Empirical Spectral Distribution of a Matrix Under Perturbation

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.

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

$$\begin{aligned} \phi (\lambda ) = \frac{1}{\pi } \int _\mathbb {C}\frac{\bar{\partial }(\widetilde{\phi }_p(z) \chi (z))}{\lambda - z} \, \mathrm {d}^2 z\,, \end{aligned}$$

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:

• for each n , $\phi \longmapsto N_n(\phi )$ is linear,

• for each n and each $\phi \in \mathcal {L}$, $\mathbb {E}[N_n(\phi )]=0$,

• there is a constant C such that for each n and each $\phi \in \mathcal {L}$, ${\text {Var}}(N_n(\phi ))\le C\Vert \phi \Vert ^2$,

• 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

$$\begin{aligned} \widehat{\phi }(k):= \int \mathrm {e}^{ \mathrm {i}kx}\phi (x)\mathrm {d}x\qquad \qquad { (k\in \mathbb {R})} \end{aligned}$$

and, for $s>0$,

$$\begin{aligned} \Vert \phi \Vert _{\mathcal {H}_s}:=\Vert k\longmapsto (1+2|k|)^s\,\widehat{\phi }(k)\Vert _{L^2}. \end{aligned}$$

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

$$\begin{aligned} {\text {Var}}({\text {Tr}}\phi (M))\le C \Vert \phi \Vert _{\mathcal {H}_s}^2\int _{y=0}^{\infty }y^{2s-1}\mathrm {e}^{-y}\int _{x\in \mathbb {R}}{\text {Var}}({\text {Tr}}((x+\mathrm {i}y-M)^{-1}))\mathrm {d}x\mathrm {d}y. \end{aligned}$$

1.3 A Density Lemma

We did not find Lemma 13 in the literature, so we provide its proof. Recall that for any $z\in \mathbb {C}\backslash \mathbb {R}$,

$$\begin{aligned} \varphi _z(x)=\frac{1}{z-x}. \end{aligned}$$

Lemma 12

For any $z\in \mathbb {C}\backslash \mathbb {R}$, we have, in the $L^2$ sense,

$$\begin{aligned} \widehat{\varphi _z}=(t\longmapsto -{\text {sgn}}(\mathfrak {Im}z)2\pi \mathrm {i}\mathbb {1}_{\mathfrak {Im}(z)t>0}\mathrm {e}^{\mathrm {i}tz}) \end{aligned}$$

(33)

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

$$\begin{aligned} \varphi _z(\xi )=\frac{-\mathrm {i}}{\mathrm {i}(\xi -z)}= -\mathrm {i}\int _{t=0}^{+\infty } \mathrm {e}^{-\mathrm {i}t(\xi -z)}\mathrm {d}t= -\mathrm {i}\int _{t=0}^{+\infty } \mathrm {e}^{-\mathrm {i}t\xi } \mathrm {e}^{\mathrm {i}tz}\mathrm {d}t . \end{aligned}$$

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 $\eta >0$,

$$\begin{aligned} P_\eta (E)=\frac{1}{\pi }\frac{\eta }{E^2+\eta ^2}=\frac{1}{2\mathrm {i}\pi }\left( \varphi _{\mathrm {i}\eta }(E)-\varphi _{-\mathrm {i}\eta }(E)\right) \end{aligned}$$

and that, by Lemma 12,

$$\begin{aligned} \widehat{P_\eta }(t)=\mathrm {e}^{-\eta |t|}. \end{aligned}$$

Hence for any $f\in \mathcal {H}_s$, we have

$$\begin{aligned} \Vert f-P_\eta *f\Vert _{\mathcal {H}_s}^2=\int (1+2|x|)^{2s}|\widehat{f}(x)|^2(1-\mathrm {e}^{-\eta |x|})^2\mathrm {d}x, \end{aligned}$$

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 $x\in \mathbb {R}$,

$$\begin{aligned} P_\eta *f(x)= & {} \frac{1}{\pi }\int f(t)\frac{\eta }{\eta ^2+(x-t)^2}\mathrm {d}t \\= & {} -\frac{1}{\pi }\int f(t)\mathfrak {Im}(\varphi _{t+\mathrm {i}\eta }(x))\mathrm {d}t \\= & {} \frac{1}{2\pi \mathrm {i}}\int f(t)(\varphi _{t-\mathrm {i}\eta }(x)-\varphi _{t+\mathrm {i}\eta }(x))\mathrm {d}t. \end{aligned}$$

Without loss of generality, one can suppose that the support of f is contained in [0, 1]. Then, for any $n\ge 1$,

$$\begin{aligned} P_\eta *f(x)= & {} \frac{1}{2n\pi \mathrm {i}}\sum _{k=1}^n f \left( \frac{k}{n}\right) \left( \varphi _{\frac{k}{n}-\mathrm {i}\eta }(x)-\varphi _{\frac{k}{n}+\mathrm {i}\eta }(x)\right) +R_n(x) \end{aligned}$$

(34)

where for $[t]_n:=\lceil nt\rceil /n$,

$$\begin{aligned} R_n(x)= \frac{1}{2\pi \mathrm {i}}\int f(t)\left( \varphi _{t-\mathrm {i}\eta }(x)-\varphi _{t+\mathrm {i}\eta }(x)\right) -f([t]_n)\left( \varphi _{[t]_n-\mathrm {i}\eta }(x)-\varphi _{[t]_n+\mathrm {i}\eta }(x)\right) \mathrm {d}t. \end{aligned}$$

The error term $R_n(x)$ rewrites

$$\begin{aligned} R_n(x)= & {} \frac{1}{2\pi \mathrm {i}}\int (f(t)-f([t]_n))(\varphi _{t-\mathrm {i}\eta }-\varphi _{t+\mathrm {i}\eta })(x)\mathrm {d}t\\&+\frac{1}{2\pi \mathrm {i}}\int f([t]_n)(\varphi _{t-\mathrm {i}\eta }-\varphi _{[t]_n-\mathrm {i}\eta }+\varphi _{t+\mathrm {i}\eta }-\varphi _{[t]_n+\mathrm {i}\eta })(x)\mathrm {d}t. \end{aligned}$$

Now, note that for any $t\in \mathbb {R}$ and $\eta \in \mathbb {R}\backslash \{0\}$, we have by Lemma 12,

$$\begin{aligned} \widehat{\varphi _{t+i\eta }}=(x\mapsto - {\text {sgn}}(\eta )2\pi \mathrm {i}\, \mathbb {1}_{\eta x>0} \, \mathrm {e}^{\mathrm {i}xz}), \end{aligned}$$

so that when, for example, $\eta >0$, for any $t\in \mathbb {R}$,

$$\begin{aligned} \Vert \varphi _{t+\mathrm {i}\eta }\Vert _{\mathcal {H}_s}^2=4\pi ^2\int _0^\infty (1+2|x|)^{2s}\mathrm {e}^{-2\eta x}\mathrm {d}x \end{aligned}$$

does not depend on t and for any $t,t'\in \mathbb {R}$,

$$\begin{aligned} \Vert \varphi _{t+\mathrm {i}\eta }-\varphi _{t'+\mathrm {i}\eta }\Vert _{\mathcal {H}_s}^2= & {} 4\pi ^2\int _0^\infty (1+2|x|)^{2s}|\mathrm {e}^{\mathrm {i}tx}-\mathrm {e}^{\mathrm {i}t' x}|^2\mathrm {e}^{-2\eta x}\mathrm {d}x\\= & {} 4\pi ^2\int _0^\infty (1+2|x|)^{2s}|\mathrm {e}^{\mathrm {i}(t-t')x}-1|^2\mathrm {e}^{-2\eta x}\mathrm {d}x \end{aligned}$$

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). $\square$