The distribution of overlaps between eigenvectors of Ginibre matrices

Abstract

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

Appendices

Appendix A: Eigenvalues dynamics

This Appendix derives the Dyson-type dynamics for eigenvalues of nonnormal matrices. More precisely, we consider the Ornstein-Uhlenbeck version so that the equilibrium measure is the (real or complex) Ginibre ensemble. These dynamics take a particularly simple form in the case of complex Gaussian addition, where the drift term shows no interaction between eigenvalues: only the correlation of martingale terms is responsible for eigenvalues repulsion.

We also describe natural dynamics with equilibrium measure given by the real Ginibre ensemble. Then, the eigenvalues evolution is more intricate.

It was already noted in [11] that eigenvectors impact the eigenvalues dynamics for nonnormal matrices, and the full dynamics in the complex case have been written down in [30].

1.1 Complex Ginibre dynamics

Let G(0) be a complex matrix of size N, assumed to be diagonalized as $YGX = \Delta = \mathrm {Diag}(\lambda _1,\dots ,\lambda _N)$, where X, Y are the matrices of the right- and left-eigenvectors of G(0). We also assume that G(0) has simple spectrum, and X, Y invertible. The right eigenvectors $(x_i)$ are the columns of X, and the left-eigenvectors $(y_j)$ are the rows of Y. They are chosen uniquely such that $XY=I$ and, for any $1\leqslant k\leqslant N$, $X_{kk}=1$.

We now consider the complex Dyson-type dynamics: for any $1\leqslant i,j\leqslant N$,

$$\begin{aligned} \mathrm{d}G_{ij}(t)=\frac{\mathrm{d}B_{ij}(t)}{\sqrt{N}}-\frac{1}{2}G_{ij}(t)\mathrm{d}t, \end{aligned}$$

(A.1)

where the $B_{ij}$’s are independent standard complex Brownian motions: $\sqrt{2}{{\,\mathrm{Re}\,}}(B_{ij})$ and $\sqrt{2}{{\,\mathrm{Im}\,}}(B_{ij})$ are standard real Brownian motions. One can easily check that G(t) converges to the Ginibre ensemble as $t\rightarrow \infty$, with normalization (1.1).

In the following, the bracket of two complex martingales M, N is defined by bilinearity: $\langle M,N\rangle =\langle {{\,\mathrm{Re}\,}}M,{{\,\mathrm{Re}\,}}N\rangle -\langle {{\,\mathrm{Im}\,}}M,{{\,\mathrm{Im}\,}}N\rangle +\mathrm {i}\langle {{\,\mathrm{Re}\,}}M,{{\,\mathrm{Im}\,}}N\rangle +\mathrm {i}\langle {{\,\mathrm{Im}\,}}M,{{\,\mathrm{Re}\,}}N\rangle$.

Proposition A.1

The spectrum $(\lambda _1(t),\dots , \lambda _n(t))$ is a semimartingale satisfying the system of equations

$$\begin{aligned} \mathrm {d}\lambda _k(t) = \mathrm{d}M_k(t) - \frac{1}{2} \lambda _k(t)\mathrm {d}t \end{aligned}$$

where the martingales $(M_k)_{1\leqslant k\leqslant N}$ have brackets $\langle M_i,M_j\rangle =0$ and

$$\begin{aligned} \mathrm{d}\langle M_i,\overline{M_j}\rangle _t={\mathscr {O}}_{ij}(t)\frac{\mathrm{d}t}{N}. \end{aligned}$$

Remark A.2

As explained below, this equation (in particular the off-diagonal brackets) is coherent with the eigenvalues repulsion observed in (1.2). Contrary to the Hermitian Dyson Brownian motion, all eigenvalues are martingales (up to the Ornstein Uhlenbeck drift term), so that their repulsion is not due to direct mutual interaction, but to correlations between these martingales at the microscopic scale.

For example, assume that G(0) is already at equilibrium. Using physics conventions, for any bulk eigenvalues $\lambda _1,\lambda _2$ satisfying $w={{\,\mathrm{O}\,}}(1)$ (remember $w=\sqrt{N}(\lambda _1-\lambda _2)$), Proposition A.1 and Theorem 1.4 imply

$$\begin{aligned}&{\mathbb {E}}\left( \mathrm{d}\lambda _1\mathrm{d}\overline{\lambda _2}\mid \lambda _1=z_1,\lambda _2=z_2\right) \sim {\mathbb {E}}({\mathscr {O}}_{12}\mid \lambda _1=z_1,\lambda _2=z_2)\\&\quad \frac{\mathrm{d}t}{N}\sim -(1-|z_1|^2)\frac{1}{|w|^4}\frac{1- (1+|\omega |^2)e^{-|\omega |^2}}{1-e^{-|\omega |^2}}\mathrm{d}t \end{aligned}$$

in the bulk. By considering the real part in this equation and denoting $\mathrm{d}\lambda _1=\mathrm{d}x_1+\mathrm {i}\mathrm{d}y_1$, $\mathrm{d}\lambda _2=\mathrm{d}x_2+\mathrm {i}\mathrm{d}y_2$, we have in particular ${\mathbb {E}}(\mathrm{d}x_1\mathrm{d}x_2+\mathrm{d}y_1\mathrm{d}y_2)<0$, and this negative correlation is responsible for repulsion: the eigenvalues tend to move in opposite directions. Moreover, as eigenvalues get closer on the microscopic scale, $w\rightarrow 0$ and the repulsion gets stronger:

$$\begin{aligned} {\mathbb {E}}\left( \mathrm{d}\lambda _1\mathrm{d}\overline{\lambda _2}\mid \lambda _1=z_1,\lambda _2=z_2\right) \sim -\frac{1-|z_1|^2}{|w|^2}\mathrm{d}t. \end{aligned}$$

On the other hand, for mesoscopic scale $N^{-1/2}\ll |\lambda _1-\lambda _2|$, Proposition A.1 and Theorem 1.4 give ${\mathbb {E}}\left( \mathrm{d}\lambda _1\mathrm{d}\overline{\lambda _2}\right) \sim -\frac{(1-|\lambda _1|^2)}{N^2|\lambda _1-\lambda _2|^4}\mathrm{d}t={{\,\mathrm{o}\,}}(\mathrm{d}t)$, so that increments are uncorrelated for large N.

For a given differential operator $f\mapsto f'$, we introduce the matrix $C=X^{-1}X'$. Along the following lemmas, all eigenvalues are assumed to be distinct. In our application, this spectrum simplicity will hold almost surely for any $t\geqslant 0$ as G(0) has simple spectrum.

Lemma A.3

We have $X'=XC$ and $Y'=-CY$.

Proof

The first equality is the definition of C. For the second one, $XY=I$ gives $XY'+X'Y=0$, hence $Y' = - X^{-1} X' Y = -CY.$$\square$

Lemma A.4

The first order perturbation of eigenvalues is given by $\lambda _k ' = y_k G' x_k$.

Proof

We have $\Delta ' = (YGX)' = Y'GX + YG'X + YGX' = YG'X + YGXC - C YGX = YG'X + \Delta C - C \Delta = YG'X + [\Delta ,C]$. Therefore $\lambda _k' = (YG'X)_{kk} + [\Delta ,C]_{kk} = y_k G' x_k$. $\square$

Lemma A.5

For any $i \ne j$, $C_{ij} = {y_i G' x_j \over \lambda _j - \lambda _i}$.

Proof

For such i, j, $\Delta '_{ij} = 0$. With the same computation as in the previous lemma, this gives $(YG'X)_{ij} + [\Delta ,C]_{ij}=0$. Thus $(\lambda _i - \lambda _j) C_{ij} = - (YG'X)_{ij} = - y_i G' x_j$, from which the result follows. $\square$

Lemma A.6

For any $1\leqslant k\leqslant N$, $C_{kk} = - \sum _{l \ne k} X_{kl} {y_l G' x_k \over \lambda _k - \lambda _l}$.

Proof

We use the assumption $X_{kk}=1$. From this, and the definition of C, we get

$$\begin{aligned} X'_{kk}=0=(XC)_{kk} = \sum _{l=1}^n X_{kl} C_{lk} = X_{kk} C_{kk} + \sum _{l \ne k} X_{kl} C_{lk}. \end{aligned}$$

As a consequence, $C_{kk} = - \sum _{l \ne k} X_{kl} C_{lk}$ and we obtain the result thanks to the previous lemma. $\square$

From now on the differential operator will be either $\partial _{{{\,\mathrm{Re}\,}}G_{ab}}$ ($G' = E_{ab}= \{ \delta _{ia} \delta _{jb} \}_{1\leqslant i,j\leqslant N}$), or $\partial _{{{\,\mathrm{Im}\,}}G_{ab}}$, ($G' = \mathrm {i}E_{ab})$. In both cases, $G''=0$. We denote $C^{{{\,\mathrm{Re}\,}}}$ and $C^{{{\,\mathrm{Im}\,}}}$ accordingly. In particular, for any k and $i\ne j$ the following holds:

$$\begin{aligned}&\partial _{{{\,\mathrm{Re}\,}}G_{ab}} \lambda _k = Y_{ka} X_{b,k},\ \partial _{{{\,\mathrm{Im}\,}}G_{ab}}\lambda _k = \mathrm {i}Y_{ka} X_{b,k} \nonumber \\&C_{ij}^{{{\,\mathrm{Re}\,}}} = {Y_{ia} X_{bj} \over \lambda _j - \lambda _i},\ C_{kk}^{{{\,\mathrm{Re}\,}}} = - \sum _{l \ne k} X_{kl} { Y_{la} X_{b,k} \over \lambda _k - \lambda _l}, \nonumber \\&C_{ij}^{{{\,\mathrm{Im}\,}}} = \mathrm {i}{Y_{ia} X_{bj} \over \lambda _j - \lambda _i}, \ C_{kk}^{{{\,\mathrm{Im}\,}}} = - \mathrm {i}\sum _{l \ne k} X_{kl} { Y_{la} X_{b,k} \over \lambda _k - \lambda _l}. \end{aligned}$$

(A.2)

Lemma A.7

We have

$$\begin{aligned} \partial _{{{\,\mathrm{Re}\,}}G_{ab}} X_{ij}&= \sum _{l \ne j} (X_{il}- X_{ij} X_{jl} ) {Y_{la} X_{bj} \over \lambda _j - \lambda _l},\\ \partial _{{{\,\mathrm{Im}\,}}G_{ab}} X_{ij}&= \mathrm {i}\sum _{l \ne j} (X_{il}- X_{ij} X_{jl} ) {Y_{la} X_{bj} \over \lambda _j - \lambda _l}. \\ \partial _{{{\,\mathrm{Re}\,}}G_{ab}} Y_{ij}&= \sum _{l \ne i} \frac{1}{ \lambda _i - \lambda _l} (X_{il} Y_{la} X_{b,i} Y_{ij} + Y_{ia} X_{bl} Y_{lj}),\\ \partial _{{{\,\mathrm{Im}\,}}G_{ab}} Y_{ij}&= \mathrm {i}\sum _{l \ne i} \frac{1}{ \lambda _i - \lambda _l} (X_{il} Y_{la} X_{b,i} Y_{ij} + Y_{ia} X_{bl} Y_{lj}). \end{aligned}$$

Proof

Below is the computation for $\partial _{{{\,\mathrm{Re}\,}}G_{ab}}X_{ij}$. We use $X'=XC$ and (A.2):

$$\begin{aligned} X'_{ij}= & {} (XC)_{ij} = \sum _{l=1}^n X_{il} C_{lj} = \sum _{l \ne j} X_{il} {Y_{la} X_{bj} \over \lambda _j - \lambda _l} \\&- X_{ij} \sum _{l \ne j} X_{jl} { Y_{la} X_{bj} \over \lambda _j - \lambda _l} = \sum _{l \ne j} (X_{il}- X_{ij} X_{jl} ) {Y_{la} X_{bj} \over \lambda _j - \lambda _l}. \end{aligned}$$

The case $\partial _{{{\,\mathrm{Im}\,}}G_{ab}}X_{ij}$ is obtained similarly, as are the formulas for Y. $\square$

Lemma A.8

The second order perturbation of eigenvalues is given by

$$\begin{aligned} \partial _{{{\,\mathrm{Re}\,}}G_{ab}}^2 \lambda _k = 2 \sum _{l \ne k} { Y_{ka} X_{bl} Y_{la} X_{b,k} \over \lambda _k - \lambda _l},\ \ \partial _{{{\,\mathrm{Im}\,}}G_{ab}}^2 \lambda _k = -2 \sum _{l \ne k} { Y_{ka} X_{bl} Y_{la} X_{b,k} \over \lambda _k - \lambda _l}. \end{aligned}$$

Proof

We compute the perturbation for $\partial _{{{\,\mathrm{Re}\,}}G_{ab}}$. Differentiating $\lambda$ a second time gives

$$\begin{aligned} \lambda _k'' = y_k' G' x_k + y_k G'' x_k+ y_k G' x_k' = Y_{ka}' X_{b,k} + Y_{ka} X_{b,k}'. \end{aligned}$$

Replacing $X'$ and $Y'$ with their expressions yields

$$\begin{aligned} \lambda _k''&= \sum _{l \ne k} \frac{1}{ \lambda _k - \lambda _l} (X_{kl} Y_{la} X_{b,k} Y_{ka} + Y_{ka} X_{bl} Y_{la}) X_{b,k} \\&\quad +\, Y_{ka} \sum _{l \ne j} (X_{bl}- X_{b,k} X_{kl} ) {Y_{la} X_{b,k} \over \lambda _k - \lambda _l} \\&= \sum _{l \ne k} \frac{1}{ \lambda _k - \lambda _l} (X_{kl} Y_{la} X_{b,k} Y_{ka} X_{b,k} + Y_{ka} X_{bl} Y_{la} X_{b,k}\\&\quad + Y_{ka} X_{bl} Y_{la} X_{b,k}- Y_{ka}X_{b,k} X_{kl} Y_{la} X_{b,k}) \\&= 2 \sum _{l \ne k} { Y_{ka} X_{bl} Y_{la} X_{b,k} \over \lambda _k - \lambda _l}, \end{aligned}$$

which concludes the proof, the other cases being similar. $\square$

For the proof of Proposition A.1, we need the following elementary lemma.

Lemma A.9

Let $\tau =\inf \{t\geqslant 0:\exists i\ne j, \lambda _i(t)=\lambda _j(t)\}$. Then $\tau =\infty$ almost surely.

Proof

The set of matrices G with Jordan form of type

$$\begin{aligned} \lambda _1\oplus \dots \oplus \lambda _{N-2}\oplus \left( \begin{array}{cc} \lambda _{N-1}&{}1\\ 0&{}\lambda _{N-1}\end{array}\right) (\mathrm{respectively}\ \lambda _1\oplus \dots \oplus \lambda _{N-2}\oplus \lambda _{N-1}\oplus \lambda _{N-1}) \end{aligned}$$

is a submanifold ${\mathcal {M}}_1$ (resp. ${\mathcal {M}}_2$) of ${\mathbb {C}}^{N^2}$ with complex codimension 1 (resp. 3), see e.g. [36, 47]. Therefore, almost surely, a Brownian motion in ${\mathbb {C}}^{N^2}$ starting from a diagonalizable matrix with simple spectrum will not hit ${\mathcal {M}}_1$ or ${\mathcal {M}}_2$. This concludes the proof. $\square$

All derivatives can therefore be calculated, as eigenvalues and eigenvectors are analytic functions of the matrix entries (see [35]).

Proof of Proposition A.1

In our context,the Itô formula will take the following form: for a function f from ${\mathbb {C}}^n$ to ${\mathbb {C}}$ of class $C^2$, where $B_t=(B_t^1,\dots ,B_t^n)$ is made of independent standard complex Brownian motions, we have

$$\begin{aligned} \mathrm {d}f (B_t)= & {} \sum _{i=1}^n \Bigg ({\partial f \over \partial {{\,\mathrm{Re}\,}}z_i } \mathrm {d}{{\,\mathrm{Re}\,}}B_t^{i} + {\partial f \over \partial {{\,\mathrm{Im}\,}}z_i} \mathrm {d}{{\,\mathrm{Im}\,}}B_t^{i} \Bigg )\nonumber \\&+ \frac{1}{ 2} \Bigg (\sum _{i=1}^n {\partial ^2 f \over \partial {{\,\mathrm{Re}\,}}{z_i}^2 } + {\partial ^2 f \over \partial {{{\,\mathrm{Im}\,}}z_i}^2 } \Bigg ) \mathrm {d}t. \end{aligned}$$

(A.3)

For any given $0<{\varepsilon }<\min \{|\lambda _i(0)-\lambda _j(0)|,i\ne j\}$, let

$$\begin{aligned} \tau _{{\varepsilon }}=\inf \{t\geqslant 0:\exists i\ne j, |\lambda _i(t)-\lambda _j(t)|<{\varepsilon }\}. \end{aligned}$$

(A.4)

Eigenvalues are smooth functions of the matrix coefficients on the domain $\cap _{i<j}\{|\lambda _i-\lambda _j|>{\varepsilon }\}$, so that Eq. (A.3) together with Lemmas A.4 and A.8 gives the following equality of stochastic integrals, with substantial cancellations of the drift term:

$$\begin{aligned} \mathrm {d}\lambda _k(t\wedge \tau _{{\varepsilon }})= & {} \sum _{i,j=1}^n Y_{ki} X_{jk} \left( \frac{\mathrm {d}B_{ij}(t\wedge \tau _{{\varepsilon }})}{\sqrt{N}} - \frac{G_{ij} }{2}\mathrm {d}(t\wedge \tau _{\varepsilon })\right) \\&+ \frac{1}{N} \sum _{i,j,\ell \ne k}\Bigg ( { Y_{ki} X_{jl} Y_{li} X_{jk} \over \lambda _k - \lambda _l} - { Y_{ki} X_{jl} Y_{li} X_{jk} \over \lambda _k - \lambda _l} \Bigg ) \mathrm {d}(t\wedge \tau _{{\varepsilon }})\\= & {} \sum _{i,j=1}^n Y_{ki} X_{jk} \frac{\mathrm {d}B_{ij}(t\wedge \tau _{{\varepsilon }})}{\sqrt{N}} -\frac{1}{2} \sum _{i,j=1}^n Y_{ki} G_{ij} X_{jk} \mathrm {d}(t\wedge \tau _{{\varepsilon }})\\= & {} \sum _{i,j=1}^n Y_{ki} X_{jk} \frac{\mathrm {d}B_{ij}(t\wedge \tau _{{\varepsilon }})}{\sqrt{N}} -\frac{1}{2} \lambda _k \mathrm {d}(t\wedge \tau _{{\varepsilon }}). \end{aligned}$$

Taking ${\varepsilon }\rightarrow 0$ in the above equation together with Lemma A.9 yields

$$\begin{aligned} \mathrm {d}\lambda _k(t)= \sum _{i,j=1}^n Y_{ki} X_{jk} \frac{\mathrm {d}B_{ij}(t)}{\sqrt{N}} -\frac{1}{2} \lambda _k \mathrm {d}t. \end{aligned}$$

The eigenvalues martingales terms are correlated. Their brackets are

$$\begin{aligned}&\mathrm {d}\langle \lambda _i, {{\bar{\lambda }}}_j \rangle _t = \frac{1}{N} \sum _{a,b,c,d=1}^n Y_{ia} X_{b,i} \overline{Y_{jc} X_{d,j}} \mathrm {d}\langle B_{ab}, \overline{ \mathrm {d}B_{cd}}\rangle _t \nonumber \\&\quad = (X^{\mathrm{t}}{\overline{X}})_{ij} (Y Y^*)_{ij} \frac{\mathrm {d}t}{N} = {\mathscr {O}}_{ij}(t) \frac{\mathrm {d}t}{N}, \end{aligned}$$

(A.5)

$$\begin{aligned}&\mathrm {d}\langle \lambda _i, \lambda _j \rangle _t=0. \end{aligned}$$

(A.6)

This concludes the proof. $\square$

1.2 Proof of Corollary 1.6

Let $A_{t,{\varepsilon }}=\{\sup _{0\leqslant s\leqslant t}|\lambda _1(s)-\lambda _1(0)|<N^{\varepsilon }t^{1/2}\}$. We start by proving that

$$\begin{aligned} {\mathbb {P}}(A_{t,{\varepsilon }})=1-{{\,\mathrm{o}\,}}(1). \end{aligned}$$

(A.7)

From Proposition A.1 and Itô’s formula, we have

$$\begin{aligned} e^{\frac{t}{2}}\lambda _1(t)-\lambda _1(0)=\int _0^te^{\frac{s}{2}}\mathrm{d}M_1(s), \end{aligned}$$

(A.8)

which is a local martingale. It is an actual martingale because

$$\begin{aligned} {\mathbb {E}}\left( \langle \int _0^{\cdot }e^{\frac{s}{2}}\mathrm{d}M_1(s)\rangle _t\right) =\int _0^t{\mathbb {E}}\left( e^{\frac{s}{2}} \frac{{\mathscr {O}}_{11}(s)}{N}\mathrm{d}s\right) ={{\,\mathrm{O}\,}}(t)<\infty , \end{aligned}$$

(A.9)

where in the last equality we used ${\mathbb {E}}({\mathscr {O}}_{11}(s))={{\,\mathrm{O}\,}}(N)$, which follows from (2.11). The estimate (A.7) follows by Doob’s and Markov’s inequalities.

For (1.16), we start with

$$\begin{aligned} |e^{\frac{t}{2}}\lambda _1(t)-\lambda _1(0)|^2=2\mathrm{Re}\int _0^t\overline{e^{\frac{s}{2}}\lambda _1(s)-\lambda _1(0)}e^{\frac{s}{2}}\mathrm{d}M_1(s) +\int _0^te^s\frac{{\mathscr {O}}_{11}(s)}{N}\mathrm{d}s. \end{aligned}$$

(A.10)

This implies

$$\begin{aligned} {\mathbb {E}}\left( e^t|\lambda _1(t)-\lambda _1(0)|^2 \mathbb {1}_{\{\lambda _1(0)\in {\mathscr {B}}\}}\right) = \int _0^t{\mathbb {E}}\left( e^s\frac{{\mathscr {O}}_{11}(s)}{N} \mathbb {1}_{\{\lambda _1(0)\in {\mathscr {B}}\}}\right) \mathrm{d}s+{{\,\mathrm{o}\,}}(t). \end{aligned}$$

(A.11)

Here, we used that $(\mathrm{Re}\int _0^t\overline{e^{\frac{s}{2}}\lambda _1(s)-\lambda _1(0)}e^{\frac{s}{2}}\mathrm{d}M_1(s) )_t$ is an actual martingale, because the expectation of its bracket is

$$\begin{aligned}&\int _0^te^{s}{\mathbb {E}}\left( |e^{\frac{s}{2}}\lambda _1(s) -\lambda _1(0)|^2\frac{{\mathscr {O}}_{11}(s)}{N}\mathbb {1}_{\{\lambda _1(0)\in {\mathscr {B}}\}}\mathrm{d}s\right) \\&\quad \leqslant 2\int _0^te^{2s}{\mathbb {E}}\left( |\lambda _1(s)|^2+1) \frac{{\mathscr {O}}_{11}(s)}{N}\mathrm{d}s\right) <\infty , \end{aligned}$$

where for the last inequality we used (2.11).

To evaluate the right hand side of (A.11), we would like to change $\lambda _1(0)\in {\mathscr {B}}$ into $\lambda _1(s)\in {\mathscr {B}}$. First,

$$\begin{aligned}&\left| {\mathbb {E}}\left( \frac{{\mathscr {O}}_{11}(s)}{N} \mathbb {1}_{A_{t,{\varepsilon }}}\left( \mathbb {1}_{\lambda _1(0)\in {\mathscr {B}}}-\mathbb {1}_{\lambda _1(s)\in {\mathscr {B}}}\right) \right) \right| \nonumber \\&\quad \leqslant {\mathbb {E}}\left( \frac{{\mathscr {O}}_{11}(s)}{N} \mathbb {1}_{\mathrm{dist}(\lambda _1(s),\partial {\mathscr {B}})\leqslant N^{\varepsilon }t^{1/2}}\right) ={{\,\mathrm{O}\,}}(N^{\varepsilon }t^{1/2}), \end{aligned}$$

(A.12)

where for the last inequality we used (2.11), again. Moreover, if $1/p+1/q=1$ with $p<2$. we have

$$\begin{aligned} {\mathbb {E}}\left( \frac{{\mathscr {O}}_{11}(s)}{N} \mathbb {1}_{(A_{t,{\varepsilon }})^c}\right) \leqslant {\mathbb {E}}\left( \left( \frac{{\mathscr {O}}_{11}(s)}{N}\right) ^p\right) ^{1/p} {\mathbb {P}}\left( (A^{(1)}_{t,{\varepsilon }})^c\right) ^{1/q}={{\,\mathrm{o}\,}}(1), \end{aligned}$$

(A.13)

where we used [24, Theorem 2.3] to obtain that uniformly in the complex plane and in N, ${\mathscr {O}}_{11}/N$ has finite moment of order $p<2$. Equations (A.11), (A.12) and (A.13) imply

$$\begin{aligned} {\mathbb {E}}\left( |\lambda _1(t)-\lambda _1(0)|^2 \mathbb {1}_{\lambda _1(0)\in {\mathscr {B}}}\right) = \int _0^t{\mathbb {E}}\left( \frac{{\mathscr {O}}_{11}(s)}{N} \mathbb {1}_{\{\lambda _1(s)\in {\mathscr {B}}\}}\right) \mathrm{d}s+{{\,\mathrm{o}\,}}(t), \end{aligned}$$

and one concludes the proof of (1.16) with (2.11).

The proof of (1.17) is identical, except that we rely on the off-diagonal bracket $\mathrm{d}\langle \lambda _1,\bar{\lambda }_2\rangle _s={\mathscr {O}}_{12}(s)\frac{\mathrm{d}s}{N}$, the estimate (1.9), and the elementary inequality

$$\begin{aligned} |{\mathscr {O}}_{12}|=|(R_j^* R_i)(L_j^* L_i)|\leqslant & {} \Vert R_j\Vert \Vert R_i\Vert \Vert L_j\Vert \Vert L_i\Vert \\\leqslant & {} \frac{1}{2}\left( \Vert R_i\Vert ^2 \Vert L_i\Vert ^2+\Vert R_j\Vert ^2\Vert L_j\Vert ^2\right) =\frac{1}{2} \left( {\mathscr {O}}_{11}+{\mathscr {O}}_{22}\right) \end{aligned}$$

to bound the (first and p-th) moment of ${\mathscr {O}}_{12}$ in the whole complex plane based on those of ${\mathscr {O}}_{11}$, ${\mathscr {O}}_{22}$.

1.3 Real Ginibre dynamics

We now consider G(0) a real matrix of size N, again assumed to be diagonalized as $YGX = \Delta = \mathrm {Diag}(\lambda _1,\dots ,\lambda _N)$, where X, Y are the matrices of the right- and left-eigenvectors of G(0). We also assume that G(0) has simple spectrum, and X, Y invertible. We keep the same notations for the right eigenvectors $(x_i)$, columns of X, and the left-eigenvectors $(y_j)$, rows of Y. They are again chosen such that $XY=I$ and, for any $1\leqslant k\leqslant N$, $X_{kk}=1$.

In this subsection, the real Dyson-type dynamics are ($1\leqslant i,j\leqslant N$),

$$\begin{aligned} \mathrm{d}G_{ij}(t)=\frac{\mathrm{d}B_{ij}(t)}{\sqrt{N}}-\frac{1}{2}G_{ij}(t)\mathrm{d}t, \end{aligned}$$

(A.14)

where the $B_{ij}$’s are independent standard Brownian motions. One can easily check that G(t) converges to the real Ginibre ensemble as $t\rightarrow \infty$.

Note that the real analogue of Lemma A.9 gives weaker repulsion: the set of real matrices with Jordan form of type

$$\begin{aligned} \lambda _1\oplus \dots \oplus \lambda _{N-2}\oplus \left( \begin{array}{cc} \lambda _{N-1}&{}1\\ 0&{}\lambda _{N-1}\end{array}\right) \end{aligned}$$

is a submanifold ${\mathcal {M}}_1$ of ${\mathbb {R}}^{N^2}$, supported on $\lambda _{N-1}\in {\mathbb {R}}$, with real codimension 1 (as proved by a straightforward adaptation of [36, Theorem 7]). Denoting $\tau =\inf \{t\geqslant 0:\exists i\ne j, \lambda _i(t)=\lambda _j(t)\}$, under the dynamics (A.14) for any $t>0$ we therefore have

$$\begin{aligned} {\mathbb {P}}(\tau <t)>0, \end{aligned}$$

so that we can only state the real version of Proposition A.1 up to time $\tau$. In fact, collisions occur transforming pairs of real eigenvalues into pairs of complex conjugate eigenvalues, a mechanism coherent with the random number of real eigenvalues in the real Ginibre ensemble [22, 41].

The overlaps (1.4) are enough to describe the complex Ginibre dynamics, and so are they for the real Ginibre ensemble, up to the introduction of the following notation: we define $\bar{i}\in \llbracket 1,N\rrbracket$ through $\lambda _{\bar{i}}=\overline{\lambda _i}$, i.e. ${\bar{i}}$ is the index of the conjugate eigenvalue to $\lambda _i$. Note that ${\bar{i}}=i$ if $\lambda _i\in {\mathbb {R}}$. For real matrices, if $L_j,R_j$ are eigenvectors associated to $\lambda _j$, ${\bar{L}}_j,{\bar{R}}_j$ are eigenvectors for ${{\bar{\lambda }}}_j$, so that

$$\begin{aligned} {\mathscr {O}}_{i{\bar{j}}}=({\bar{R}}_j^* R_i)({\bar{L}}_j^* L_i)=(R_j^{\mathrm{t}} R_i)(L_j^{\mathrm{t}} L_i). \end{aligned}$$

Proposition A.10

The spectrum $(\lambda _1(t),\dots , \lambda _n(t))$ evolves according to the following stochastic equations, up to the first collision:

$$\begin{aligned} \mathrm {d}\lambda _k(t\wedge \tau ) = \mathrm{d}M_k(t\wedge \tau )+ \left( \sum _{l\ne k}\frac{ {{\mathscr {O}}}_{k\bar{l}}}{\lambda _k-\lambda _\ell } -\frac{1}{2} \lambda _k\right) \mathrm {d}(t\wedge \tau ) \end{aligned}$$

where the martingales $(M_k)_{1\leqslant k\leqslant N}$ have brackets

$$\begin{aligned} \mathrm{d}\langle M_i,M_j\rangle _{t\wedge \tau }={{\mathscr {O}}}_{i\bar{j}}(t)\frac{\mathrm{d}(t\wedge \tau )}{N},\ \ \mathrm{d}\langle M_i,\overline{M_j}\rangle _{t\wedge \tau }={\mathscr {O}}_{ij}(t)\frac{\mathrm{d}(t\wedge \tau )}{N}. \end{aligned}$$

Note that the real eigenvalues have associated real eigenvectors. For those, ${{\mathscr {O}}}_{k{\bar{l}}}={\mathscr {O}}_{kl}$, and the variation is real: real eigenvalues remain real as long as they do not collide.

Remark A.11

Proposition A.10 is coherent with the attraction between conjugate eigenvalues exhibited in [45]. In fact, if $\eta ={{\,\mathrm{Im}\,}}(\lambda _k)>0$, the drift interaction term with $\bar{\lambda }_k$ is ${\mathscr {O}}_{kk}/(\lambda _k-{{\bar{\lambda }}}_k)=-\mathrm {i}{\mathscr {O}}_{kk}/(2\eta )$, so that these eigenvalues attract each other stronger as they approach the real axis.

For the proof, we omit the details and only mention the differences with respect to Proposition A.1. We apply the Itô formula for a ${\mathscr {C}}^2$ function f from ${\mathbb {R}}^n$ to ${\mathbb {C}}$, with argument $U_t=(U_t^1,\dots ,U_t^n)$ is made of independent Ornstein-Uhlenbeck processes. Together with the perturbation formulas for $\lambda _k$, Lemmas A.4 and A.8 , we obtain (remember the notation (A.4))

$$\begin{aligned} \mathrm {d}\lambda _k(t\wedge \tau _{{\varepsilon }})= & {} \sum _{i,j=1}^n Y_{ki} X_{jk} \left( \frac{\mathrm {d}B_{ij}(t\wedge \tau _{{\varepsilon }})}{\sqrt{N}} - \frac{1}{2}G_{ij} \mathrm {d}(t\wedge \tau _{\varepsilon }\right) \\&+ \sum _{i,j} \sum _{l \ne k} { Y_{ki} X_{jl} Y_{li} X_{jk} \over \lambda _k - \lambda _l} \mathrm {d}(t\wedge \tau _{{\varepsilon }}) \\= & {} \sum _{i,j=1}^n Y_{ki} X_{jk} \frac{\mathrm {d}B_{ij}(t\wedge \tau _{{\varepsilon }})}{\sqrt{N}}\\&+\left( \sum _{l\ne k}\frac{ (X^tX)_{lk} (YY^t)_{kl}}{\lambda _k-\lambda _\ell } -\frac{1}{2} \lambda _k \right) \mathrm {d}(t\wedge \tau _{{\varepsilon }}). \end{aligned}$$

We can take ${\varepsilon }\rightarrow 0$ in the above formulas and the brackets are calculated as follows, concluding the proof:

$$\begin{aligned}&\mathrm {d}\langle \lambda _i, {{\bar{\lambda }}}_j \rangle _{t\wedge \tau } = \frac{1}{N} \sum _{a,b,c,d=1}^n Y_{ia} X_{b,i} \overline{Y_{jc} X_{d,j}} \mathrm {d}\langle B_{ab}, \overline{ \mathrm {d}B_{cd}}\rangle _{t\wedge \tau }\nonumber \\&\quad = (X^{\mathrm{t}}{\overline{X}})_{ij} (Y Y^*)_{ij} \frac{\mathrm {d}(t\wedge \tau )}{N} = {\mathscr {O}}_{ij}(t) \frac{\mathrm {d}(t\wedge \tau )}{N}, \end{aligned}$$

(A.15)

$$\begin{aligned}&\mathrm {d}\langle \lambda _i, \lambda _j \rangle _{t\wedge \tau }= \frac{1}{N} \sum _{a,b,c,d=1}^n Y_{ia} X_{b,i} {Y_{jc} X_{d,j}} \mathrm {d}\langle B_{ab},{ \mathrm {d}B_{cd}}\rangle _{t\wedge \tau }\nonumber \\&\quad = (X^{\mathrm{t}}{X})_{ij} (Y Y^{\mathrm{t}})_{ij} \frac{\mathrm {d}(t\wedge \tau )}{N} = {\mathscr {O}}_{i{\bar{j}}}(t) \frac{\mathrm {d}(t\wedge \tau )}{N}. \end{aligned}$$

(A.16)

Appendix B: Normalized eigenvectors

This paper focuses on the condition numbers and off-diagonal overlaps, but the Schur decomposition also easily gives information about other statistics such as the angles between eigenvectors. We include these results for the sake of completeness. We denote the complex angle as

$$\begin{aligned} \arg (\lambda _1,\lambda _2)={R_1^{*} R_2 \over \Vert R_1 \Vert \Vert R_2 \Vert }, \end{aligned}$$

where the phases of $R_1(1)$ and $R_2(1)$ can be chosen independent uniform on $[0,2\pi )$. We also define

$$\begin{aligned} \Phi (z) = {z \over \sqrt{1+|z|^2} }. \end{aligned}$$

Proposition B.1

Conditionally on $\lambda _1=z_1,\lambda _2=z_2$, we have

$$\begin{aligned} \arg (\lambda _1,\lambda _2) \overset{(\mathrm d)}{=}\Phi \left( {X\over \sqrt{N} |z_1 - z_2| }\right) \end{aligned}$$

where $X \sim {\mathscr {N}}_{{\mathbb {C}}}(0,\frac{1}{2} \mathrm{Id})$.

In particular, for $\lambda _1, \lambda _2$ at mesoscopic distance, the complex angle converges in distribution to a Dirac mass at 0. Therefore in such a setting eigenvectors strongly tend to be orthogonal: matrices sampled from the Ginibre ensemble are not far from normal, when only considering eigenvectors angles. The limit distribution becomes non trivial in the microscopic scaling $|\lambda _1 - \lambda _2| \sim N^{-1/2}$, it is the pushforward of a complex Gaussian measure by $\Phi$.

Proof

From Proposition 2.1 we know that $R_1^*R_2=R_{T,1}^*R_{T,2}$, $\Vert R_1\Vert =\Vert R_{T,1}\Vert$ and $\Vert R_2\Vert =\Vert R_{T,2}\Vert$, where $R_{T,i}$ (and $L_{T,i}$) are the normalized bi-orthogonal bases of right and left eigenvectors for T, defined as (2.2). The first eigenvectors are written $R_{T,1}=(1,0,\dots ,0)$ and $R_{T,2}=(a,1,0\dots ,0)$ where $a=-\bar{b}_2=-\frac{T_{12}}{\lambda _1-\lambda _2}$, with $T_{12}$ complex Gaussian ${\mathscr {N}}\left( 0,\frac{1}{2N}\mathrm{Id}\right)$, independent of $\lambda _1$ and $\lambda _2$. This gives

$$\begin{aligned} \arg (\lambda _1,\lambda _2) =-\frac{{\bar{b}}_2}{\sqrt{1+|b_2|^2}} \end{aligned}$$

and concludes the proof. $\square$

From Proposition B.1, the distribution of the angle for fixed $\lambda _1$ and random $\lambda _2$ can easily be inferred. For example, if $\lambda _2$ is chosen uniformly among eigenvalues in a macroscopic domain $\Omega \subset \{|z|<1\}$ with nonempty interior, we obtain the convergence in distribution ($X_\Omega$ is uniform on $\Omega$, independent of ${\mathscr {N}}$)

$$\begin{aligned} N | \arg (\lambda _1 ,\lambda _2) |^2 \underset{N\rightarrow \infty }{\rightarrow }\frac{|{\mathscr {N}}|^2}{|z_1-X_\Omega |^2}. \end{aligned}$$

When $z_1=0$ and $z_2$ is free, the following gives a more precise distribution, for finite N and in the limit.

Corollary B.2

Conditionally on $\{ \lambda _1 =0 \}$ we have

$$\begin{aligned} N | \arg (\lambda _1 ,\lambda _2) |^2 \overset{(\mathrm d)}{=} N\beta _{1,U_N}\overset{(\mathrm d)}{\longrightarrow }X \end{aligned}$$

where $U_N$ is an independent random variable uniform on $\{ 2,\dots ,N\}$, and X has density ${1 - (1+t) e^{-t} \over t^2} {\mathbf {1}}_{{\mathbb {R}}_+} (t).$

Proof

From Corollary 5.6, $N |\lambda _2|^2 \sim \gamma _{U_N}$. Together with Lemma 2.5, this gives

$$\begin{aligned} | \arg (\lambda _1 ,\lambda _2) |^2 = { {|{\mathscr {N}}|^2 \over N | \lambda _2|^2 } \over 1+ {|{\mathscr {N}}|^2 \over N | \lambda _2|^2 } } \overset{(\mathrm d)}{=} {\gamma _1 \over \gamma _1 + \gamma _{U_N}} \overset{(\mathrm d)}{=} \beta _{1,U_N}. \end{aligned}$$

The limiting density then follows from the explicit distribution of $\beta$ random variables. $\square$