Additive/Multiplicative Free Subordination Property and Limiting Eigenvectors of Spiked Additive Deformations of Wigner Matrices and Spiked Sample Covariance Matrices

Abstract

When some eigenvalues of a spiked additive deformation of a Wigner matrix or a spiked multiplicative deformation of a Wishart matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the subordination function relative to the free (additive or multiplicative) convolution plays an important part in the asymptotic behavior.

Appendix A

1.1 A.1 Poincaré Inequality and Concentration Inequalities

We first derive in this section concentration inequalities based on the Poincaré inequality. We refer the reader to the book [2]. A probability measure μ on $\mathbb{R}$ is said to satisfy the Poincaré inequality with constant $C_{\mathrm{PI}}$ if for any $\mathcal{C}^{1}$ function $f: \mathbb{R}\rightarrow\mathbb{C}$ such that f and f′ are in L ²(μ),

$$\mathbf{V}(f)\leq C_{\mathrm{PI}}\int\vert f' \vert^2\,d\mu,$$

with $\mathbf{V}(f) = \int\vert f-\int f \,d\mu\vert^{2} \,d\mu$.

We refer the reader to [19] for a characterization of the measures on $\mathbb{R}$ which satisfy a Poincaré inequality.

Remark A.1

If the law of a random variable X satisfies the Poincaré inequality with constant $C_{\mathrm{PI}}$ then, for any fixed α≠0, the law of αX satisfies the Poincaré inequality with constant $\alpha^{2} C_{\mathrm{PI}}$.

If a probability measure μ on $\mathbb{R}$ satisfies the Poincaré inequality with constant $C_{\mathrm{PI}}$ then the product measure μ ^⊗M on $\mathbb{R}^{M}$ satisfies the Poincaré inequality with constant $C_{\mathrm{PI}}$ in the sense that for any differentiable function F such that F and its gradient ∇F are in L ²(μ ^⊗M),

$$\mathbf{V}(f)\leq C_{\mathrm{PI}} \int\Vert\nabla F \Vert_2^2\,d\mu^{\otimes M}$$

with $\mathbf{V}(f) = \int\vert f-\int f \,d\mu^{\otimes M} \vert^{2} \,d\mu^{\otimes M}$ (see Theorem 2.5 in [31]).

An important consequence of the Poincaré inequality is the following concentration result.

Lemma A.1

(Lemma 4.4.3 and Exercise 4.4.5 in [1] or Chap. 3 in [39])

Let $\mathbb{P}$ be a probability measure on $\mathbb{R^{M}}$ which satisfies a Poincaré inequality with constant $C_{\mathrm{PI}}$. Then there exists K ₁>0 and K ₂>0 such that, for any Lipschitz function F on $\mathbb{R}^{M}$ with Lipschitz constant $\vert F \vert_{\mathrm{Lip}}$,

$$\forall\epsilon> 0, \quad\mathbb{P} \bigl(\big \vert F-\mathbb {E}_{\mathbb {P}}(F)\big\vert> \epsilon \bigr) \leq K_1 \exp \biggl(-\frac{\epsilon }{K_2 \sqrt{C_{\mathrm{PI}}} \vert F \vert_{\mathrm{Lip}}}\biggr).$$

1.2 A.2 Technical Tools

We need the following result on the extension of Lipschitz functions on $\mathbb{R}$ to the Hermitian matrices.

Lemma A.2

(See [24])

Let f be a real $C_{\mathcal{L}}$-Lipschitz function on $\mathbb{R}$. Then its extension on the N×N Hermitian matrices is $C_{\mathcal{L}}$-Lipschitz with respect to the norm $\Vert M\Vert_{2}=\{\operatorname{Tr} (MM^{*})\}^{\frac{1}{2}}$.

Proof

Let A and B be N×N Hermitian matrices. Let us consider their spectral decompositions

$$A=\sum_i \lambda_i(A)P^{(A)}_i$$

and

$$B=\sum_i \lambda_i(B)P^{(B)}_i.$$

We have

Now, since $\operatorname{Tr}( P^{(A)}_{i} P^{(B)}_{j})\geq0$, we can deduce that

$$\big\Vert f(B) -f(A) \big\Vert_2^2 \leq\sum _{i,j} C_\mathcal{L}^2\bigl(\lambda_i(A)-\lambda_j(B)\bigr)^2\operatorname{Tr}\bigl(P^{(A)}_i P^{(B)}_j\bigr)=C_\mathcal{L}^2 \Vert B-A \Vert_2^2.$$

 □

We recall here some useful properties of the resolvent (see [20, 37]).

Lemma A.3

For a N×N Hermitian or symmetric matrix M, for any $z \in\mathbb{C}\setminus \operatorname{Spect}(M)$, we denote by G(z):=(zI _N −M)⁻¹ the resolvent of M.

Let $z \in\mathbb{C}\setminus\mathbb{R}$,

• (i) ∥G(z)∥≤|ℑz|⁻¹ where ∥.∥ denotes the operator norm.

• (ii) |G(z) _ij |≤|ℑz|⁻¹ for all i,j=1,…,N.

• (iii) Let $z \in\mathbb{C}$ such that |z|>∥M∥; we have

  $$\big\Vert G(z)\big \Vert\leq\frac{1}{|z| - \Vert M \Vert}.$$

We recall here the following classical result due to Weyl.

Lemma A.4

(Theorem 4.3.7 of [35]) Let B and C be two N×N Hermitian matrices. For any pair of integers j,k such that 1≤j,k≤N and j+k≤N+1, we have

$$\lambda_{j+k-1} (B+C) \leq\lambda_{j}(B) +\lambda_{k}(C).$$

For any pair of integers j,k such that 1≤j,k≤N and j+k≥N+1, we have

$$\lambda_j(B) + \lambda_k(C) \leq\lambda_{j+k-N}(B+C).$$

The following result on quadratic forms is of basic use in the sample covariance matrix setting. Note that, a complex random variable x will be said standardized if $\mathbb{E}(x)=0$ and $\mathbb {E}(\vert x \vert^{2})=1$.

Proposition A.1

(Lemma 2.7 of [5])

Let B=(b _ij ) be a N×N matrix and Y _N be a vector of size N which contains i.i.d. standardized entries with bounded fourth moment. Then there is a constant K>0 such that

$$\mathbb{E}\big\vert Y_N^* B Y_N - \operatorname{Tr} B\big\vert^2 \leq K \operatorname{Tr} \bigl(BB^*\bigr).$$

The following technical lemma is fundamental in this paper. We refer the reader to the Appendix of [20] where it is proved using the ideas of [33].

Lemma A.5

Let h be an analytic function on $\mathbb{C}\setminus\mathbb{R}$ which satisfies

$$ \big\vert h(z)\big\vert\leq\bigl(\vert z\vert+K\bigr)^\alpha P\bigl(\vert\Im z\vert^{-1}\bigr)$$

and φ be in $\mathcal{C}^{\infty}(\mathbb{R}, \mathbb{R})$ with compact support. Then,

$$\limsup_{y\rightarrow0^+}\bigg\vert\int_\mathbb{R}\varphi(x)h(x+iy)\,dx\bigg\vert< + \infty.$$