Eigenvectors of non normal random matrices

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v},\mathbf{v}'$ associated with distinct eigenvalues $\lambda,\lambda'$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product $$\sqrt{n}(\lambda'-\lambda)\langle\mathbf{v},\mathbf{v}'\rangle$$ is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.


Setup and main results
Let V : R + → R be a function such that the following holds.
(1.1) Let X be an n × n complex matrix with law ∝ e −n Tr V (M * M ) dM, (1.2) where dM is the standard Lebesgue measure on n × n complex matrices. In particular, all eigenvalues of X are distinct, almost surely (see Remark 3.1). Let z, z ∈ C and let λ and λ denote the eigenvalues of X that are the closest to respectively z and z (if z = z or λ is the closest eigenvalue to both z and z , then λ is the second closest to z ). Let v and v denote some associated eigenvectors of unit 2 norm. We want to study the quantity n|λ − λ|| v, v | 2 , which leads us to introduce the random variable Y , defined through any of the two following equivalent equations Since eigenvalues are almost surely distinct and | v, v | is invariant under multiplication of the eigenvectors by a complex scalar of norm 1, the random variable Y is well defined. Recall α from (1.1). Our first main result is the following.

Remark 1.2.
In the case of the Ginibre ensemble (V (x) = x, so that the entries of X are i.i.d. standard complex Gaussian variables with variance n −1 ), the random variable Y has an exponential law of mean 1. This fact is probably well known, and follows from Equations (2.3) and (2.7)-(2.8) below. In particular, √ Y is distributed like the norm of a standard complex Gaussian variable.
(We recall that a standard complex Gaussian variable is a centered complex Gaussian random variable Z such that E Z 2 = 0 and E |Z| 2 = 1.) Our second main result is concerned with Ginibre matrices, for which we extend an asymptotic version of Remark 1.2 to the multivariate framework. Theorem 1.3. Suppose that V (x) = x. For a fixed k ≥ 2, let z 1 , . . . , z k be (deterministic) points in the unit disk, possibly dependent on n, such that for a certain ε > 0, uniformly in n, √ n min 1≤i<j≤k |z j − z i | ≥ n ε . (1.6) For each i, let λ i be the eigenvalue that is the closest to z i and let v i be an associated eigenvector. Let θ i , i = 1, . . . , k, be i.i.d. variables uniformly distributed on the [0, 2π], independent of X. Then the distribution of the triangular array converges, as n → ∞, to the distribution of a triangular array of independent standard complex Gaussian variables.
Remark 1.4. The typical distance between two eigenvalues of X that are "neighbors" of each other in the spectrum of X has order n −1/2 . Hence, because of Hypothesis (1.6), this result is well adapted for most pairs of eigenvalues, but not for those that are as close as possible (in our proof, Hypothesis (1.6) is necessary for estimates (2.11) to Eigenvectors of non normal random matrices

Background
The study of eigenvectors of random Ginibre matrices seems to have been initiated in [10]. For a matrix X, let v i (respectively, w i ) denote the left (respectively, right) eigenvectors corresponding to eigenvalues λ i , where the normalization v i , w j = δ ij is imposed. Using the Schur representation X = U T U * with T upper triangular and U unitary, they computed, for the Ginibre ensemble, the correlations of eigenvectors and cross correlations of right and left eigenvectors, with special emphasis on the correlator (1.7) Using the joint density of entries of T , the evaluation of the latter correlations reduce to the evaluation of certain Green functions. This point of view was recently significantly expanded to more general models in [13] (using diagrammatic methods), as well as in [5], where multi-points correlations are evaluated and related to two point correlations. We refer the reader to the introduction of [5] for further details and an extensive bibliography. Recent works [7,4] study distributional limits for condition numbers, as well as refined estimates for overlaps in the microscopic and mesoscopic regime.
Another very relevant recent work is [3], which deals with matrices X with joint density of entries of the form (1.2) (whithout assuming the convexity of V ). In this general setup, the correlator O 12 from (1.7) is computed.
Our results, as well as [11,5], build upon the evaluation of the joint distribution of the entries of T , see [8,15,12]; these derivations do not address explicitely the ordering of the diagonal elements in T ; in our approach, we choose the ordering as function of the full set of diagonal elements. For this reason, we provide explicitly a proof of the joint distribution of entries.
Finally, we mention that general delocalization results for eigenvectors of random non-Hermitian matrices with independent entries appear in [14].

A conjecture
The inner products appearing in Theorem 1.3 can be written in terms of the off diagonal entries of the upper triangular matrix T in the Schur decomposition of X, see the proof of Theorem 1.1 below. In the Ginibre case, these entries (scaled by √ n) are iid standard complex Gaussians, and for general V , it is still the case that Y = n|t 12 | 2 , see (2.3) below. This leads us to the following. Conjecture 1.5. Under the assumptions of Theorem 1.1, the sequence Y / E Y converges in distribution, as n → ∞, to the exponential law of parameter one. Some preliminary computations make the conjecture plausible. In addition, the simulations in Figure 1 are in agreement with the conjecture.

Proofs
In the proofs below, we use the joint distributions derived in Theorem 3.3 from the appendix.
Proof of Theorem 1.1. Set O := O z,z ,z ,z ,...,z (see (3.2) for the definition of this set). By Theorem 3.3, we know that X can be written X = U T U * with U unitary and T = [t ij ] upper triangular having the density ∝ 1 (t11,...,tnn)∈O |∆(t 11 , . . . , t nn )| 2 e −n Tr V (T * T ) , Hence by definition of O, λ and λ are the two first diagonal entries of T . Thus the vectors w := (1, 0, . . . , 0) and w := |t 12 Figure 1: Universality of eigenvector angles: density 2xe −x 2 (in red) vs the his- with distribution ∝ e −n Tr V (M * M ) (n = 150, sample of size 40, sampled thanks to Langevin Monte Carlo), with 2 different choices for V . The sample on the right has been rescaled so that its empirical second moment is 1 (because the distribution of can only be universal up to a rescaling of the matrix).
Before the proof of Theorem 1.3, we prove two preliminary lemmas. We suppose here Lemma 2.1. Let z 0 in the unit disk, possibly depending on n and s ∈ (0, 1/2). Then with probability tending to one as n → ∞, X has at least n 1−2s 5 eigenvalues at distance ≤ n −s from z 0 .
Proof. Let N s denote the number of eigenvalues in the disk D(z 0 , n −s ). Let f be a smooth non negative function with value 1 on the disk D(0, 1/2) and with support contained in the disk D(0, 1). Then we have where the λ j denote the eigenvalues. By the local circular law by Yin [17, Th. 1.2] (see also [16,Th. 9] for the case where |z 0 | < 1), we know that with probability tending to one, where L denotes the Lebesgue measure on C and σ := (1 − 2s)/2. We deduce that with probability tending to one, Let z, z in the unit disk, possibly depending on n, such that for a certain fixed ε ∈ (0, 1/2), uniformly in n, Let λ, λ be the eigenvalues of X that are the closest to respectively z and z (if λ is the closest eigenvalue to both z and z , then λ is the second closest to z ). Then for any fixed δ ∈ (0, ε), we have √ n|λ − λ| ≥ n δ with probability tending to one as n → ∞.
Proof. Let s ∈ (1/2 − ε, 1/2). By the previous lemma, we have |λ − z| ≤ n −s and |λ − z | ≤ n −s with probability tending to one as n → ∞. Thus which allows to conclude, as −s < −1/2 + ε and ε > δ. tending to one but with probability at least 1 − Cn −D for any D > 0 (and for C a constant depending only on D, not on z 0 , z, z ), which can be useful when using a union bound.
The proof is the same and follows from the fact that in [17], the error probability is ≤ Cn −D .
Proof of Theorem 1.3. For u, v some random variables implicitly depending on n, we use the notation u ∼ v (resp. u = O(v), u = o(v)) when u/v tends in probability to one (resp. u/v is tight, u/v tends in probability to 0) as n → ∞.
By definition of O, λ 1 , . . . , λ k are the k first diagonal entries of T . Besides, as U is unitary, where the w i are the eigenvectors of T associated to the λ i (multiplied by independent uniform phases e iθi , independent of T ). For each i, w i is in the kernel of T − λ i , hence has only its i first coordinates non zero, and these coordinates are proportional to the vector (x i (1), . . . , x i (i)) ∈ C i , satisfying (λ 1 − λ i )x i (1) + t 1,2 x i (2) + · · · · · · · · · · · · · · · · · · · · · + t 1,i We solve this linear system: (2.10) To analyse the asymptotic behavior of these inner products, let us analyse the asymptotic behavior of each variable x i ( ), 1 ≤ ≤ i ≤ k. By (1.6) and Lemma 2.2, using the fact that k is fixed, we have (2.11) By the previous equations and the estimate (2.11), using the fact that the random variables t ij = √ nt ij are independent standard complex Gaussian variables, we have, for any i = 1, . . . , k, we obtain successively the following estimates: 14) It implies that for all 1 ≤ ≤ i − 1, and that, as x i (i) = 1, .
where we used (2.11). It follows that and, as ( t ij ) 1≤i<j≤k is a collection of independent standard complex Gaussian variables independent of the θ i 's, the result is proved.

Change of variables in the Schur decomposition
We endow the sets M n (C) and T n (C) of n × n respectively complex matrices and upper-triangular complex matrices with the Euclidian structures defined by and let dM (resp. dT ) denote the associated Lebesgue measure on M n (C) (resp. on T n (C)). We also denote by U n the group of unitary n × n matrices and by dU the Haar measure on U n .

Remark 3.1.
It is useful to note that the set of matrices in M n (C) with multiple eigenvalues is the set of matrices whose characteristic polynomial has null discriminant (see [1,Def. A.10]), so that this set is a level set of a non constant polynomial function on M n (C), hence has zero Lebesgue measure (the last fact can be checked by applying Fubini's theorem).
An important example of admissible set is the following one. Fix z 1 , . . . , z n ∈ C. Then the set of n-tuples (t 1 , . . . , t n ) ∈ C n where for each i, the i-th entry t i is strictly closer to z i than all the forthcoming ones, i.e. the set O z1,...,zn := {(t 1 , . . . , t n ) ∈ C n ; ∀i < j, |z i − t i | < |z i − t j |}, (3.2) is admissible.  and C n is a constant depending only on n (and not on ρ). Proof of Theorem 3.3. Some statements which are very close to Theorem 3.3 are proved in various texts, as [8,12,6,15]. However, firstly, these results are a bit less general and written in slightly different languages and, secondly and more importantly, they do not treat the question of the ordering the diagonal entries of T (which is the cornerstone of our approach in this paper). For this reason, we provide a complete proof.
Proof. Let f : T → R be a test function. Then 1 What we call here the Jacobian of a smooth function between two Euclidian spaces with the same dimension is the absolute value of the determinant of the matrix of its derivative in any pair of orthogonal bases.

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http://www.imstat.org/ecp/ Eigenvectors of non normal random matrices Definition 3.6. Let U be the set of unitary matrices whose entries are all non zero, whose diagonal entries are positive and whose principal minors are all invertible.
The following lemma can be found in [1,Lemma 2.5.6].
Lemma 3.7. The map (without the zeros) is diffeomorphism from U onto a subset of R n(n−1)/2 with closed null mass complementary. We denote its inverse by Θ.   |Jϕ(T, x)|dx dT.
Then, the following lemma concludes the proof of Theorem 3.3. Proof. Let Θ : T n (C) × R n(n−1) → T n (C) × U be defined by Θ(T, x) := (T, Θ(x)) and F : T n (C) × U → M n (C) be defined by F (T, U ) := U T U * (note that F is defined on a manifold and not on an open subset of an Euclidian space). We have ϕ = F • Θ on T O × R n(n−1) , so we have hence it suffices to prove that on T O × U , Note that the tangent space of U at U is the space  of strictly lower triangular n × n matrices, it is easy to see that the determinant of this map is the one of the map C T of Lemma 3.11 below, which concludes the proof of Lemma 3.10.
Lemma 3.11. Let T sl be as in (3.10), iH 0 be as in (3.9) and π : M n (C) → T sl be the canonical projection. Then for any T = [t ij ] ∈ T n (C), the map C T : iH 0 → T sl defined by C T (M ) := π(M T − T M ), has Jacobian all spaces being endowed with the Euclidian structure induced by (3.1).
Proof. To prove this lemma, we shall first fix some orthonormal bases of iH 0 and T sl , order them and then prove that the matrix of C T on these (conveniently ordered) bases is lower triangular by 2 × 2 blocs with diagonal blocs having determinants |t jj − t ii | 2 , 1 ≤ j < i ≤ n. Let us denote the elementary matrices by E ij and let B iH0 be the family and let B T sl be the family (E ij , iE ij ) 1≤j<i≤n . It follows that the matrix of C T on the bases B iH0 and B T sl (ordered as above) is lower diagonal by 2 × 2 blocs, with 2 × 2 diagonal blocs the matrices of the linear maps C → C (C considered as a real vector space) m −→ (t jj −t ii )×m, 1 ≤ j < i ≤ n. The determinant of such a map is |t jj − t ii | 2 , so the result follows.

Klein's lemma and consequences
The following lemma can be found in [1,Lemma 4.4.12].  the probability measure P V,R n ∝ e −V (x) dx satisfies a LSI with constant α −1 . This implies that for any 1-Lipschitz function f : R n → R and any δ > 0, we have P V,R n |f (x) − E P V,R n f | ≥ δ ≤ 2e −αδ 2 /2 .