Further Theory and Applications

Abstract

In addition to an analogy with the Boltzmann factor of the OCP with inverse temperature $\beta = 2$, the GinUE eigenvalue PDF permits an interpretation as the absolute value squared of the ground state wave function for spinless fermions in the plane subject to a perpendicular magnetic field. More generally, for $\beta$ twice an odd positive integer m the OCP Boltzmann factor occurs as proportional to the absolute value squared of Laughlin trial wavefunction in the theory of the fractional quantum Hall effect with filling fraction 1/m. In relation to the topic of chaotic behaviour in dissipative quantum systems, statistical properties of the now complex energy spectrum can be compared with those of bulk GinUE eigenvalues.

In addition to an analogy with the Boltzmann factor of the OCP with inverse temperature $\beta = 2$, the GinUE eigenvalue PDF permits an interpretation as the absolute value squared of the ground state wave function for spinless fermions in the plane subject to a perpendicular magnetic field. More generally, for $\beta$ twice an odd positive integer m the OCP Boltzmann factor occurs as proportional to the absolute value squared of Laughlin trial wavefunction in the theory of the fractional quantum Hall effect with filling fraction 1/m. In relation to the topic of chaotic behaviour in dissipative quantum systems, statistical properties of the now complex energy spectrum can be compared with those of bulk GinUE eigenvalues. This leads to consideration of the so-called dissipative spectral form factor, the second moment of the absolute value of the statistics $\textrm{Tr} \, X^k$, and simple dissipators associated with Lindblad operators, all of which can be calculated exactly for GinUE matrices. The remaining two sections of this chapter are on the singular values, and eigenvectors, respectively of GinUE matrices. Consideration of the singular values is natural in relation to the questions relating to the determinant of GinUE matrices, while there have been recent applications of the statistical properties of eigenvectors to the eigenstate thermalisation hypothesis for non-Hermitian Hamiltonians.

1 Fermi Gas Wave Function Interpretation

We have seen that the rewrite of (1.7), written in the exponential form (1.12), allows for the GinUE eigenvalue PDF to be interpreted as the Boltzmann factor for a particular Coulomb gas. If instead of an exponential form we use (2.12) to rewrite (1.7) as

$$\begin{aligned} \Big | \prod _{j=1}^N e^{- | z_j|^2/2} \det [ z_j^{k-1} ]_{j,k=1,\dots ,N} \Big |^2, \end{aligned}$$

(6.1)

then we are led to an interpretation as the absolute value squared of a ground state free Fermi quantum many-body wave function. Thus inside the absolute value of (6.1) is a Slater determinant of single-body wave functions $\{\phi _l(z) \}_{l=0,\dots ,N-1}$ with $\phi _l(z) = e^{- | z|^2/2} z^l$. What remains then is to identify the corresponding one-body Hamiltonian for quantum particles in the plane which have these single-body wave functions for the lowest energy states.

The appropriate setting for this task is a quantum particle confined to the xy-plane subject to a perpendicular magnetic field, (0, 0, B), $B > 0$. Fundamental to this setting is the vector potential $\textbf{A}$, related to the magnetic field by $\nabla \times \textbf{A}= (0,0,B)$. The so-called symmetric gauge corresponds to the particular choice $\textbf{A}= (-By,Bx,0)=:(A_x,A_y,0)$, which henceforth will be assumed. Physical quantities in this setting are m (the particle mass), e (particle charge), $\hbar$ (Planck’s constant), c (speed of light), which together with B are combined to give $\omega _c := e B/mc$ (cyclotron frequency) and $\ell := \sqrt{\hbar c/eB}$ (magnetic length).

Defining the generalised momenta and corresponding raising and lowering operators by

$$ \Pi _u = - i \hbar {\partial \over \partial u} + {e \over c} A_u \, \, (u = x,y), \qquad a^\dagger = {\ell \over \sqrt{2} \hbar } (\Pi _x + i \Pi _y), \qquad a = (a^\dagger )^\dagger , $$

allows the quantum Hamiltonian to be written in the harmonic oscillator like form $H_B = \hbar \omega _c(a^\dagger a + {1 \over 2})$ [166]. Important too are the quantum centre of orbit operators and associated raising and lowering operators

$$ U = u - {\ell ^2 \over \hbar } \Pi _u \, \, (U = X,Y; \, u=x,y), \qquad b^\dagger = {1 \over \sqrt{2} \ell } (X - i Y), \qquad b = (b^\dagger )^\dagger , $$

for which $X^2 + Y^2 = 2 \ell ^2 (b^\dagger b + {1 \over 2})$. The operators $\{a, a^\dagger \}$ commute with $\{b, b^\dagger \}$, implying that H and $X^2 + Y^2$ permit simultaneous eigenstates. A complete orthogonal set can be constructed using the raising operators according to

$$\begin{aligned} |n,m \rangle = {(a^\dagger )^n (b^\dagger )^m \over \sqrt{n! m!}} |0,0\rangle , \end{aligned}$$

(6.2)

with eigenvalues of H equal to $(n + {1 \over 2}) \hbar \omega _c$ and eigenvalue of $X^2 + Y^2$ equal to $(2m+1) \ell ^2$. The ground state $|0,0\rangle$ is characterised by $a |0,0\rangle = b|0,0\rangle = 0$, which can be checked to have the unique solution $|0,0\rangle \propto e^{-(x^2+y^2)/4 \ell ^2}$. From this, application of $(b^\dagger )^m$ gives $|0,m\rangle \propto \bar{z}^m e^{-|z|^2/4 \ell ^2}$, $z = x + i y$. Forming a Slater determinant with respect to the first N eigenstates of this type gives (6.1) with $\ell ^2 = 1/2$. Generally, states with quantum number $n=0$ and thus belonging to the ground state are said to be in the lowest Landau level. One remarks that the largest eigenvalue of $X^2 + Y^2$ is then $N - 1/2$, which is in keeping with the squared radius of the leading-order support in the circular law.

The above theory of a quantum particle in the plane subject to a perpendicular magnetic field can be recast to apply to a rotating quantum particle in the plane [325, 384]. There, for an appropriate rotation frequency, the infinite degenerate lowest Landau level in the magnetic interpretation becomes a unique ground state wave function. It is further true that the elliptic GinUE PDF (2.35) admits an interpretation as the absolute value squared of state in the lowest Landau level, and furthermore the corresponding orthogonal polynomials (2.37) can be constructed using a Bogolyubov transformation of $\{b, b^\dagger \}$ [257]. Also, the PDF on the sphere (2.52) permits an interpretation as the absolute value squared of the ground state wave function for a free Fermi gas on the sphere subject to a perpendicular magnetic field [311]. Another point of interest relates to the N-body Fermi ground state corresponding to the quantum Pauli Hamiltonian in the plane with a perpendicular inhomogeneous magnetic field B(x, y). The Hamiltonian $H_B$ defined above then is to be multiplied by the $2 \times 2$ identity matrix, and the spin coupling term $-(g\hbar /2m) B(x,y) \textrm{diag}\,(1/2,-1/2)$ added. With $B(x,y) = - {1 \over 2} \nabla ^2 W(x,y)$ for some real-valued W, and with the assumption $\Phi := \int B(x,y) \, dx dy < \infty$, the ground state for this model (which is spin polarised with all spins up) permits an exact solution for $g=2$ [8]. The ground state of normalisable eigenfunctions has degeneracy $[\Phi /2\pi \hbar ] =: N$, with basis of eigenfunctions $\{ z^j e^{W(x,y)/2 \hbar } \}_{j=0}^{N-1}$. This implies the Fermi many-body ground state (6.1) with $e^{-|z_j|^2/2}$ replaced by $e^{W(x_j,y_j)/2 \hbar }$ [7].

A single particle state formed as a linear superposition of $\{ | 0, m \rangle \}_{m=0}^N$ has the form

$$ \phi (z) = {1 \over \mathcal N \pi ^{1/2}} e^{-|z|^2/2} p(\bar{z}), \quad p(z) := \sum _{n=0}^N {\alpha _n \over \sqrt{n!} } z^n, $$

with $\mathcal N = (\sum _{j=0}^N | \alpha _n |^2 )^{1/2}$. Following [252] and choosing each $\alpha _n$ as an independent standard complex Gaussian one has that p(z) is an example of a complex Gaussian random polynomial, where the variance of each coefficient is 1/n!. Nearly ten years before Ginibre’s calculation of the GinUE eigenvalue PDF (1.7), Hammersley [315] obtained the explicit functional form of the zero PDF of p(z), which was found to be proportional to

$$\begin{aligned} {\prod _{1 \le j < k \le N} | z_k - z_j|^2 \over (\sum _{j=0}^N |e_j|^2)^N}, \end{aligned}$$

(6.3)

where $e_j$ is the j-th elementary symmetric polynomial in $\{ z_j \}$. One immediately notices that the product term $\prod _{1 \le j < k \le N} | z_k - z_j|^2$ is common to both (1.7) and (6.3). However, while (1.7) gives rise to a determinantal point process, it turns out that the point process implied by (6.3) is permanental [316]. Nonetheless, there are many striking similarities, including a circular law for the global eigenvalue density, a rapidly decaying simple functional form for the bulk truncated two-particle correlation

$$ \rho _{(2)}^T(z_1,z_2) = {1 \over \pi ^2} \Big ( f(|z_1-z_2|^2/2) - 1 \Big ), \quad f(x) = {1 \over 2} {d^2 \over d x^2} (x^2 \coth x ), $$

and a slowly decaying truncated two-point correlation at the boundary; see [252, 316]. In the limit $N \rightarrow \infty$ this Gaussian random polynomial forms what is referred to as the Gaussian analytic function [326].

Remark 6.1

1. Upon stereographic projection of the sphere to the plane, it is possible to write the quantum Hamiltoniacge particle in a constant perpendicular magnetic field in a form unified with the original planar case [203]. This involves the Kähler metric and potential, and permits a viewpoint which carries over to further generalise the space to higher-dimensional complex manifolds in $\mathbb C^m$. A point of interest is that doing so gives, for the bulk scaling limit of the corresponding N particle lowest Landau level state, the natural higher-dimensional analogue of the kernel (2.19) [89, 90].

2. The squared wavefunction for higher Landau levels (say the r-th) has been shown to give rise to the determinantal point process with bulk-scaled kernel

$$ K_\infty ^{r}(w,z) = L_r^0(|w-z|^2) e^{w \bar{z}} e^{-(|w|^2+|z|^2)/2}; $$

see e.g. [497, Proposition 2.5]. Allowing for mixing between Landau levels up to and including level r leads to squared wave functions giving rise to the same determinantal point process except for the replacement of Laguerre polynomials $L_r^0 \mapsto L_r^1$ in the kernel [309]. For finite N both kernels are built up from particular polyanalytic functions; see e.g. [2]. Extending [384], the precise mapping between the rotating fermions in the higher Landau levels and the polyanalytic Ginibre ensemble was established in [380]. Furthermore, its full counting statistics and generalisations to finite temperature were obtained in [381, 505], while [1, 148] relates the variance of the counting statistics to the entanglement entropy of the region.

3. In the theory of the fractional quantum Hall effect, constructing an anti-symmetric state with filling fraction of the lowest Landau level $\nu = 1/m$, for m an odd integer, plays a crucial role. To accomplish this, Laughlin [389] proposed the ground state wave function proportional to

$$\begin{aligned} \prod _{l=1}^N e^{-|z_l|^2/4 \ell ^2} \prod _{1 \le j < k \le N} (\bar{z}_k - \bar{z}_j)^m; \end{aligned}$$

(6.4)

note that with the assumption that m is odd, this is anti-symmetric as required for fermions. Moreover, it belongs to the lowest Landau level as follows from the theory in the text below (6.2). The absolute value squared of (6.4) coincides with the Boltzmann factor (1.12) with $\beta = 2m$, and the scaling $z_l \mapsto z_l/\sqrt{2m\ell ^2}$. From potential theoretic/Coulomb gas reasoning, the bulk density is therefore $1/(2m\pi \ell ^2)$. The factor of m in the denominator is in precise agreement with the requirement that the filling fraction be equal to 1/m.

4. The ground state N-body free spinless Fermi gas in the plane, without a magnetic field but confined by a radial harmonic potential, is also an example of a determinantal point process for which exact calculations are possible; see the recent review [179]. However, its statistical state is distinct from that of GinUE. Thus with a global scaling so that the support is the unit disk, the density profile is the $d=2$ Thomas–Fermi functional form ${2 \over \pi } (1 - |z|^2) \chi _{|z|<1}$, in contrast to the circular law (2.17). The bulk-scaled two-point correlation function (bulk density $1/4\pi$) is given in terms of the $J_1$ Bessel function

$$ \rho _{(2),\infty }^\textrm{hF}(z_1,z_2) = \Big ( {1 \over 4 \pi } \Big )^2 \bigg ( 1 - \Big ( {2J_1(|z_1 - z_2|) \over |z_1 - z_2|}\Big )^2 \bigg ), $$

in contrast to (2.24). This gives a decay proportional to $1/|z_1 - z_2|^3$ of $\rho _{(2),\infty }^{\textrm{hF}, T}$. Also, the edge-scaled correlation kernel now involves Airy functions [178], rather than the error function seen in (2.19). Universality results relating to many-body free Fermi ground states in dimension $d \ge 2$ have recently been obtained [184]. We highlight in particular the macroscopic fluctuation theorem for the linear statistic $G = \sum _j g(\textbf{r}_j/R)$, with g assumed sufficiently smooth and absolutely integrable, in the $R \rightarrow \infty$ limit [184, Th. III.2]:

$$\begin{aligned} {G - {R^d \omega _d \over (2 \pi )^d} \int _{\mathbb R^d} g(\textbf{r}) \, d^d \textbf{r}\over \sigma _d R^{(d-1)/2}} \mathop {\rightarrow }\limits ^\textrm{d} \textrm{N}[0,\Sigma (g)], \quad (\Sigma (g))^2 = \int _{\mathbb R^d} | \hat{g}(\textbf{r})|^2 |\textbf{r}| \, d^d \textbf{r}. \end{aligned}$$

(6.5)

Here $\omega _d = \pi ^{d/2}/\Gamma (1+d/2)$ is the volume of the Euclidean ball in $\mathbb R^d$, $\omega _d/(2 \pi )^d$ is the bulk density, $\sigma _d^2 := \omega _{d-1}/(2 \pi )^d$ and the Fourier transform has the definition $\hat{g}(\xi ) = {1 \over (2 \pi )^{d/2}} \int _{\mathbb R^d} e^{ -i \xi \cdot \textbf{r}} g(\textbf{r}) \, d^d \textbf{r}$. Note in particular that in contrast to (3.19), the variance of G now diverges with the scale R.

5. The functional form (6.3) consists of the product over differences times a many-body term, whereas the GinUE eigenvalue PDF (1.7) is the same product over differences times a single-body term. Another occurrence of the former structure, now with the product over differences raised to the power of $\beta$ as for the OCP, has recently appeared as the eigenvalue PDF for a particular tridiagonal non-Hermitian random matrix [438].

2 Quantum Chaos Applications

The pioneering works of Wigner and Dyson relating to the Hermitian random matrix ensembles was, as noted in Chap. 1, motivated by seeking a model for the (highly excited) energy levels of a complex quantum system. Later, in the 1980s, as a fundamental contribution to the then emerging subject of quantum chaos, Bohigas et al. [98] identified the correct meaning of a complex quantum system not by the number of particles but rather as one for which the underlying classical mechanics is chaotic. To test this prediction on say the numerically generated spectrum of a quantum billiard system, the energy levels (beyond some threshold to qualify as being highly excited) were first unfolded so that their local density became unity, and then their numerically determined statistical properties were compared against random matrix predictions for the appropriate symmetry class; see e.g. [307]. Most popular among the statistical properties have been the variance for the number of eigenvalues in a large interval, and the distribution of the spacing between successive eigenvalues.

A natural extension of these advances is to inquire about the spectrum of a dissipative chaotic quantum system, which due to the loss of energy need not be real. This question was taken up by Grobe, Haake and Sommers [304] for the specific model of a damped periodic kicked top. The quantum dynamics are specified by a subunitary density operator. It is the spectrum of this operator, which after unfolding, and considering only those eigenvalues in the upper half plane away from the real axis (there is a symmetry which requires that the eigenvalues come in complex conjugate pairs—see the recent paper [32] for a discussion of this point in a random matrix context) that was compared in [304] in a statistical sense to GinUE. Following from precedents in the Hermitian case, in the statistical quantity measured was the distribution of the radial spacing between closest eigenvalues, to be denoted by $P^\textrm{s,GinUE}(r)$ with the normalisation $\int _0^\infty P^\textrm{s,GinUE}(r) \, dr = 1$. This is the quantity $F_\infty (0;D_r)$ of Remark 3.1.1. Recalling (3.10) we therefore have

$$\begin{aligned} P^\textrm{s,GinUE}(r) = - {d \over d r} e^{r^2} \prod _{j=1}^\infty \Big ( 1 - {\gamma (j;r^2) \over \Gamma (j)} \Big ). \end{aligned}$$

(6.6)

It follows that for small r, $P^\textrm{s,GinUE}(r) \sim 2 r^3$, while it follows from (3.11) and the statement in the final sentence of the first paragraph of Sect. 3.1.2 that for large r, $\log P^\textrm{s,GinUE}(r) \sim -r^4/4$. A numerical plot can be obtained from the functional form (6.6). For the moments the formula $\langle r^p \rangle = p \int _0^\infty r^{p-1} e^{r^2} E_\infty (0;r) \, dr$ holds true. In particular, for the mean we calculate $\langle r \rangle = 1.142929\dots$.

A variation of the closest neighbour spacing for an eigenvalue at z is the complex ratio $(z^\textrm{c} - z)/(z^\textrm{nc} - z)$, where $z^\textrm{c}$ is the closest neighbour to z, and $z^\textrm{nc}$ is the next closest neighbour [477]. An approximation, with fast convergence properties to the large N form, has been given recently in [204].

Also very recently a non-Hermitian Hamiltonian realisation of GinUE has been obtained in the context of a proposed non-Hermitian q-body Sachdev–Ye–Kitaev (SYK) model, with N Majorana fermions—N large and tuned $\textrm{mod} \, 8$—and $q>2$ and tuned $\textrm{mod} \, 4$ [281]. On another front, again very recently, the emergence of GinUE behaviours in certain models of many-body quantum chaotic systems in the space direction has been demonstrated [499]. Of interest in both these lines of study is the so-called dissipative (connected) spectral form factor

$$ \textrm{K}_N^\textrm{c}(t,s) = {1 \over N} \textrm{Cov} \, \Big ( \sum _{j=1}^N e^{i (x_j t + y_j s)}, \sum _{j=1}^N e^{-i (x_j t + y_j s)} \Big ). $$

Making use of the first formula in (3.2) and the finite N form of (2.23), this can be evaluated in terms of the hypergeometric function $_1F_1$ [404, Eq. (3)] (corrected in [282, Appendix A]; note too that both those references use global-scaled variables, whereas we do not).

Proposition 6.1

We have

$$\begin{aligned} \textrm{K}_N^\textrm{c}(t,s) & = 1 - {1 \over N} \sum _{m,n=0}^{N-1} {(t^2+s^2)^{ |m-n|/2 } \over n! m! 2^{|m-n|}} \nonumber \\ &\quad \times \bigg ( {\textrm{max}(m,n)! \over |m-n|!} \, _1F_1\Big (\textrm{max}(m,n)+1,|m-n|+1;- {t^2 + s^2 \over 4 }\Big ) \bigg )^2. \end{aligned}$$

(6.7)

In particular,

$$\begin{aligned} \lim _{N \rightarrow \infty } \textrm{K}_N^\textrm{c}( t, s) = 1 - e^{-(t^2+s^2)/4}; \end{aligned}$$

(6.8)

cf. (3.17).

Remark 6.2

From [404] one has the large N form

$$ {1 \over N} \textrm{K}_N^\textrm{c}\Big ( {t \over \sqrt{N}}, {s \over \sqrt{N}} \Big ) \sim {1 \over N} + 4 {J_1(|\tau |^2) \over |\tau |^2} - {1 \over N} e^{-|\tau |^2/4N}, \quad \tau :=t+is, $$

extending the limit formula (6.8) and exhibiting a so-called slope-dip-ramp-plateau graphical form; see also [163].

Also of interest in the many-body quantum chaos application is the GinUE average of $|\textrm{Tr} \, X^k|^2$ for positive integer k [499].

Proposition 6.2

We have

$$ \Big \langle |\textrm{Tr} \, X^k|^2 \Big \rangle _\textrm{GinUE} = {1 \over (k+1) (N-1)!} \Big ( (k+N)! - {N! (N-1)! \over (N - k - 1)!} \Big ). $$

In particular

$$ \lim _{N,k \rightarrow \infty \atop k/N=x} {1 \over k N^k} \Big \langle |\textrm{Tr} \, X^k|^2 \Big \rangle _\textrm{GinUE}= {2 \sinh (x^2/2) \over x^2}. $$

Proof

(Sketch) In [499] the average is reduced to $\int _{\mathbb C} dz_1 \int _{\mathbb C} dz_2 \, \rho _{(2),N}(z_1,z_2) z_1^k \bar{z}_1^k$; see too the earlier work [374]. The evaluation of a more general quantity can be found in [268, Corollary 4].    $\square$

We conclude this section with a brief account of the use of the GinUE in an ensemble theory of Lindblad dynamics [133, 189, 478]. This relates to the evolution of the density matrix $\rho _t$ for an N-level dissipative quantum system in the so-called Markovian regime, specified by the master equation $\dot{\rho }_t = \mathcal L (\rho _t)$. Here the operator $\mathcal L$ assumes a special structure identified by Lindblad [406], and by Gorini, Kossakowski, and Sudarshan [299]. Specifically, $\mathcal L$ consists of the sum of two terms, the first corresponding to the familiar unitary von Neumann evolution, and the second to a dissipative part, being the sum over operators $D_L$ (referred to as simple dissipators), represented as $N^2 \times N^2$ matrices according to

$$ D_L = 2 L \otimes _T L^\dagger - L^\dagger L \otimes _T \mathbb I_N - \mathbb I_N \otimes _T L^\dagger L, $$

for some $N \times N$ matrix L. Here $A \otimes _T B := A \otimes B^T$, where $\otimes$ is the usual Kronecker product. In an ensemble theory, there is interest in $F_N(t):= {1 \over N^2} \langle \textrm{Tr} \, e^{t D_L } \rangle _L$ [133].

Proposition 6.3

Let L be chosen from GinUE with global scaling. We have

$$ \lim _{N \rightarrow \infty } F_N(t) = e^{-4t} \Big ( I_0(2t) + I_1(2t) \Big )^2. $$

Proof

(Sketch) Following Can [133], using a diagrammatic calculus, it is first demonstrated that

$$ \lim _{N \rightarrow \infty } {1 \over N^2} \langle D_L^k \rangle = \lim _{N \rightarrow \infty } {(-1)^k \over N^2} \Big \langle \Big ( L^\dagger L \otimes _T \mathbb I_N + \mathbb I_N \otimes _T L^\dagger L \Big )^k \Big \rangle _{L \in \textrm{GinUE}}. $$

The average on the RHS, in terms of the eigenvalues $\{x_j\}$ of $L^\dagger L$, reads

$$ \Big \langle \sum _{j,l=1}^N (x_j + x_l)^k \Big \rangle _{L^\dagger L} $$

and consequently

$$ \lim _{N \rightarrow \infty } F_N(t)= \lim _{N \rightarrow \infty } \Big \langle {1 \over N^2} \sum _{j,l=1}^N e^{-t(x_j + x_l)} \Big \rangle _{L^\dagger L} = \lim _{N \rightarrow \infty } \bigg ( \Big \langle {1 \over N} \sum _{j=1}^N e^{-tx_j} \Big \rangle _{L^\dagger L} \bigg )^2. $$

The latter is the mean of a linear statistic in the ensemble $\{ L^\dagger L \}$ (complex Wishart matrices; see e.g. [237, Sect. 3.2]). Using the Marchenko–Pastur law for the global density of this ensemble (see e.g. [237, Sect. 3.4.1]), the stated result follows.    $\square$

Remark 6.3

(Classification of non-Hermitian matrices) It was commented in the Introduction that, in distinction to Dyson’s viewpoint based on symmetry considerations, Ginibre’s study [293] was not similarly motivated. Nowadays however, it is recognised that a symmetry viewpoint is fundamental to topologically driven effects in non-Hermitian quantum physics [67]. Starting with [91, 418] and continuing in [359], a classification scheme based on symmetries with respect to the involutions of transpose, complex conjugation and Hermitian conjugation, and in which the (anti-)commutation relation involves unitary matrices satisfying certain quadratic relations in terms of these involution, has been given. For example, defining the block unitary matrix $P=\textrm{diag} \, (\mathbb I_N, - \mathbb I_N)$, and requiring that the matrix ensemble $\{A \}$ have the (anti-)symmetry $A = - P A P$, gives that each A has the form

$$\begin{aligned} A = \begin{bmatrix} 0_{N \times N} & X \\ Y & 0_{N \times N} \end{bmatrix} \end{aligned}$$

(6.9)

for some square matrices X, Y. Denoting the eigenvalues of the matrix product XY as $\{ -z_j^2 \}$, one sees that the eigenvalues of A are $\{ \pm i z_j \}$. In the realm of non-Hermitian classifications, this is referred to as the chiral class [456]. Additionally, the chiral models corresponding to GinSE and GinOE were studied in [11] and [38], respectively.

In keeping with the viewpoint of this section, a basic question concerns signatures of the symmetry in the eigenvalue spectrum. For example, in (6.9), with X, Y GinUE matrices, are the bulk-scaled eigenvalues of A statistically distinct from individual GinUE matrices? We know from the results quoted in the paragraph above Remark 2.6 that the answer in this case is no. However, the answer to this question is yes, if instead the symmetry is that $A = A^T$, for the independent entries of A standard complex Gaussians. This was demonstrated in [313] by a numerical study of the nearest neighbour spacing distribution, and the relevance to Lindblad dynamics discussed.

3 Singular Values

One recalls that for a complex square matrix X the squared singular values are the eigenvalues of $X^\dagger X$. For a general ensemble of non-Hermitian matrices $\{X\}$, motivation to study the singular values comes from various viewpoints. For example, in Remark 2.6.4, singular values (specifically of product matrices) appeared in the context of Lyapunov exponents. As other example, one recalls that plus/minus of the singular values are the eigenvalues of the $2N \times 2N$ Hermitian matrix

$$\begin{aligned} H = \begin{bmatrix} 0_{N \times N}& X \\ X^\dagger & 0_{N \times N} \end{bmatrix}. \end{aligned}$$

(6.10)

The importance of this in relation to the eigenvalues of X is that the resolvent of the modification of (6.10) obtained by replacing each X by $X - z \mathbb I_N$ is fundamental to the study of the circular law for the spectral density beyond the Gaussian case; see e.g. [100, Sect. 4.1]. Another piece of theory is that the condition number $\kappa _N$ associated with X is equal to the ratio of the smallest to the largest singular value [210]. We remark too that from the identity $| \det X| = | \det X^\dagger X|^{1/2}$ the distribution of the modulus of $\det X$ is determined by the singular values.

For the GinUE, the squared singular values $\{s_j\}_{j=1}^N$ say are known to have for their joint distribution a PDF proportional to

$$\begin{aligned} \prod _{j=1}^N e^{-s_j} \prod _{1 \le j < k \le N} (s_k - s_j)^2, \quad s_j \in \mathbb R_+; \end{aligned}$$

(6.11)

see e.g. [237, Proposition 3.2.2 with $\beta = 2$, $n=m=N$]. This is an example of the Laguerre unitary ensemble (LUE). After scaling by N, almost surely the largest squared singular value has the limiting value 4 [459]. However, after the same scaling, a simple change of variables in (6.11) integrated from $(s,\infty )$ in each variable reveals that the smallest singular value is an exponential random variable with rate parameter $N^2$. Putting these facts together implies that for large N, $\kappa _N/N$ is distributed according to the heavy tailed distribution with PDF ${8 \over x^3} e^{-4/x^2}\chi _{x > 0}$ [210]. Also, for $n \times N$ ($n \ge N$) rectangular GinUE matrices it is proved in [153] that $\langle \log \kappa _N \rangle < {N \over |n-N|+1} +c$, where $c=2.24$, for any $N \ge 2$.

Let $P_N(t)$ denote the PDF for the distribution of $|\det X|^2$ for GinUE matrices. Making use of knowledge of the PDF of squared singular values (6.11) shows that the Mellin transform of $P_N(t)$ is equal to the multiple integral

$$\begin{aligned} {1 \over C_N} \int _0^\infty ds_1 \cdots \int _0^\infty ds_N \, \prod _{j=1}^N s_j^{s-1} e^{-s_j} \prod _{1 \le j < k \le N} (s_k - s_j)^2 = \prod _{j=0}^{N-1}{ \Gamma (s+j) \over \Gamma (1+j)}. \end{aligned}$$

(6.12)

Here the normalisation $C_N$ is such that the expression equals unity for $s=1$, while the evaluation of the multiple integral follows as a special case of the Laguerre weight Selberg integral; see e.g. [237, Proposition 4.7.3]. As noted in [269, Eq. (2.17)] (see also [475, Proposition 2.2]), it follows immediately from this that

$$\begin{aligned} | \det X |^2 \mathop {=}\limits ^\textrm{d} \prod _{l=1}^N {1 \over 2} \chi _{2l}^2. \end{aligned}$$

(6.13)

(Proposition 2.14 regarding the independent distribution of the absolute values of the eigenvalues.) In words this says that the absolute value squared of the determinant of GinUE matrices is equal in distribution to the product of N independent chi-squared distributions, with degrees of freedom $2,4,\dots ,2N$, each scaled by a factor of 2. Starting from (6.13), and defining the global-scaled GinUE matrices $X^\textrm{g}$ by $X^\textrm{g} = {1 \over \sqrt{N}} X$, the distribution of $\log |\det X^\textrm{g}|^2$ can be shown to have leading-order mean $-N$, variance $\log N$, and after recentring and rescaling satisfy a central limit theorem [475, Th. 3.5]. For a general linear statistic $\sum _{j=1}^N f(z_j)$ of global-scaled GinUE matrices, the leading-order mean is ${N \over \pi } \int _{|z|<1} f(z) \, d^2z$. For $f(z) = \log |z|^2$, this gives the stated value of $-N$. Also, we notice that substituting this choice of f(z) in the variance formula implied by (3.21) gives ${1 \over \pi } \int _{|\textbf{r}|<1} {1 \over x^2+y^2} \, dx dy$, which is not integrable at the origin, in keeping with the variance actually diverging as $\log N$.

There is an alternative viewpoint on the result (6.13) which does not require knowledge of the joint distribution of the singular values (6.11), nor the evaluation of the multiple integral (6.12). The idea, used in both [269, 475] and which goes back to Bartlett [74] in the case of real Gaussian matrices, is to decompose X in terms of its QR (Gram–Schmidt) decomposition. The matrix of orthonormal vectors Q constructed from the columns of X will for $X \in \textrm{GinUE}$, be a Haar distributed unitary matrix, which we denote by U. The matrix $R=[r_{jk}]_{j,k=1}^N$ is upper triangular with diagonal elements real and positive. One notes

$$\begin{aligned} \det X^\dagger X = \prod _{j=1}^N r_{jj}^2, \end{aligned}$$

(6.14)

and so it suffices to have knowledge on the distribution of $\{r_{jj}\}_{j=1}^N$ for X.

Proposition 6.4

Let $\{r_{jj}\}_{j=1}^N$ denote the diagonal elements in the QR decomposition of a GinUE matrix X. We have

$$\begin{aligned} r_{jj}^2 \mathop {=}\limits ^\textrm{d} {1 \over 2} \chi _{2j}^2. \end{aligned}$$

(6.15)

Proof

The QR decomposition $X = UR$ gives the corresponding decomposition of measure (see e.g. [237, Proposition 3.2.5])

$$ (dX) = \prod _{j=1}^N r_{jj}^{2(N-j)+1} (dR) (U^\dagger dU), $$

where as anticipated $(U^\dagger dU)$ is recognised as the Haar measure on the space of complex unitary matrices. The element distribution of GinUE matrices is proportional to $e^{-\textrm{Tr} \, X^\dagger X} = e^{- \sum _{1 \le j \le k \le N} |r_{jk}|^2}$. The various factorisations imply that integrating over U and the off-diagonal elements of R only changes the normalisation. We then read off that each $r_{jj}$ has a distribution with PDF proportional to $r^{2(N-j)+1}e^{-r^2}$, which implies (6.15).    $\square$

Using (6.15) in (6.14) reclaims (6.13).

Remark 6.4

1. The fact that the QR decomposition of a Ginibre matrix gives rise to a Haar distributed random unitary matrix provides a practical realisation of the latter. However, the QR decomposition of in-built mathematical software may not generate the triangular matrix R with strictly positive entries, which affects the properties of Q. Further details, and how this can be fixed, are discussed in [213].

2. Since with $\{z_j\}$ the eigenvalues of X, $| \det X |^2 = \prod _{j=1}^N | z_j|^2$, the fact that the Mellin transform of the distribution of this quantity is given by the product of gamma functions in (6.12) implies

$$\begin{aligned} \Big \langle \prod _{l=1}^N |z_l|^{2(s-1)} \Big \rangle _\textrm{GinUE}^\textrm{g} = N^{N(s-1)}\prod _{j=0}^{N-1} {\Gamma (s+j) \over \Gamma (1+j)}. \end{aligned}$$

(6.16)

Here the superscript “g” indicates the use of global scaling coordinates $z_l \mapsto \sqrt{N} z_l$. We observe that the knowledge of the induced GinUE normalisation $C_{n,N}$ in Proposition 2.8 provides a direct derivation of (6.16). For large N this ratio of gamma functions can be written in terms of the Barnes G-function according to ${G(N+s) \over G(N+1) G(s)}$; see [237, Eq. (4.183)]. Known asymptotics for ratios of the Barnes G-function (see e.g. [237, Eq. (4.185)]) then gives that for large N, and with $s = \gamma /2+1$ for convenience,

$$\begin{aligned} \Big \langle \prod _{l=1}^N |z_l|^{\gamma } \Big \rangle _\textrm{GinUE}^\textrm{g} \sim N^{\gamma ^2/8} e^{-(\gamma /2)N} {(2 \pi )^{\gamma /4} \over G(1 + \gamma /2)}. \end{aligned}$$

(6.17)

This is the special case $z=0$ of an asymptotic formula for $\langle \prod _{l=1}^N |z-z_l|^{\gamma } \rangle _\textrm{GinUE}^\textrm{g}$ given by Webb and Wong [531, Th. 1.1].

3. It is a standard result in random matrix theory (see e.g. [459]) that the density of singular values in (6.11), after the global scaling $s_j \mapsto s_j N$, has the particular Marchenko–Pastur form $\rho ^\textrm{MP}_{(1),\infty }(x) = {1 \over 2 \pi } ({4 - x \over x} )^{1/2} \chi _{0 < x < 4}$. The k-th moment of the density is given in terms of the specific mixed moment of a global-scaled Ginibre matrix $\tilde{G}$, $\langle \textrm{Tr} (\tilde{G}^\dagger G)^k \rangle$. The calculation of these moments for large N relates to free probability—see the recent introductory text [464] for the main ideas—and to combinatorics as is seen from the fact that $\int _0^4 x^k \rho ^\textrm{MP}_{(1),\infty }(x) \, dx= C_k$, where $C_k$ denotes the k-th Catalan number. Works on mixed moments of Ginibre matrices include [176, 191, 312, 530]. An explicit example is the average in Proposition 6.2. The scaling limit therein where both N and k become large with their ratio fixed is called the BMN large N limit; see e.g. [40, Sect. 6]. Mixed moments of the pseudo inverse of rectangular GinUE are calculated in [168].

4. The squared singular values as specified by the PDF (6.11) form a determinantal points process, being a special case of the LUE; see e.g. [237, Chaps. 3 and 5]. This is similarly true of the squared singular values of the various extensions of GinUE considered above: for example in the case of the spherical model and truncated unitary matrices, it is the classical Jacobi unitary ensemble (JUE) which arises, while the singular values of products of GinUE matrices, or of truncated unitary matrices, gives rise to a class of determinantal point processes called Pólya ensembles [270, 365, 366, 378]. A determinantal point process also results from the squared singular value of the so-called shifted GinUE—matrices $G+A$ for $G \in \textrm{GinUE}$ and A fixed [81, 263, 377]. A notable exception is the singular values of elliptic GinUE matrices, which form a Pfaffian point process [351]. Similarly, with $G \in \textrm{GinUE}$ and $U \in U(N)$ chosen with Haar measure, the squared singular values of $X;=(\mathbb I_N + U)G$ form a Pfaffian point process [259]. The random matrix X occurs in the construction of the Bures–Hall measure on the space of density matrices [541].

5. Remark 6.3 drew attention to a symmetry based classification of non-Hermitian matrices. The recent work [360] discusses properties of the corresponding singular values.

4 Eigenvectors

Associated with the set of eigenvalues $\{z_j\}$ of a Ginibre matrix G are two sets of eigenvectors—the left eigenvectors $\{ \boldsymbol{\ell }_j \}$ such that $\boldsymbol{\ell }_j^T G = x_j \boldsymbol{\ell }_j^T$, and the right eigenvectors $\{ \textbf{r}_j \}$ such that $G \textbf{r}_j = x_j \textbf{r}_j$. These are not independent, but rather (upon suitable normalisation), form a biorthogonal set

$$\begin{aligned} \boldsymbol{\ell }_i^T \textbf{r}_j = \delta _{i,j}. \end{aligned}$$

(6.18)

This property follows from the diagonalisation formula $G = X D X^{-1}$, where the columns of X are the right eigenvectors, D the diagonal matrix of eigenvectors, and the rows of $X^{-1}$ are the left eigenvectors. For nonzero scalars $\{c_i\}$ we see that (6.18) is unchanged by the rescalings $\textbf{r}_j \mapsto c_j \textbf{r}_j$ and $\boldsymbol{\ell }_j \mapsto (1/c_j) \boldsymbol{\ell }_j$.

For $N \times 1$ column vectors $\textbf{u}, \textbf{v}$, define the inner product $\langle \textbf{u}, \textbf{v}\rangle := \bar{ \textbf{u}}^T \textbf{v}$. The so-called overlap matrix has its elements $\mathcal O_{ij}$ expressed in terms of this inner product according to

$$\begin{aligned} \mathcal O_{ij} := \langle \boldsymbol{\ell }_i, \boldsymbol{\ell }_j \rangle \langle \textbf{r}_i, \textbf{r}_j \rangle . \end{aligned}$$

(6.19)

Note that this is invariant under the mappings noted in the final sentence of the above paragraph, and for fixed i and summing over j gives 1. Also, it follows from (6.19) that the diagonal entries relate to the lengths

$$\begin{aligned} \mathcal O_{jj} = || \boldsymbol{\ell }_j ||^2 || \textbf{r}_j ||^2. \end{aligned}$$

(6.20)

The square root of this quantity is known as the eigenvalue condition number; see the introduction to [107] and [173, Sect. 1.1] for further context and references. Significant too is the fact that the overlaps (6.20) appear in the specification of a Dyson Brownian motion extension of GinUE [107, 220, 303].

Statistical properties of $\{ O_{ij} \}$ for GinUE were first considered by Chalker and Mehlig [146, 147]. By the Schur decomposition (2.2), instead of a GinUE matrix G, we may consider an upper triangular matrix Z with the eigenvalues $\{z_j\}$ of G on the diagonal, and standard complex Gaussians, as off-diagonal entries. For the eigenvalue $z_1$, the triangular structure shows that $\boldsymbol{\ell }_1 =(1,b_2,\dots ,b_N)^T$ and $\textbf{r}_1=(1,0,\dots ,0)^T$ where for $p > 1$ and $b_1 = 1$, $b_p = {1 \over z_1 - z_p} \sum _{q=1}^{p-1} b_q Z_{qp}$. From this last relation, it follows that with $\boldsymbol{\ell }_1^{(n)} =(1,b_2,\dots ,b_n)^T$ for $n < N$ we have

$$\begin{aligned} || \boldsymbol{\ell }_1^{(n+1)} ||^2 = || \boldsymbol{\ell }_1^{(n)} ||^2 \Big ( 1 + {1 \over | z_1 - z_{n+1}|^2} \Big | \sum _{q=1}^n \tilde{b}_q Z_{q (n+1)} \Big |^2 \Big ), \qquad \tilde{b}_q := {b_q \over \sqrt{\sum _{q=1}^n |b_q|^2} }. \end{aligned}$$

(6.21)

This has immediate consequences in relation to $\mathcal O_{11}$ as shown by Bourgade and Dubach [107].

Proposition 6.5

Let the eigenvalues $\{z_j \}$ be given. We have

$$\begin{aligned} \mathcal O_{11} \mathop {=}\limits ^\textrm{d} \prod _{n=2}^N \Big ( 1 + { |X_n|^2 \over | z_1 - z_n|^2} \Big ), \end{aligned}$$

(6.22)

where each $X_n$ is an independent complex standard Gaussian. Furthermore, it follows from this that after averaging over $\{z_2,\dots ,z_N\}$

$$\begin{aligned} \mathcal O_{11} \Big |_{z_1=0} \mathop {=}\limits ^\textrm{d} {1 \over \textrm{B}[2,N-1]}, \end{aligned}$$

(6.23)

where $\textrm{B}[\alpha ,\beta ]$ refers to the beta distribution.

Proof

A product formula for $\mathcal O_{11}$ follows from (6.20), the fact that $|| \textbf{r}_1 || = 1$, and by iterating (6.21). This product formula is identified with the RHS of (6.22) upon noting that a vector of independent standard complex Gaussians dotted with any unit vector (here $(\tilde{b}_1,\dots , \tilde{b}_n)$) has a distribution equal to a standard complex Gaussian.

In relation to (6.23) a minor modification of the proof of Proposition 2.14 shows that conditioned on $z_1 = 0$, the ordered squared moduli $\{ | z_j |^2 \}_{j=2}^N$ are independently distributed as $\{ \Gamma [j;1] \}_{j=2}^N$. Noting too that with each $X_j$ a standard complex Gaussian, $|X_j|^2 \mathop {=}\limits ^\textrm{d} \Gamma [1;1]$ it follows that

$$ \mathcal O_{11} \Big |_{z_1=0} \mathop {=}\limits ^\textrm{d} \prod _{n=2}^N \Big ( 1 + {\tilde{X}_n \over Y_n} \Big ), \qquad \tilde{X}_n \mathop {=}\limits ^\textrm{d} \Gamma [1;1], \, Y_n \mathop {=}\limits ^\textrm{d} \Gamma [n;1]. $$

Next, we require the knowledge of the standard fact that $Y_n/(Y_n + \tilde{X}_n) \mathop {=}\limits ^\textrm{d} \textrm{B}[n,1]$. Furthermore (see e.g. [237, Exercises 4.3 q.1]), for $x \mathop {=}\limits ^\textrm{d} \textrm{B}[\alpha +\beta ,\gamma ]$, $y \mathop {=}\limits ^\textrm{d} \textrm{B}[\alpha ,\beta ]$, we have that $xy \mathop {=}\limits ^\textrm{d} \textrm{B}[\alpha ,\beta +\gamma ]$, which tells us that with $b_n \mathop {=}\limits ^\textrm{d} \textrm{B}[n,1]$ we have $\prod _{n=2}^N b_n \mathop {=}\limits ^\textrm{d} \textrm{B}[2,N-1]$.

Dividing both sides of (6.23) by N we see that the $N \rightarrow \infty$ is well defined since $N \textrm{B}[2,N-1] \rightarrow \Gamma [2,1]$. After scaling the GinUE matrix $G \mapsto G/\sqrt{N}$ so that the leading eigenvalue support is the unit disk, an analogous limit formula has been extended from $z_1 = 0$ to any $z_1 = w$, $|w| < 1$ in [107]. Thus

$$\begin{aligned} { \mathcal O_{11} \Big |_{z_1=w_1} \over N (1 - |w_1|^2) } \mathop {\rightarrow }\limits ^\textrm{d} {1 \over \Gamma [2;1]}. \end{aligned}$$

(6.24)

In words, with $\mathcal O_{11}$ corresponding to the condition number, one has that the instability of the spectrum is of order N and is more stable towards the edge. Another point of interest is that the PDF for $1/ \Gamma [2,1]$ is $\chi _{t > 0} e^{-1/t}/t^3$, which is heavy tailed, telling us that only the zeroth and first integer moments are well defined. We remark that limit theorems of the universal form (6.24) have been proved in the case of the complex spherical ensemble of Sect. 2.5, and for a sub-block of a Haar distributed unitary matrix [200]; see also [451].

The $1/t^3$ tail implied by (6.24) has been exhibited from another viewpoint in the work of Fyodorov [272]. There the joint PDF for the overlap non-orthogonality $\mathcal O_{jj} -1$, and the eigenvalue position $z_j$, was computed for finite N. The global-scaled limit of this quantity, $\mathcal P^\textrm{g}(t,w)$ say, was evaluated as [272, Eq. (2.24)]

$$\begin{aligned} \mathcal P^\textrm{g}(t,w) = {(1 - |w|^2)^2 \over \pi t^3} e^{-(1-|w|^2)/t}, \quad |w|<1. \end{aligned}$$

(6.25)

Note that for the first moment in t this gives

$$\begin{aligned} \int _0^\infty t \mathcal P^\textrm{g}(t,w) \, dt = {1 \over \pi } (1 - |w|^2), \end{aligned}$$

(6.26)

in keeping with a prediction from  [146, 147]. This was first proved in [530].

An explicit formula for the large N form of the average of the overlap (6.19) in the off-diagonal case (say $(i,j)=(1,2)$), with the GinUE matrix scaled $G \mapsto G/\sqrt{N}$, and conditioned on $z_1 = w_1$, $z_2 = w_2$ with $|w_1|, |w_2| < 1$ is also known [39, 107, 146, 147, 173]:

$$\begin{aligned} \left\langle \mathcal O_{12} \Big |_{z_1 = w_1, z_2 = w_2} \right\rangle \mathop {\sim }\limits _{N \rightarrow \infty } - {1 \over N} {1 - w_1 \bar{w}_2 \over \pi ^2 | w_1 - w_2|^4} \bigg ( {1 - (1 + N |w_1 - w_2|^2) e^{- N | w_1 - w_2|^2} \over 1 - e^{- N | w_1 - w_2|^2} } \bigg ). \end{aligned}$$

(6.27)

This formula is uniformly valid down to the scale $N | w_1 - w_2| = \textrm{O}(1)$. The large N form of the average value of the diagonal overlap for products of M global-scaled GinUE matrices has been considered in [78, 113], with the result

$$\begin{aligned} \lim _{N \rightarrow \infty }{1 \over N} \left\langle \mathcal O_{11} \Big |_{z_1 = w} \right\rangle = {1 \over \pi } |z|^{-2+2/M} (1 - |z|^{2/M}) \chi _{|z|<1}; \end{aligned}$$

(6.28)

cf. (2.79). We remark that the average values of $\mathcal {O}_{11}$ and $\mathcal {O}_{12}$, conditioned on multiple eigenvalues are shown to have a determinantal form in [39, Th. 1], thus exhibiting an integrable structure for these eigenvector statistics.

For general complex non-Hermitian matrices with independently distributed entries of the form $\xi _{jk} + i {\zeta }_{jk}$, where each $\xi _{jk}, \tilde{\zeta }$ is an identically distributed zero mean real random variable of unit variance (this class is sometimes referred to as complex non-Hermitian Wigner matrices; see e.g. [28] and Sect. 6.5 below), a line of research in relation to the normalised eigenvectors is to quantify the similarity with a complex vector drawn from the sphere embedded in $\mathbb C^N$ with uniform distribution. Recent references on this include [413, 417]. For random vectors on the sphere, there are bounds on the size of the components which rule out gaps in the spread of the size of the components, referred to in [476] as no-gaps localisation. As noted in [417, Sect. 1.1], the bi-unitary invariance of GinUE matrices implies individual eigenvectors are distributed uniformly at random from the complex sphere, and thus with probability close to one have that the j-th largest modulus of the entries is bounded above and below by a positive constant times $\sqrt{N-j}/N$, for j in the range from N/2 up to N minus a constant time $\log N$.

Remark 6.5

1. The overlap $\mathcal O_{11} |_{z = w}$ as specified by (6.24) shows itself in a construction of a reduced density matrix $\rho _A$ associated with Ginibre eigenvectors, conditioned on the event that the eigenvalue z is equal to w. Thus from [164, Eq. (18)] we have

$$\begin{aligned} \rho _A = {\mathbb I_A \over N_A} + \sqrt{1 - |w|^2 \over \Gamma [2,1]} G, \end{aligned}$$

(6.29)

where $1 \ll N_A \ll N_B$, $N_A + N_B = N$, and G is an $N \times N$ global-scaled Ginibre matrix. Relevant to the study of [164] is the eigenvalue density implied by (6.29). Appropriately shifting the functional form of the circular law (2.32) and averaging over $\Gamma [2,1]$ shows that in the variable $x=|N_A^{-1} - z|/\sqrt{1 - |w|^2}$ the density is given by [164, Eq. (21)]

$$\begin{aligned} \mu (x) = {1 \over \pi } \bigg ( 2 - e^{-1/x^2} \Big ( {1 + 2 x^2 + 2 x^4 \over x^4} \Big ) \bigg ). \end{aligned}$$

(6.30)

2. Another appearance of $\mathcal O_{11}$ (and $\mathcal O_{12}$) is in the study of the eigenstate thermalisation hypothesis (ETH) for non-Hermitian Hamiltonians [165]. In this topic, with A a fixed $N \times N$ matrix, of interest is the statistical properties of $\langle \boldsymbol{\ell }_i, A \textbf{r}_j \rangle$. With $\mathbb E_U$ denoting the expectation with respect to the matrix U in the Schur decomposition (2.2), it is found, restricting attention to the case $i=j$ for simplicity, that

$$\begin{aligned} & \mathbb E_U \langle \boldsymbol{\ell }_i, A \textbf{r}_i \rangle = {\textrm{Tr}\, A \over N}, \\ & \mathbb E_U | \langle \boldsymbol{\ell }_i, A \textbf{r}_i \rangle |^2 = {|\textrm{Tr}\, A|^2 \over N} + {\mathcal O_{11} \over N^2} \Big ( \textrm{Tr} \, A^\dagger A - {| \textrm{Tr} \, A|^2 \over N} \Big ). \end{aligned}$$

With $\textrm{Tr} \, A = O(N)$, in light of (6.24) this shows that the fluctuations are of the same order as the mean, which is interpreted as a violation of the ETH.

3. A so-called non-Hermitian Rosenweig–Porter model can be defined by weighting each of the off-diagonal entries of a GinUE matrix by $N^{-\gamma /2}$, where $\gamma > 0$. As for the Hermitian case, of interest is the inverse partition ratio $I_q(j)$, defined as

$$ I_q(j) = \sum _{p=1}^N \Big | \langle \textbf{e}_p, r_j \rangle \langle \ell _j, \textbf{e}_p\rangle \Big |^q, \qquad q > 1/2, $$

where $\textbf{e}_p$ denotes the p-th standard basis vector in $\mathbb C^N$. It is found in [177] that

$$ I_q(j) \asymp N^{1-q}, \, \, \gamma <~\hbox {1} \qquad I_q(j) = O(1), \, \, \gamma > 1, $$

independent of j. In particular, $I_q(j) = O(1)$ has the interpretation of the eigenstates being localised.

5 Non-Hermitian Wigner Ensembles

As remarked at the beginning of the paragraph below (6.28), the term (complex) non-Hermitian Wigner matrices refers to the class of non-Hermitian random matrices with independently distributed entries of the form $\xi _{jk} + i {\zeta }_{jk}$, where each $\xi _{jk}, \tilde{\zeta }$ is an identically distributed zero mean real random variable of unit variance. When the latter is the real normal $\textrm{N}[0,1/\sqrt{2}]$, the GinUE is obtained. However, more generally, non-Hermitian Wigner matrices do not share with GinUE the property of allowing for explicit formulas in relation to the joint eigenvalue PDF or the correlation functions. Nonetheless, for large N many statistical properties of non-Hermitian Wigner matrices are in fact independent of the detail of the particular distribution of the elements, beyond it having mean zero and unit variance. This property is referred to as universality, and generally is a prominent theme in random matrix theory [219].

In the specific case of the universality of the circular law functional form for the global density, references on this have been given below (2.17). As noted in the first paragraph of Sect. 6.4, important to establishing this result is control of the resolvent associated with a shifted version of the matrix H (6.10). The work [46] establishes the near optimal convergence of the spectral radius $s_N$ say, improved on in [162] to the statement that for any $\epsilon > 0$ there is a $C_\epsilon > 0$ such that

$$ \mathop {\limsup }\limits _{N} \textrm{Pr} \Big (\Big |s_N - 1 - \sqrt{\gamma _N \over 4 N} \Big | \ge {C_\epsilon \over \sqrt{N \log N}} \Big ) \le \epsilon , $$

where $\gamma _N$ is as in (3.13). An analysis based on the resolvent associated with (6.10) has been used to establish local universality of the edge correlation functional form associated with the correlation kernel (2.19), where it is required that the expected value of the square of the entries vanish. In such work it is not convergence to the explicit functional form that is established, but rather convergence to the case of standard complex Gaussian entries.

Consider now the global-scaled covariance of smooth linear statistics $f, \bar{g}$, which in the case of GinUE is given by (3.21), with a normal Gaussian fluctuation as specified by Proposition 3.4. In the case of complex non-Hermitian Wigner matrices, with the expected value of the square of the entries random variable $\xi + i \zeta$ equally zero, the RHS of (3.21) aquires the extra term [160] (see also the review [156])

$$ \kappa _4 \Big ( {1 \over \pi } \int _{|\textbf{r}|<1} f(\textbf{r}) \, dx dy - f_0 \Big ) \Big ( {1 \over \pi } \int _{|\textbf{r}|<1} \bar{g}(\textbf{r}) \, dx dy - \bar{g}_0 \Big ). $$

Here $\kappa _4$ is the fourth cumulant of a single entry, $\kappa _4 := \langle | \xi + i \zeta |^4 \rangle - 2$ (note that this vanishes for GinUE) and $f_0, \bar{g}_0$ are as those in the statement of Proposition 3.3. Note that this extra term vanishes if f or g are real analytic. Similarly, the $\beta = 2$ case of the expansion (4.18) must be modified by the additive term [160]

$$ - {\kappa _4 \over \pi N} \int _{|z| < 1} f(\textbf{r}) (2 |z|^2 - 1) \, d^2 z. $$

Should the variance of the entries of a non-Hermitian Wigner matrix not be well defined due to the entries being heavy tailed, distinct statistical properties to those described above are expected. While explicit formulas are generally not available, introducing a parameter $\alpha$ as in the theory of Lévy $\alpha$-stable laws it is known that the limiting global density decays proportionally to $|z|^{2(\alpha -1)} e^{-(\alpha /2) |z|^\alpha }$ for large |z|; see [100, Sect. 6].

Aspects of universality in relation to eigenvector overlaps are investigated at a numerical level in [107]. In particular, evidence is presented for the validity of the inverse gamma distribution (6.24) for a general class of complex non-Hermitian Wigner ensembles.