N_n(\phi )\) is linear,
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for each n and each \(\phi \in \mathcal {L}\), \( \mathbb {E}[N_n(\phi )]=0 \),
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there is a constant C such that for each n and each \(\phi \in \mathcal {L}\), \({\text {Var}}(N_n(\phi ))\le C\Vert \phi \Vert ^2\),
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there is a quadratic form \(V:\mathcal {L}_1\rightarrow \mathbb {R}_+\) such that for any \(\phi \in \mathcal {L}_1\), we have the convergence in distribution \(N_n(\phi )\underset{n\rightarrow \infty }{\longrightarrow }\mathcal {N}(0, V(\phi ))\).
Then, V is continuous on \(\mathcal {L}_1\), can (uniquely) be continuously extended to \(\mathcal {L}\) and for any \(\phi \in \mathcal {L}\), we have the convergence in distribution \(N_n(\phi )\underset{n\rightarrow \infty }{\longrightarrow }\mathcal {N}(0, V(\phi ))\).
One of the assumptions of previous lemma concerns a variance domination. The next proposition provides a tool in order to check it. Let us first remind the definition of the Sobolev space \(\mathcal {H}_s\). For \(\phi \in L^1(\mathbb {R},\mathrm {d}x)\), we define
and, for \(s>0\),
We define the Sobolev space \(\mathcal {H}_s\) as the set of functions with finite \(\Vert \cdot \Vert _{\mathcal {H}_s}\) norm. Let us now state Proposition 2 of the paper [21] of Shcherbina and Tirozzi.
Proposition 11
For any \(s>0\), there is a constant \(C=C(s)\) such that for any n, any \(n\times n\) Hermitian random matrix M, and any \(\phi \in \mathcal {H}_s\), we have
1.3 A Density Lemma
We did not find Lemma 13 in the literature, so we provide its proof. Recall that for any \(z\in \mathbb {C}\backslash \mathbb {R}\),
Lemma 12
For any \(z\in \mathbb {C}\backslash \mathbb {R}\), we have, in the \(L^2\) sense,
and \(\varphi _z\) belongs to each \(\mathcal {H}_s\) for any \(s\in \mathbb {R}\).
Proof
It is well known that if \(\mathfrak {Re}z>0\), then \(\displaystyle \frac{1}{z}= \int _{t=0}^{+\infty } \mathrm {e}^{-tz}\mathrm {d}t.\)
Let \(z=E+\mathrm {i}\eta \), \(E\in \mathbb {R}, \eta >0\). For any \(\xi \in \mathbb {R}\), we have
We deduce (33) for \(\mathfrak {Im}z>0\). The general result can be deduced by complex conjugation. \(\square \)
Lemma 13
Let \(\mathcal {L}_1\) denote the linear span of the functions \(\varphi _z(x):=\frac{1}{z-x}\), for \(z\in \mathbb {C}\backslash \mathbb {R}\). Then the space \(\mathcal {L}_1\) is dense in \(\mathcal {H}_s\) for any \(s\in \mathbb {R}\).
Proof
We know, by Lemma 12, that \(\mathcal {L}_1\subset \mathcal {H}_s\). Recall first the definition of the Poisson kernel, for \(E\in \mathbb {R}\) and \(\eta >0\),
and that, by Lemma 12,
Hence for any \(f\in \mathcal {H}_s\), we have
so that, by dominated convergence, \(P_\eta *f\longrightarrow f\) in \(\mathcal {H}_s\) as \(\eta \rightarrow 0\).
To prove Lemma 13, it suffices to prove that any smooth compactly supported function can be approximated, in \(\mathcal {H}_s\), by functions of \(\mathcal {L}_1\). So let f be a smooth compactly supported function. By what precedes, it suffices to prove that for any fixed \(\eta >0\), \(P_\eta *f\) can be approximated, in \(\mathcal {H}_s\), by functions of \(\mathcal {L}_1\). For \(x\in \mathbb {R}\),
Without loss of generality, one can suppose that the support of f is contained in [0, 1]. Then, for any \(n\ge 1\),
where for \([t]_n:=\lceil nt\rceil /n\),
The error term \(R_n(x)\) rewrites
Now, note that for any \(t\in \mathbb {R}\) and \(\eta \in \mathbb {R}\backslash \{0\}\), we have by Lemma 12,
so that when, for example, \(\eta >0\), for any \(t\in \mathbb {R}\),
does not depend on t and for any \(t,t'\in \mathbb {R}\),
depends only on \(t'-t\) end tends to zero (by dominated convergence) when \(t'-t\rightarrow 0\).
We deduce that \(\Vert R_n\Vert _{\mathcal {H}_s}\longrightarrow 0\) as \(n\rightarrow \infty \), which closes the proof, by (34). \(\square \)
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Benaych-Georges, F., Enriquez, N. & Michaïl, A. Empirical Spectral Distribution of a Matrix Under Perturbation. J Theor Probab 32, 1220–1251 (2019). https://doi.org/10.1007/s10959-017-0790-0
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DOI: https://doi.org/10.1007/s10959-017-0790-0