Statistical Properties of GinSE and Elliptic GinSE

Abstract

Eigenvalues of GinSE matrices occur in complex conjugate pairs as for complex GinOE eigenvalues, but now with zero probability of an eigenvalue being real. Also in common with complex GinOE eigenvalues is that the GinSE eigenvalues form a Pfaffian point process, and that their PDF admits a Coulomb gas interpretation with image terms. While the determination of the required skew-orthogonal polynomials is straightforward, the particular form gives a new challenge in relation to analysing the various scaling limits. This is overcome by making use of a certain inhomogeneous partial differential equation satisfied by the pre-kernel. Such an approach carries over to the analysis of a weakly non-Hermitian regime. Also contained in this chapter is the large N analysis of the partition function and gap probability associated with GinSE eigenvalues, and results relating to the singular values and eigenvectors. Furthermore, a study is undertaken of an elliptic generalisation, which interpolates between GinSE and Hermitian GSE matrices.

Eigenvalues of GinSE matrices occur in complex conjugate pairs as for complex GinOE eigenvalues, but now with zero probability of an eigenvalue being real. Also in common with complex GinOE eigenvalues is that the GinSE eigenvalues form a Pfaffian point process, and that their PDF admits a Coulomb gas interpretation with image terms. While the determination of the required skew-orthogonal polynomials is straightforward, the particular form gives a new challenge in relation to analysing the various scaling limits. This is overcome by making use of a certain inhomogeneous partial differential equation satisfied by the pre-kernel. Such an approach carries over to the analysis of a weakly non-Hermitian regime. Also contained in this chapter is the large N analysis of the partition function and gap probability associated with GinSE eigenvalues, and results relating to the singular values and eigenvectors. Furthermore, a study is undertaken of an elliptic generalisation, which interpolates between GinSE and Hermitian GSE matrices.

1 Eigenvalue PDF

Using a similarity transformation, the symplectic Ginibre matrix G can be identified via the matrix-valued version of the quaternion realisation (1.1) as

$$\begin{aligned} \begin{bmatrix} A & B \\ -\bar{B} & \bar{A} \end{bmatrix} \in \mathbb {C}^{2 N \times 2 N}, \end{aligned}$$

(10.1)

where A, B are independent copies of GinUE. Due to this form, the matrix G has 2N eigenvalues that come in complex conjugate pairs $\pm z_j$, where we take $\textrm{Im} \, z_j > 0$. Importantly from the viewpoint of calculating the Jacobians associated with a change of variables involving eigenvalues, the eigenvectors of complex conjugate eigenvalues are related by a linear transformation. This can be encoded by forming a block Schur decomposition $G = U Z U^\dagger$ with U a $2N \times 2N$ unitary matrix with $2 \times 2$ block entries of the form (1.1)—a conjugation equivalent symplectic unitary matrix—and Z a block triangular matrix, with diagonal blocks

$$\begin{aligned} \Big \{ \begin{bmatrix} 0 & z_j \\ \bar{z}_j & 0 \end{bmatrix} \Big \}_{j=1}^N \end{aligned}$$

containing the eigenvalues of G, and off-diagonal elements having the quaternion form (1.1); see [331, Proposition A.25]. As in the proof of Proposition 2.1, the Jacobian calculation is carried out by forming the wedge product of the matrix of differentials of $U^\dagger dG U$, in the order of the block indices (j, k) with j decreasing from N to 1, and k increasing from 1 to N. This gives the eigenvalue dependent factor

$$\begin{aligned} \prod _{j=1}^N e^{ -2 W(z_j, \bar{z}_j) } |z_j-\bar{z}_j|^2 \prod _{1 \le j < k \le N} | z_k - z_j|^2 |z_k-\bar{z}_j|^2, \end{aligned}$$

and moreover (after some working) the product of differentials can be shown to factorise as in (2.6). The eigenvalue PDF now follows by integrating over the independent off-diagonal entries of Z. The resulting functional form (1.8) is formally the same as the GinUE eigenvalue PDF (1.7) with $N \mapsto 2N$, and where $z_{j+N}$ is identified with $\bar{z}_j$, but with terms involving only differences of $\{ \bar{z}_j \}$ ignored (due to dependencies in the quaternion structure these are not associated with independent differentials). As previously mentioned, the eigenvalue PDF (1.8) of GinSE was derived already in the original work [293] of Ginibre starting from the eigenvalue–eigenvector decomposition. For the details of the above outlined method using the quaternionic Schur decomposition, see [331, proof of Proposition A.25].

In general, an eigenvalue PDF of the non-Hermitian random matrices in the same symmetry class of the GinSE is of the form

$$\begin{aligned} \frac{1}{ N!\, Z_N^{ \mathbb {H} }(W) } \prod _{j=1}^N e^{ -2 W(z_j, \bar{z}_j) } |z_j-\bar{z}_j|^2 \prod _{1 \le j < k \le N} | z_k - z_j|^2 |z_k-\bar{z}_j|^2, \quad \textrm{Im} \, z_j > 0, \end{aligned}$$

(10.2)

where $Z_N^{ \mathbb {H} } \equiv Z_N^{ \mathbb {H} }(W)$ is the partition function, which turns (10.2) into a probability measure. Here W can be an arbitrary real function such that $Z_N^{ \mathbb {H} }$ exists, which furthermore is assumed to satisfy the complex conjugation symmetry $W(z,\bar{z})=W(\bar{z},z)$. The ensemble of the form (10.2) is called the planar symplectic ensemble [22].

2 Coulomb Gas Perspective

The eigenvalue PDF (10.2) can be rewritten in terms of the Hamiltonian

$$\begin{aligned} H_{\mathbb {H}}(z_1,\dots ,z_N)&:= \sum _{1 \le j < l \le N} \log \frac{1}{|z_j-z_l|^2 |z_j-\bar{z}_l|^2} \nonumber \\ &\quad + \sum _{ j=1}^N \bigg (2 W(z_j,\bar{z}_j) + \log \frac{1}{ |z_j-\bar{z}_j|^2 } \bigg ) \end{aligned}$$

(10.3)

as

$$\begin{aligned} \frac{1}{ N!\, Z_N^{ \mathbb {H} }(W) }\,e^{-H_\mathbb {H}(z_1,\dots ,z_N)}. \end{aligned}$$

(10.4)

Analogous to the discussion of Sect. 7.2, this can be regarded as a two-dimensional Coulomb gas [245] in the upper half plane $\mathbb {H}$ with image charges in the lower half plane; cf. [367]. As such, this is an image system counterpart of the eigenvalue PDF of the normal matrix model, which by way of comparison is given by

$$\begin{aligned} \frac{1}{ N!\,Z_N^{ \mathbb {C} }(W) }\,e^{-H_\mathbb {C}(z_1,\dots ,z_N)}, \end{aligned}$$

(10.5)

where

$$\begin{aligned} H_{ \mathbb {C} }(z_1,\dots ,z_N):= \sum _{1 \le j < l \le N} \log \frac{1}{|z_j-z_l|^2}+ \sum _{j=1}^N W(z_j,\bar{z}_j); \end{aligned}$$

see Sect. 5 and references therein.

The Coulomb gas interpretation (10.4) allows one to describe the limiting spectral distribution using logarithmic potential theory. In the scaling $W(z,\bar{z})=N Q(z)$, chosen to make the interaction and the potential term in (10.3) of the same order, one can observe that the continuum limit of the Hamiltonian (10.3) divided by $2N^2$ is given by

$$\begin{aligned} I_Q[\mu ]&:= \frac{1}{4} \int _{ \mathbb {C}^2 } \log \frac{1}{ |z-w| |z-\bar{w}| }\, d\mu (z)\, d\mu (w) +\int _{ \mathbb {C} } Q \,d\mu \nonumber \\ &= \int _{ \mathbb {C}^2 } \log \frac{1}{ |z-w| }\, d\mu (z)\, d\mu (w) +\int _{ \mathbb {C} } Q \,d\mu . \end{aligned}$$

(10.6)

Here, we have used the complex conjugation symmetry $Q(z)=Q(\bar{z})$ for the second line. Thus it is natural to expect that the empirical measure $\mu _Q$ of (10.4) converges to Frostman’s equilibrium measure, a unique probability measure minimising the energy functional (10.6). This convergence was shown by Benaych-Georges and Chapon [82] for a general potential Q. In particular it shows that the limiting spectral distributions of (10.4) and (10.5) are identical Sect. 5.2. In particular, for $Q(z)=|z|^2$, this gives rise to the universal appearance of the circular law for the GinUE and GinSE.

With regards to quantitative features, let us first recall that under minor assumptions on Q, the equilibrium measure $\mu _Q$ is absolutely continuous and takes the form

$$\begin{aligned} d\mu _Q(z)= \frac{ \partial _z \partial _{ \bar{z} } Q(z)}{\pi } \chi _{ z\in S_Q } \,d^2 z, \end{aligned}$$

(10.7)

where the compact set $S_Q$ is called the droplet. For a radially symmetric potential $q(r)=Q(|z|=r)$ which is strictly subharmonic in $\mathbb {C}$, the droplet is of the form $S_Q=\{ R_1 \le |z| \le R_2 \}$, where the pair of constant $(R_1,R_2)$ is characterised by

$$\begin{aligned} R_1 q'(R_1)=0, \quad R_2 q'(R_2)=2, \end{aligned}$$

(10.8)

see [479, Sect. IV.6]. In particular, for $Q(z)=|z|^2$, it follows that $\partial _z \partial _{ \bar{z} } Q(z)=1$ and $R_0=0,R_1=1$, which coincides with the circular law of the GinSE.

Remark 10.1

Underlying the image charge viewpoint of (10.3) is the pair potential (7.2), which we know is the solution of the two-dimensional Poisson equation in Neumann boundary conditions along the x-axis. If instead we take the point $\vec{r}$ to be inside a disk of radius R and require Neumann boundary conditions on the boundary of the disk, the pair potential is [237, Eq. (15.188) with $\epsilon = 0$]

$$\begin{aligned} \phi (\vec{r},\vec{r}') = - \log \Big ( |z - z'||R - z z'/R| \Big ). \end{aligned}$$

Imposing a smeared out charge neutral background of density $1/\pi$ (and hence taking $R=\sqrt{N}$) the corresponding charge neutral Boltzmann factor is [237, Eq. (15.190)]

$$\begin{aligned} A_{N,\beta } e^{-\beta \sum _{j=1}^N | z_j|^2/2} \prod _{1 \le j < k\le N} |z_k - z_j|^\beta |1 - z_j \bar{z}_k/N|^\beta \prod _{j=1}^N (1 - |z_j|^2/N)^{\beta /2}, \end{aligned}$$

(10.9)

where $A_{N,\beta } = e^{-\beta N^2((1/4) \log N - 3/8)}$. Here $\beta > 0$ is the inverse temperature, with the case $\beta = 2$ being the disk analogue of (10.3), although this viewpoint gives the self-energy term of exponent 1 rather than 2 as in (1.8); see [245, Sect. 2.1] for more on this point.

3 Skew-Orthogonal Polynomials

We define the skew-symmetric form $\langle \cdot , \cdot \rangle _{s,S}$ by

$$\begin{aligned} \langle f, g \rangle _{s,S} := \int _{\mathbb {C}} \Big ( f(z) g(\bar{z}) - g(z) f(\bar{z}) \Big ) (z - \bar{z}) e^{-2 W(z, \bar{z} )} \,d^2z, \end{aligned}$$

(10.10)

where in keeping the notation of (7.16) the subscripts indicate skew and GinSE. Note the similarity with the second term in (7.16). As discussed in Sect. 7.4, a family $\{q_{m}\}_{m \ge 0}$ of monic polynomials $q_m$ of degree m is said to be a family of skew-orthogonal polynomials if the following skew-orthogonality conditions hold: for all $k, l \in \mathbb {N}$

$$\begin{aligned} &\langle q_{2k}, q_{2l} \rangle _{s,S} = \langle q_{2k+1}, q_{2l+1} \rangle _{s,S} = 0, \nonumber \\ &\langle q_{2k}, q_{2l+1} \rangle _{s,S} = -\langle q_{2l+1}, q_{2k} \rangle _{s,S} = r_k \,\delta _{k, l}. \end{aligned}$$

(10.11)

We mention that the matrix averages formulas (7.18) are again valid; see [353]. In distinction to ensembles based on GinOE, there are also alternative methods which in fact have a broader scope, so these instead will be discussed below.

Proposition 10.1

For a radially symmetric potential $W(z,\bar{z})=\omega (|z|),$ let

$$\begin{aligned} h_k=2\pi \int _0^\infty r^{2k+1} e^{-2\omega (r)}\,dr \end{aligned}$$

(10.12)

be the squared orthogonal norm. Then

$$\begin{aligned} q_{2k+1}(z)=z^{2k+1}, \qquad q_{2k}(z)=z^{2k}+\sum _{l=0}^{k-1} z^{2l} \prod _{j=0}^{k-l-1} \frac{h_{2l+2j+2} }{ h_{2l+2j+1} } \end{aligned}$$

(10.13)

forms a family of skew-orthogonal polynomials. Furthermore, the skew-norm is given by $r_k=2h_{2k+1}$.

Proof

Since the monomials $\{z^k\}$ are orthogonal polynomials with respect to a rotationally symmetric weight function, it follows that

$$\begin{aligned} \langle z^k, z^l \rangle _{s,S} &= \int _{ \mathbb {C} } \Big ( z^{k+1} \bar{z}^l-z^{l+1} \bar{z}^k- z^k \bar{z}^{l+1}+z^l \bar{z}^{k+1}\Big ) e^{-2\omega (|z|)}\,d^2 z \\ &= 2\delta _{k+1,l} h_{k+1} - 2\delta _{l+1,k} h_k . \end{aligned}$$

Note here that the indices in the Kronecker delta differ by one, which in turn immediately leads to $\langle q_{2k+1}, q_{2l+1} \rangle _{s,S} = 0$. Furthermore, it follows that $\langle q_{2k}, q_{2l+1} \rangle _{s,S}=0$ if $l>k.$ Let us write

$$\begin{aligned} a_l= \prod _{j=0}^{k-l-1} \frac{h_{2l+2j+2} }{ h_{2l+2j+1} }. \end{aligned}$$

(10.14)

Then we have

$$\begin{aligned} \langle q_{2k}, q_{2k+1} \rangle _{s,S} &= \Big \langle z^{2k}+\sum _{l=0}^{k-1} a_l z^{2l} \, , \, z^{2k+1} \Big \rangle _{s,S} = 2 h_{2k+1}. \end{aligned}$$

On the other hand, for the case $k<l$, after straightforward computations, we obtain

$$\begin{aligned} \langle q_{2k+1}|q_{2l} \rangle _{s,S}= 2 ( a_k h_{2k+1}-a_{k+1} h_{2k+2} )=0. \end{aligned}$$

Therefore, we have shown that $\langle q_{2k+1}, q_{2l+1} \rangle _{s,S} = r_k \delta _{k,j}$. The other cases follow from similar computations with minor modifications.    $\square$

As an example of Proposition 10.1, for $W^\textrm{g}(z,\bar{z})=|z|^2$, the associated skew-orthogonal polynomials $q_k^{ \mathrm g }$ are given by

$$\begin{aligned} q_{2k+1}^{ \mathrm g }(z)=z^{2k+1}, \qquad q_{2k}^{ \mathrm g }(z)= \sum _{l=0}^k \frac{k!}{ l! }z^{2l}, \qquad r_k^\textrm{g}= \frac{(2k+1)!}{2^{2k+1}}\pi . \end{aligned}$$

(10.15)

These are a special case of the skew-orthogonal polynomials obtained in [353] for the elliptic GinSE; see the sentence below (10.23). We also refer to [237, Exercises 15.9 q.2] and [240, Proposition 1] for a derivation of (10.15).

The crux of Proposition 10.1 is that one can construct the skew-orthogonal polynomials using the associated (monic) orthogonal polynomials $p_j$ with respect to the same weight, i.e.

$$\begin{aligned} \int _{\mathbb {C} } p_j(z) \overline{p_k(z)} e^{-2 W(z,\bar{z})}\,d^2z = h_k \delta _{j,k}. \end{aligned}$$

(10.16)

A setting beyond the radially symmetric case in which we can construct the skew-orthogonal polynomials is when the associated orthogonal polynomials satisfy a three-term recurrence relation.

Proposition 10.2

Suppose that the sequence of monic orthogonal polynomials $(p_j)$ satisfies the three-term recurrence relation

$$\begin{aligned} z p_k(z)= p_{k+1}(z) +b_k p_k(z)+c_k p_{k-1}(z), \qquad b_k, c_k \in \mathbb {R}. \end{aligned}$$

(10.17)

Then

$$\begin{aligned} & q_{2k+1}(z)= p_{2k+1}(z), \nonumber \\ & q_{2k}(z)= \sum _{l=0}^k a_{l} z^l, \qquad a_l:= \prod _{ j=0 }^{ k-l-1 } \frac{ h_{2l+2j+2}-c_{2l+2j+2}h_{2l+2j+1} }{ h_{2l+2j+1}-c_{2l+2j+1} h_{2l+2j} } \end{aligned}$$

(10.18)

satisfies (10.11) with $r_k=2(h_{2k+1}-c_{2k+1}h_{2k})$. Conversely, if the skew-orthogonal polynomials have the form (10.18), then the three-term recurrence (10.17) holds.

Proof

(Sketch) The essential idea of the proof has already been given in the proof of Proposition 10.1 above. The notable difference is that while computing the skew-symmetric form $\langle q_k, q_l \rangle _{s,S}$, we expand the term

$$\begin{aligned} \Big ( q_k(z) \overline{ q_l(z) } - \overline{ q_k(z) } q_l(z) \Big ) (z-\bar{z}) \end{aligned}$$

(10.19)

in the integrand using the three-term recurrence relation (10.17).    $\square$

The idea of constructing skew-orthogonal polynomials in Proposition 10.2 first appeared in [353], where the Hermite polynomials were considered. Later, this was extended to the Laguerre polynomials in [11]. The general statement in Proposition 10.2 was given in [26]. As an example, we consider the elliptic GinSE potential

$$\begin{aligned} W^{ \mathrm e }(z,\bar{z})= \frac{1}{1-\tau ^2} (|z|^2-\tau \, \mathrm{{Re}} \,z^2), \qquad \tau \in [0,1). \end{aligned}$$

(10.20)

The associated monic orthogonal polynomials and norms are given by

$$\begin{aligned} p_k^{ \mathrm e }(z) = \Big ( \frac{\tau }{4} \Big )^{k/2} H_k \Big ( \frac{z}{ \sqrt{\tau } } \Big ), \qquad h_k^{ \mathrm e }= \sqrt{1-\tau ^2} \frac{k!}{2^{k+1}}\pi , \end{aligned}$$

(10.21)

see e.g. [24, Lem. 7]. Then by using the recurrence relation

$$\begin{aligned} z p_k^{ \mathrm e }(z) = p_{k+1}^{ \mathrm e }(z) + \frac{\tau }{2}\,k\, p_{k-1}^{ \mathrm e }(z) \end{aligned}$$

(10.22)

and Proposition 10.2, we have

$$\begin{aligned} & q_{2k+1}^{ \mathrm e }(z)=p_{2k+1}^{ \mathrm e }(z), \qquad q_{2k}^{ \mathrm e }(z)= \sum _{l=0}^k \frac{k!}{ l! } p_{2l}^{ \mathrm e }(z), \nonumber \\ &r_k^\textrm{e}= (1-\tau ) \sqrt{1-\tau ^2} \frac{(2k+1)!}{2^{2k+1}}\pi . \end{aligned}$$

(10.23)

Note that (10.15) can be recovered by taking the $\tau \rightarrow 0$ limit of (10.23).

4 Correlation Functions and Sum Rules

The k-point correlation function $\rho _{(k),N}^\textrm{s}$ of the ensemble (10.4) is given by

$$\begin{aligned} \rho _{(k),N}^\textrm{s}(z_1,\dots , z_k) := \frac{N!}{(N-k)!} \frac{1}{ Z_N^{ \mathbb {H} }(W) } \int _{\mathbb {C}^{N-k}} e^{-H_{ \mathbb {H} }(z_1,\dots ,z_N) }\prod _{j=k+1}^N \, d^2 z_j. \end{aligned}$$

(10.24)

As an analogue of Proposition 7.5, a Pfaffian formula for $\rho _{(k),N}^\textrm{s}$ follows [353].

Proposition 10.3

Let $q_j$ be the skew-orthogonal polynomials as specified in (10.11). Using these polynomials, define

$$\begin{aligned} \kappa _N^\textrm{s}(z,w)=\sum _{k=0}^{N-1} \frac{q_{2k+1}(z) q_{2k}(w) -q_{2k}(z) q_{2k+1}(w)}{r_k}. \end{aligned}$$

(10.25)

Then we have

$$\begin{aligned} \rho _{(k),N}^\textrm{s}(z_1,\dots , z_k) =\prod _{j=1}^{k} (\overline{z}_j-z_j) \textrm{Pf} \, [\mathcal K_N^\textrm{s}(z_j,z_l)]_{j,l=1,\dots ,k}, \end{aligned}$$

(10.26)

where

$$\begin{aligned} \mathcal {K}_N^\textrm{s}(z,w)= e^{ -W(z,\bar{z})-W(w,\bar{w}) } \begin{bmatrix} \kappa _N^\textrm{s}(z,w) & \kappa _N^\textrm{s}(z,\bar{w}) \\ \kappa _N^\textrm{s}(\bar{z},w) & \kappa _N^\textrm{s}(\bar{z},\bar{w}) \end{bmatrix}. \end{aligned}$$

(10.27)

Using Proposition 10.3 and (10.15), it follows that the matrix entry $\kappa _N^{ \mathrm g }(z,w)$—referred to as the pre-kernel—of the GinSE is given by

$$\begin{aligned} \kappa _N^{ \mathrm g }(z,w) &= \frac{ \sqrt{2} }{ \pi } \bigg ( \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k+1}}{(2k+1)!!} \sum _{l=0}^k \frac{( \sqrt{2} w)^{2l} }{(2l)!!} - \sum _{k=0}^{N-1} \frac{( \sqrt{2} w)^{2k+1}}{(2k+1)!!} \sum _{l=0}^k \frac{( \sqrt{2} z)^{2l} }{(2l)!!} \bigg ). \end{aligned}$$

(10.28)

One strategy for analysing the double summation is to derive a suitable differential equation for the kernel. This idea essentially goes back to an early work [437] of Mehta and Srivastava.

As an analogue of Proposition 7.6, the following holds true; see [22].

Proposition 10.4

Letting $\widehat{\kappa }_N^\textrm{g}(z,w):=e^{-2zw} \kappa _N^\textrm{g}(z,w)$, we have

$$\begin{aligned} \partial _z \widehat{\kappa }_N^{ \mathrm g }(z,w) = 2(z-w) \widehat{\kappa }_N^{ \mathrm g }(z,w) + \frac{2}{\pi } \frac{ \Gamma (2N;2zw) }{(2N-1)!} -\frac{1}{\pi } (2z)^{2N} e^{-z^2} \frac{ \Gamma (N;w^2) }{ (2N-1)! }. \end{aligned}$$

(10.29)

Proof

Note that

$$\begin{aligned} & \partial _z \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k+1}}{(2k+1)!!} \sum _{l=0}^k \frac{( \sqrt{2} w)^{2l} }{(2l)!!} = \sqrt{2} \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k}}{(2k-1)!!} \sum _{l=0}^k \frac{( \sqrt{2} w)^{2l} }{(2l)!!} \\ &\quad = \sqrt{2} \sum _{k=1}^{N-1} \frac{( \sqrt{2}z )^{2k}}{(2k-1)!!} \sum _{l=0}^{k-1} \frac{( \sqrt{2} w)^{2l} }{(2l)!!} + \sqrt{2} \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k}}{(2k-1)!!} \frac{( \sqrt{2} w)^{2k} }{(2k)!!}. \end{aligned}$$

Rearranging the terms, we have

$$\begin{aligned} & \partial _z \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k+1}}{(2k+1)!!} \sum _{l=0}^k \frac{( \sqrt{2} w)^{2l} }{(2l)!!} = 2 z \sum _{k=0}^{N-1} \frac{(\sqrt{2}z )^{2k+1}}{(2k+1)!!} \sum _{l=0}^{k} \frac{( \sqrt{2} w)^{2l} }{(2l)!!} \\ &\quad +\sqrt{2} \sum _{k=0}^{N-1} \frac{( \sqrt{2}z )^{2k}}{(2k-1)!!} \frac{( \sqrt{2} w)^{2k} }{(2k)!!}-\sqrt{2} \frac{( \sqrt{2}z )^{2N}}{(2N-1)!!} \sum _{l=0}^{N-1} \frac{( \sqrt{2} w)^{2l} }{(2l)!!}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} & \partial _z \sum _{k=0}^{N-1} \frac{( \sqrt{2} w)^{2k+1}}{(2k+1)!!} \sum _{l=0}^k \frac{( \sqrt{2} z)^{2l} }{(2l)!!} \\ & \quad = 2 z \sum _{k=0}^{N-1} \frac{( \sqrt{2} w)^{2k+1}}{(2k+1)!!} \sum _{l=0}^{k} \frac{( \sqrt{2} z)^{2l} }{(2l)!!} - 2z \sum _{k=0}^{N-1} \frac{(\sqrt{2} w)^{2k+1}}{(2k+1)!!} \frac{(\sqrt{2} z)^{2k} }{(2k)!!}. \end{aligned}$$

Combining above identities, we conclude (10.29).    $\square$

As a consequence of Proposition 10.4, the uniform asymptotic expansion (3.31) can be used to derive a linear inhomogeneous differential equation of order one satisfied by the limiting correlation kernels. Then the anti-symmetry of the limiting pre-kernels characterises a unique solution, which in turn determines the limiting kernels; see [22, 118].

For the bulk case, we have

$$\begin{aligned} \rho _{(k),\infty }^\textrm{s, b}(z_1,\dots , z_k) \\ := \lim _{N \rightarrow \infty } \rho _{(k),N}^\textrm{s}(z_1,\dots , z_k) =\prod _{j=1}^{k} (\overline{z}_j-z_j) \textrm{Pf} \, [\mathcal K_\infty ^\textrm{s,b}(z_j,z_l)]_{j,l=1,\dots ,k}, \end{aligned}$$

(10.30)

where $\mathcal {K}_\infty ^\textrm{s,b}$ is of the form

$$\begin{aligned} \mathcal {K}_\infty ^\textrm{s,b}(z,w):= e^{-|z|^2-|w|^2} \begin{bmatrix} \kappa _\infty ^\textrm{s,b}(z,w) & \kappa _\infty ^\textrm{s,b}(z,\bar{w}) \\ \kappa _\infty ^\textrm{s,b}(\bar{z},w) & \kappa _\infty ^\textrm{s,b}(\bar{z},\bar{w}) \end{bmatrix}, \quad \kappa _\infty ^\textrm{s,b}(z,w):=\frac{ e^{ z^2+w^2 } }{\sqrt{\pi }} \mathrm{{erf}} (z-w). \end{aligned}$$

(10.31)

In particular, this gives the bulk-scaled density

$$\begin{aligned} \rho _{(1),\infty }^\textrm{s,b}(x+iy) = \frac{4}{\pi } y F(2y), \end{aligned}$$

(10.32)

where $F(z):=e^{-z^2} \int _0^z e^{t^2}\,dt$ is Dawson’s integral function. As $y \rightarrow 0^+$, this tends to zero with leading term ${8y^2 \over \pi }$, while for $y \rightarrow \infty$ this limits to ${1 \over \pi }$ which is the interior bulk density away from the real axis. There is a peak in this profile, as is consistent with the sum rule (10.39) below. It can be checked from (10.31) that the truncated two-point correlation implied by (10.30) decays like a Gaussian in all directions. The limiting pre-kernel (10.31) first appeared in the second edition of Mehta’s book [436] using a different approach; cf. [416, Lem. 3.5.2]. Later, this was rederived by Kanzieper in [353] using a similar method described above.

Generally, if each $\textrm{Im} (z_j)$, $(j=1,\dots ,k)$ in (10.32) is taken to infinity, but with the relative distance between each $z_i$ still finite, the RHS simplies to the Pfaffian of the anti-symmetrix matrix (8.1), as for GinOE. We know that this in turn reduces to the determinantal formula implied by Proposition 2.3 for bulk-scaled GinUE.

Contrary to the bulk case, the analogous result for the edge case appeared in the literature [22] only recently. The same Pfaffian structure was found

$$\begin{aligned} \lim _{N \rightarrow \infty } \rho _{(k),N}^\textrm{s}(\sqrt{N}+z_1,\dots , \sqrt{N}+z_k) =\prod _{j=1}^{k} (\overline{z}_j-z_j) \textrm{Pf} \, [\mathcal K_\infty ^\textrm{s,e}(z_j,z_l)]_{j,l=1,\dots ,k}, \end{aligned}$$

(10.33)

with $\mathcal K_\infty ^\textrm{s,e}$ as for $\mathcal K_\infty ^\textrm{s,b}$ in (10.33) but with $\kappa _\infty ^\textrm{s,b}$ replaced by $\kappa _\infty ^\textrm{s,e}$ throughout, where

$$\begin{aligned} \kappa _\infty ^\textrm{s,e}(z,w):=\frac{ e^{2zw} }{\pi } \int _{-\infty }^{0} e^{-t^2} \sinh ( 2t(w-z) ) \mathrm{{erfc}}(z+w-t)\,dt. \end{aligned}$$

(10.34)

Setting $w=\bar{z}$, this gives for the density

$$\begin{aligned} \rho _{(1),\infty }^\textrm{s,e}(x+iy) = -\frac{2y}{\pi } \int _{-\infty }^{0}e^{-s^2}\sin (4sy) \mathrm{{erfc}}(2x-s) \, ds. \end{aligned}$$

(10.35)

Note here that unlike the bulk case, the edge scaling limit does not have the translation invariance along the horizontal x-direction. We also remark that

$$\begin{aligned} \rho _{(1),\infty }^\textrm{s,e}(x+iy) \sim \frac{\mathrm{{erfc}}(2x)}{2\pi } + {e^{-4 x^2} \over 8 \pi ^{3/2} y^2} + \cdots , \qquad y \rightarrow \infty . \end{aligned}$$

(10.36)

The leading term of the RHS of (10.36) as a function of x coincides with the limiting edge density of the GinUE up to scaling. As in Sect. 8.1, such a limiting relation holds too for the general k-point function, which in particular exhibits a deformation from a Pfaffian to a determinant; see [22, Cor. 2.2].

A structural feature, highlighted in [22, 119], is that the kernels (10.31) and (10.34) can be expressed in a unified way

$$\begin{aligned} \kappa _\infty ^\textrm{s}(z,w):=\frac{ e^{z^2+w^2} }{\sqrt{\pi }} \int _{E} W(f_{w},f_{z})(u) \, du, \end{aligned}$$

(10.37)

where $W(f,g):=fg'-gf'$ is the Wronskian, and

$$\begin{aligned} f_z(u):=\frac{ \textrm{erfc}(\sqrt{2}(z-u)) }{2}, \qquad E:={\left\{ \begin{array}{ll} (-\infty ,\infty ) & for the bulk case , \\ (-\infty ,0) & for the edge case . \end{array}\right. } \end{aligned}$$

(10.38)

As in Sect. 7.5, the expression (10.37) allows one to observe the bulk limiting form from the edge limiting form with $z,w \rightarrow -\infty$.

We now discuss sum rules for the limiting densities (10.32) and (10.35); cf. Propositions 4.3 and 4.4.

Proposition 10.5

We have

$$\begin{aligned} \int _{-\infty }^\infty \Big (\rho _{(1),\infty }^\textrm{s,b}(x+iy) - \frac{1}{\pi } \Big )\,dy=0 \end{aligned}$$

(10.39)

and

$$\begin{aligned} \int _{ (-\infty ,\infty )^2 } \Big ( \rho _{(1),\infty }^\textrm{s,e}(x+iy) - \rho _{(1),\infty }^\textrm{s,b}(x+iy) \chi _{ x<0 } \Big )\,dx\,dy =- \frac{1}{8}. \end{aligned}$$

(10.40)

Proof

The first identity (10.39) immediately follows from the property of Dawson’s integral $F(2y)/2= \int (1- 4y F(2y)) \,dy$. For the second identity (10.40), letting $D_R$ be a disk of radius $R>0$, we first observe that by the change of variables,

$$\begin{aligned} & \int _{ D_R } \Big (\rho _{(1),\infty }^\textrm{s,b}(x+iy) \chi _{ x<0 }-\rho _{(1),\infty }^\textrm{s,e}(x+iy) \Big )\,dx\,dy \\ &\qquad - \int _{ D_R } \Big (\rho _{(1),\infty }^\textrm{s,b}(x+iy) - {1 \over \pi } \Big ) \chi _{ x<0 }\,dx\,dy \\ &\quad = \int _{ D_R } \Big ( \frac{\chi _{ x<0 }}{\pi }-\rho _{(1),\infty }^\textrm{s,e}(x+iy) \Big )\,dx\,dy = \int _{ D_R } \Big ( \frac{1}{2\pi }-\rho _{(1),\infty }^\textrm{s,e}(x+iy) \Big )\,dy\,dx \\ &\quad = \frac{1}{\pi } \int _{ D_R } \Big ( \frac{1}{2}+2y\int _{-\infty }^{0}e^{-s^2}\sin (4sy)\mathrm{{erfc}}(2x-s) \, ds \Big )\,dy\,dx \\ &\quad = \frac{1}{\pi }\int _{ D_R } \Big ( \frac{1}{2}-2y\int _{0}^{\infty }e^{-s^2}\sin (4sy)\Big (2-\mathrm{{erfc}}(2x-s)\Big ) \, ds \Big )\,dy\,dx. \end{aligned}$$

By adding the last two expressions and using

$$\begin{aligned} 4y \int _0^\infty e^{-s^2} \sin (4s y)= 4y F(2y)= \pi \rho _{(1),\infty }^\textrm{s,e}(x+iy), \end{aligned}$$

(10.41)

it follows that

$$\begin{aligned} & \int _{ D_R } \Big (\rho _{(1),\infty }^\textrm{s,b}(x+iy) \chi _{ x<0 }-\rho _{(1),\infty }^\textrm{s,e}(x+iy) \Big )\,dx\,dy \\ &\qquad = \frac{1}{\pi } \int _{ D_R } y \Big ( \int _{ -\infty }^\infty \, e^{-s^2} \sin (4sy)\mathrm{{erfc}}(2x-s)\,ds \Big ) \,dy\,dx. \end{aligned}$$

Furthermore, by using $\int _{-\infty }^\infty e^{-s^2} \sin (4sy)\,ds=0$ and the principal value integral

$$\begin{aligned} \lim _{ R \rightarrow \infty } \int _{-R}^R \mathrm{{erf}}(s-2x) \,dx = -s, \end{aligned}$$

(10.42)

we obtain

$$\begin{aligned} & -\lim _{ R \rightarrow \infty } \frac{1}{\pi } \int _{D_R} y \Big ( \int _{-\infty }^\infty \, e^{-s^2} \sin (4sy) \mathrm{{erf}}(s-2x) \,ds \Big ) \,dy\,dx \\ &\quad =\frac{1}{\pi } \int _{-\infty }^\infty y \Big ( \int _{-\infty }^\infty e^{-s^2} \sin (4sy) s \,ds \Big ) \,dy =\frac{2}{\sqrt{\pi }} \int _{-\infty }^\infty y^2 \,e^{-4y^2}\,dy =\frac{1}{8}, \end{aligned}$$

which leads to (10.40).    $\square$

Note that compared to the analogous identity (4.12) for the GinUE edge scaling limit, the RHS of (10.40) takes on the nonzero value $-1/8$. Curiously, the companion identity to (4.12), namely Proposition 4.4 with $\beta =2$, which relates to the dipole moment of the edge density profile takes on the nonzero value $-1/8 \pi$.

We remark that the bulk multipole screening sum rule analogous to (8.17)

$$\begin{aligned} \int _{\mathbb C_+} w^{2p} \rho _{(2),\infty }^{\textrm{s,b} \,T} (z,w) \, d^2 w = - z^{2p} \rho _{(1),\infty }^{\textrm{s, b} }(z), \quad p \in \mathbb Z_{\ge 0}, \end{aligned}$$

(10.43)

has been established in [244].

5 Elliptic GinSE

The elliptic GinSE is defined in a similar way to Sect. 7.9. Namely, for a parameter $\tau$, $0\le \tau <1$, set

$$\begin{aligned} X= \sqrt{1 + \tau } \, S + \sqrt{1 - \tau } \, A. \end{aligned}$$

(10.44)

Here S is a member of Gaussian symplectic ensemble (GSE) of Hermitian matrices (see e.g. [237, Sect. 1.3.2], whereas A is a member of anti-symmetric GSE. Its eigenvalue PDF is of the form (10.4) with the elliptic GinSE potential (10.20) previously mentioned.

As discussed in Sect. 10.2, the global-scaled eigenvalues $z_j \rightarrow \sqrt{N} z_j$ distribute in a way to minimise the energy (10.6) with the potential (10.20). This minimisation problem can be exactly solved, which gives that the limiting spectrum is given by the ellipse

$$\begin{aligned} \Big \{ (x,y) \in \mathbb {R}^2 : \Big ( \frac{x}{1+\tau } \Big )^2+ \Big ( \frac{y}{1-\tau } \Big )^2 \le 1 \Big \}; \end{aligned}$$

(10.45)

see e.g. Sect. 2.3 and [114]. As in the elliptic GinU/OE, this is the elliptic law for the elliptic GinSE.

Turning to the correlation functions, by combining (10.25) and (10.23), one can show that the associated pre-kernel $\kappa _N^{ \mathrm e }(z,w)$ is evaluated as

$$\begin{aligned} \kappa _N^{ \mathrm e }(z,w) &= \frac{ \sqrt{2} }{ \pi (1-\tau ) \sqrt{1-\tau ^2} }\sum _{k=0}^{N-1} \frac{ (\tau /2)^{k+1/2} }{(2k+1)!!} H_{2k+1} \Big ( \frac{z}{ \sqrt{\tau } } \Big ) \sum _{l=0}^k \frac{(\tau /2)^l}{(2l)!!} H_{2l} \Big ( \frac{w}{\sqrt{\tau }} \Big ) \nonumber \\ & \quad - \frac{ \sqrt{2} }{ \pi (1-\tau ) \sqrt{1-\tau ^2} }\sum _{k=0}^{N-1} \frac{ (\tau /2)^{k+1/2} }{(2k+1)!!} H_{2k+1} \Big ( \frac{w}{ \sqrt{\tau } } \Big ) \sum _{l=0}^k \frac{(\tau /2)^l}{(2l)!!} H_{2l} \Big ( \frac{z}{\sqrt{\tau }} \Big ). \end{aligned}$$

(10.46)

It reduces to the pre-kernel $\kappa _N^{ \mathrm g }$ in (10.28) in the limit $\tau \rightarrow 0^+$. In the weakly non-Hermitian regime when $\tau =1-\alpha ^2/N$, the scaling limit of the correlation functions at the origin was derived in [353]. The analogous result at the edge of the spectrum was later obtained in [35]. (We also remark that a mapping between the elliptic GinSE with a fermion field theory was suggested by Hastings [318].) Fairly recently, it was shown in [26] that for a fixed $\tau$ (also called the regime of strong non-Hermiticity), the universal scaling limit (10.31) appears at the origin. Let us mention that the analysis in [35, 353] was based on proper Riemann sum approximations, whereas a double contour integral representation was used in [26]. These methods provide a short way to find an explicit formula of the limiting pre-kernel, but it is not easy to perform the asymptotic analysis in a more general setup or to precisely control the error term. For these purposes, extending Proposition 10.4, the idea of using a proper differential equation was established in [118, 209], which reads as follows.

Proposition 10.6

We have

$$\begin{aligned} \partial _z \kappa _N^{ \mathrm g }(z,w)& = \frac{2z}{1+\tau } \, \kappa _N^{ \mathrm g }(z,w) + \frac{2}{\pi (1-\tau ^2)^{3/2}} \sum _{k=0}^{2N-1} \frac{ (\tau /2)^{k} }{k!} H_{k}\Big ( \frac{z}{\sqrt{\tau }} \Big ) H_{k} \Big ( \frac{w}{\sqrt{\tau }} \Big ) \nonumber \\ &\quad - \frac{2}{\pi (1-\tau ^2)^{3/2}} \frac{ (\tau /2)^{N} }{(2N-1)!!} H_{2N} \Big ( \frac{z}{\sqrt{\tau }} \Big ) \sum _{l=0}^{N-1} \frac{(\tau /2)^l}{(2l)!!} H_{2l} \Big ( \frac{w}{\sqrt{\tau }} \Big ) . \end{aligned}$$

(10.47)

Proof

(Sketch) The general idea to derive such an identity is the same as that used in the proof of Proposition 10.4; differentiate the pre-kernel and properly rearrange the indices in the summations to extract the pre-kernel itself multiplied by z up to proportionality, and then collect all the remaining additive terms. Contrary to the proof of Proposition 10.4, the well-known functional relations

$$\begin{aligned} H_j'(z)=2j H_{j-1}(z), \qquad H_{j+1}(z)=2zH_j(z)-H_j'(z) \end{aligned}$$

(10.48)

of the Hermite polynomials are crucially used in the computations and we refer to [118] for more details.    $\square$

We also refer to [127] for the Laguerre version of such an identity.

We now bring to attention the fact that the first inhomogeneous term in (10.47) corresponds to the kernel of the elliptic GinUE with $N \mapsto 2N$; see Proposition 2.5. (A similar feature for the elliptic GinOE is highlighted above Proposition 7.12.) As will be discussed below, such a relation can be observed in further extensions to GinSE. We also refer the reader to [6] for a similar relation for the Hermitian random matrix models.

Using Proposition 10.6, one can derive the scaling limits of the elliptic GinSE correlation functions in various regimes. For $\tau$ fixed, it was shown in [118] that for the real axis centred bulk and edge of the spectrum, the universal scaling limits (10.31) and (10.34) arise. Furthermore, in the edge scaling limits, as a counterpart of Proposition 2.6, the subleading correction term was derived. In the weakly non-Hermitian regime when $\tau =1-\alpha ^2/N$, the bulk and edge scaling limits were obtained in [119], extending previous results [35, 353]; see also [209]. The first paper on this topic [369] used supersymmetry techniques to deduce the density profile perpendicular to the real axis in the bulk, and was motivated by numerical findings in quantum chromodynamics (QCD) [310]. A structural finding in [119] shows that the limiting pre-kernel in the bulk scaling limit is of the unified Wronskian form (10.37) with

$$\begin{aligned} f_z(u):=\frac{1}{2\pi } \int _{-C(\alpha )}^{C(\alpha )} e^{-t^2/2} \sin (2t(z-u))\,\frac{dt}{t}, \qquad E:=\mathbb {R}, \end{aligned}$$

(10.49)

where $C(\alpha )$ is an explicit constant depending on $\alpha$ and on the position where we zoom the point process. In the same spirit, the edge scaling limit is again of the form (10.37) with

$$\begin{aligned} f_z(u):=2\alpha \int _{0}^{u} e^{ \alpha ^3(z-t)+\frac{\alpha ^6}{12} } \mathrm{{Ai}}\Big (2\alpha (z-t)+\frac{\alpha ^4}{4}\Big )\,dt, \qquad E:=(-\infty ,0). \end{aligned}$$

(10.50)

Remark 10.2

Note that the bulk scaling limit (10.49) has again the translation invariance. Conversely, it was shown in [22] that if a scaling limit of (10.4) satisfies the translation invariance, then it is of the form (10.49). The main idea for this characterisation was a use of Ward’s identity for the ensemble (10.4), which says that $\mathbb {E}_N W_N^+[ \psi ]=0$, where $\psi$ is a test function and

$$\begin{aligned} W_N^+[\psi ]&:= \sum _{ j \not =k } \psi (z_j) \Big (\frac{ 1 }{ z_j-z_k }+\frac{1}{z_j-\bar{z}_k} \Big ) +2 \sum _{j=1}^N \frac{\psi (z_j)}{z_j-\bar{z}_j} \nonumber \\ &\quad - 2 \sum _{ j=1 }^N [ \partial _z W \cdot \psi ](z_j) + \sum _{j=1}^N \partial \psi (z_j); \end{aligned}$$

(10.51)

cf. (5.21). Here, $[\partial _z W \cdot \psi ](z_j):=\partial _z W(z)|_{z=z_j} \psi (z_j)$ and $\partial \psi (z_j)= \partial _z \psi (z)|_{z=z_j}.$

6 Partition Functions and Gap Probabilities

Recall that the normal matrix ensemble (10.5) forms a determinantal point process. This integrable structure allows an explicit expression of the partition function; see (5.15). Similarly, using the Pfaffian structure (10.26) and de Bruijn type formulas [110], one can express the partition function $Z_N^{ \mathbb {H} }(W)$ in terms of the skew norms (10.11) as

$$\begin{aligned} Z_N^{ \mathbb {H} }(W)= \prod _{j=0}^{N-1} r_j; \end{aligned}$$

(10.52)

see e.g. [26, Remark 2.5]. For instance, for the elliptic GinSE, it follows from (10.23) that the associated partition function $Z_N^{ \mathbb {H} }(W^\textrm{e})$ is given by

$$\begin{aligned} Z_N^{ \mathbb {H} }(W^\textrm{e}) = \frac{((1-\tau )\sqrt{1-\tau ^2} \pi )^N}{ 2^{N^2} } \prod _{k=0}^{N-1} (2k+1)!. \end{aligned}$$

(10.53)

This explicit expression leads to the following asymptotic expansion; cf. Proposition 4.1 for its counterpart for $Z_N^{ \mathbb {C} }(W^\textrm{g})$.

Proposition 10.7

We have

$$\begin{aligned} \log Z_N^{ \mathbb {H} }(W^\textrm{e}) &= N^2 \log N - \frac{3}{2} N^2 +\frac{1}{2} N \log N +\Big ( \frac{\log ( 4\pi ^{3} (1-\tau )^{3}(1+\tau ) )}{2} -\frac{1}{2} \Big ) N \nonumber \\ &\quad -\frac{1}{24}\log N + \frac{5\log 2}{24} +\frac{1}{2} \zeta '(-1)-\frac{1}{48N}-\frac{1}{1920N^2}+O(\frac{1}{N^3}). \end{aligned}$$

(10.54)

In particular, we have

$$\begin{aligned} \log Z_N^{ \mathbb {H} }(NW^\textrm{g}) &= - \frac{3}{2} N^2 -\frac{1}{2} N \log N +\Big ( \frac{\log ( 4\pi ^{3} )}{2} -\frac{1}{2} \Big ) N \nonumber \\ &\quad -\frac{1}{24}\log N + \frac{5\log 2}{24} +\frac{1}{2} \zeta '(-1)-\frac{1}{48N}-\frac{1}{1920N^2}+O(\frac{1}{N^3}). \end{aligned}$$

(10.55)

Proof

One can rewrite (10.53) in terms of the Barnes G-function as

$$\begin{aligned} Z_N^{ \mathbb {H} }(W^\textrm{e})& = \frac{((1-\tau )\sqrt{1-\tau ^2} \pi )^N}{ 2^{3N^2/2-N/2} } \frac{G(2N+1)}{ G(N+1) } \\ &= ((1-\tau )^3(1+\tau ) \pi )^{N/2} G(N+1) \frac{ G(N+\frac{3}{2}) }{G(\frac{3}{2})}. \end{aligned}$$

Then the asymptotic behaviour (10.54) follows from the knowledge of the known asymptotic expansion of the G-function (see e.g. [226, Th. 1]). The second expansion (10.55) immediately follows from (10.54) with $\tau =0$, where the additional difference $(N^2+N) \log N$ is due to the simple scaling $z_j \mapsto \sqrt{N} z_j.$    $\square$

We now discuss the asymptotics of the partition functions in a more general setup. For a fixed Q, the asymptotic expansion of the partition function $Z_N^{ \mathbb {C} }(NQ)$ in (10.5) was discussed in Sect. 5.3 in detail. For radially symmetric potentials, the use of (10.52) and (10.13) was made in [125] to show that

$$\begin{aligned} \log Z_N^{ \mathbb {H} }(NQ) & =-2N^2 I_Q[\mu _Q] - \frac{1}{2}N\log N \nonumber \\ &\quad + \Big ( \frac{\log (4\pi ^2)}{2}- \frac{1}{2} E_Q[\mu _Q] - U_{\mu _Q}(0) \Big ) \, N \end{aligned}$$

(10.56)

$$\begin{aligned} &\quad - \frac{\chi }{24}\log N +\Big ( \frac{\chi }{2} \zeta '(-1)+\frac{5}{24}\log 2\Big )+ O(1), \end{aligned}$$

(10.57)

where

$$\begin{aligned} E_Q[\mu _Q] := \int _{ \mathbb {C} } \mu _Q(z)\,\log \mu _Q(z) \, d^2z \end{aligned}$$

(10.58)

is the entropy of the equilibrium measure $\mu _Q$. Here $\chi$ is the Euler index of the droplet; for instance $\chi =1$ for the disk and $\chi =0$ for the annulus. Compared to the expansion (5.17) of $Z_N^{\mathbb {C}}(NQ)$, a notable difference is the additional $U_{\mu _Q}(0)$ in the O(N) term, which is the logarithmic potential

$$\begin{aligned} U_\mu (z) = \int \log \frac{1}{|z-w|}\, d\mu (w) \end{aligned}$$

(10.59)

evaluated at the origin. This term is closely related to the notion of renormalised energy of the Hamiltonian (10.3) (cf. [394]) since it can be checked that

$$\begin{aligned} U_{\mu _Q}(0)=- \int \log |w-\bar{w}| \,d\mu _Q(w). \end{aligned}$$

(10.60)

For a radially symmetric potential $q(r)=Q(|z|=r)$ with the droplet specified by the radii (10.8), we have

$$\begin{aligned} &I_Q[\mu _Q]=q(R_1)-\log R_1 -\frac{1}{4} \int _{R_0}^{R_1} rq'(r)^2\,dr, \nonumber \\ & U_{\mu _Q}(0) = - \log R_1 + \frac{q(R_1)-q(R_0)}{2}. \end{aligned}$$

(10.61)

This gives that for $Q=W^{ \mathrm g }$,

$$\begin{aligned} I_Q[\mu _Q]= \frac{3}{4}, \qquad U_{ \mu _Q }(0)= \frac{1}{2}, \qquad E_Q[\mu _Q]= -\log \pi . \end{aligned}$$

(10.62)

Substituting these in the formula (10.57) reclaims (10.55).

We now discuss a relation between $Z_N^{\mathbb {C}}$ and $Z_N^{\mathbb {H}}$.

Proposition 10.8

For a radially symmetric potential $W_{ \mathbb {C} }(z,\bar{z}) \equiv \omega _{ \mathbb {C} }(|z|)$, let

$$\begin{aligned} W_{ \mathbb {H} }(z,\bar{z}) \equiv \omega _{ \mathbb {H} }(|z|) := \frac{1}{2} \omega _{ \mathbb {C} }(|z|^2). \end{aligned}$$

(10.63)

Then we have

$$\begin{aligned} Z_N^{ \mathbb {H} }( W_{ \mathbb {H} } )= Z_N^{ \mathbb {C} }( W_{ \mathbb {C} } ). \end{aligned}$$

(10.64)

Proof

By the change of variables,

$$\begin{aligned} 4 \int _0^\infty r^{4k+3} e^{ -2\omega _{ \mathbb {H} } (r) } \,dr = 2 \int _0^\infty r^{2k+1} e^{ -2\omega _{ \mathbb {H} } ( \sqrt{r} ) } \,dr= 2 \int _0^\infty r^{2k+1} e^{ -\omega _{ \mathbb {C} } ( r ) } \,dr. \end{aligned}$$

Then result now follows from Proposition 10.1, (5.15) and (10.52).    $\square$

As an example, note that by (5.15), we have

$$\begin{aligned} Z_N^{ \mathbb {C} }( 2|z| )= \prod _{k=0}^{N-1} 2\pi \int _0^\infty r^{2k+1}e^{-2r}\,dr= \prod _{k=0}^{N-1} \frac{(2k+1)!}{2^{2k+1}}\pi . \end{aligned}$$

(10.65)

Then one can observe that this coincides with (10.53) with $\tau =0.$ We also mention that if the droplet $S_{\mathbb {H}}$ associated with $W_{ \mathbb {H} }$ is

$$\begin{aligned} S_{\mathbb {H}}=\{ z \in \mathbb {C}: R_0 \le |z| \le R_1 \}, \end{aligned}$$

(10.66)

then its counterpart $S_{ \mathbb {C} }$ for $W_{ \mathbb {C} }$ is given by

$$\begin{aligned} S_{\mathbb {C}}=\{ z \in \mathbb {C}: R_0^2 \le |z| \le R_1^2 \}. \end{aligned}$$

(10.67)

This can be directly checked using (10.8). Such a relation holds in general beyond the radially symmetric potentials; see [73, Lem. 1].

As a consequence of Proposition 10.8, one can obtain various statistics of the ensemble (10.4) from the analogous results for (10.5). To be more concrete, let us focus on the gap probabilities.

Proposition 10.9

For a radially symmetric domain D, let $E_N^{ W_\mathbb {H} }(0;D)$ be the probability that the ensemble (10.4) with a potential $W_{\mathbb {H}}$ has no particle inside D. We define $E_N^{ W_\mathbb {C} }$ in a same way for (10.5). Then we have

$$\begin{aligned} E_N^{ W_\mathbb {H} }(0;D)= E_N^{ W_\mathbb {C} }(0;\tilde{D}), \end{aligned}$$

(10.68)

where $\tilde{D}$ is the image of D under the map $z \mapsto z^2$.

Proof

Let us write

$$\begin{aligned} W_{ \mathbb {H},D }(z,\bar{z}):= {\left\{ \begin{array}{ll} W_{ \mathbb {H} } (z,\bar{z}) & if z \in D^c, \\ +\infty & otherwise , \end{array}\right. }, \qquad W_{ \mathbb {C},\tilde{D} }(z,\bar{z}):= {\left\{ \begin{array}{ll} W_{ \mathbb {C} } (z,\bar{z}) & if z \in \tilde{D}^c, \\ +\infty & otherwise . \end{array}\right. } \end{aligned}$$

Then by Proposition 10.8,

$$\begin{aligned} E_N^{ \mathbb {H} }(0;D)= \frac{ Z_N^{\mathbb {H}}(W_{ \mathbb {H},D }) }{ Z_N^{\mathbb {H}}(W_{ \mathbb {H} }) }= \frac{ Z_N^{\mathbb {C}}(W_{ \mathbb {C},\tilde{D} }) }{ Z_N^{\mathbb {C}}(W_{ \mathbb {C} }) } = E_N^{ W_\mathbb {C} }(0;\tilde{D}), \end{aligned}$$

(10.69)

which completes the proof.    $\square$

This proposition is particularly helpful in the context of the Mittag–Leffler ensembles. They are two-parameter generalisations of the GinU/SE for which the associated potential is of the form

$$\begin{aligned} \omega ^\textrm{ML}(|z|)=|z|^{2b}-2\alpha \log |z|, \qquad b>0, \quad \alpha >-1. \end{aligned}$$

(10.70)

For the complex Mittag–Leffler ensemble, the precise asymptotic behaviours of the gap probabilities were obtained in [149]; cf. see Sect. 3.1.1 for a summary and further references for the GinUE case when $\alpha =0, b=1$. Then as a consequence of Proposition 10.9, the analogous results for the symplectic Mittag–Leffler ensemble immediately follow. In particular, the gap probabilities of the GinSE can be obtained from the result of a complex Mittag–Leffler ensemble with $\alpha =0, b=1/2$. (See also [36] for an earlier work.) Beyond the gap probabilities, Proposition 10.8 can be used to investigate various counting statistics [20, 116, 150] as well as fluctuations of the maximal modulus [199, 469].

Remark 10.3

1. The Neumann boundary conditions disk Coulomb gas with Boltzmann factor (10.9) is exactly solvable for $\beta = 2$ [504]. In distinction to the expansion (10.55), it is found that the large N form of the logarithm of the partition function is at order N and order $\log N$ the same for the GinUE (4.3), although now there is also an $O(\sqrt{N})$ surface tension term [518, Eq. (3.42)]. Let us also mention that a generalisation to a two-component Coulomb gas and its Pfaffian structure have been studied in [346].

2. The simple explicit formula involving (10.12) of Proposition 10.1 for the skew-orthogonal polynomials normalisations in the radial case facilitates the derivation of a central limit theorem for a radially symmetric linear statistic in GinSE [120, Appendix B]. The covariance is again given by (3.21) except that the boundary term therein is not present (as a consequence of the restriction to a radial potential), and with the bulk term now multiplied by a factor of 1/2.

7 GinSE Singular Values

Consider a $p \times N$ ($p \ge N$) rectangular GinSE matrix X. As a complex matrix, X is of size $2n \times 2N$. Forming $X^\dagger X$ gives the well known construction of quaternion Wishart matrices [237, Sect. 3.2.1]. The 2N eigenvalues of $X^\dagger X$—which are the square singular values of X—are doubly degenerate. Let the N independent eigenvalues be denoted by $\{s_j\}$. Their PDF is proportional to (see [237, Proposition 3.2.2])

$$\begin{aligned} \prod _{l=1}^N s_l^{2(p-N) + 1} e^{-2s_l} \prod _{1 \le j < k \le N} (s_k - s_j)^4. \end{aligned}$$

(10.71)

Upon the global scaling $X^\dagger X \mapsto {1 \over N} X^\dagger X$, and with $p = \alpha N$ ($\alpha > 1$), as for (8.18) the smallest and largest tend almost surely to $(\sqrt{\alpha \mp 1}+1)^2$, as for the real case. This coincidence can be understood as being a consequence of $\{s_j\}$ in both (9.43) and (10.71) as being well approximated by the zeros of the Laguerre polynomials $L_N^{p-N}(px)$ [192, 328]. As in the real case, we thus have that the condition number tends to a constant in the circumstance that $\alpha > 1$. On the other hand, this breaks down for $\alpha = 1$. Specifically, we will consider the square case $p=N$. It seems that the limiting condition number has not previously been reported in the literature. The starting point is to calculate the PDF for the smallest eigenvalue. This is equal to the differentiation operation $-{d \over ds}$ applied to the gap probability $E_N^\textrm{q W}(0,(0,s))$ of there being no eigenvalues from the origin to a point s. The latter is defined by integrating the PDF (10.71) over $s_l \in (s,\infty )$, $(l=1,\dots ,N)$. Changing variables $s_l \mapsto s_l+s$ then  shows

$$\begin{aligned} E_N^\textrm{q W}(0,(0,s)) = {e^{-2Ns} \over C_N} \int _0^\infty ds_1 \cdots \int _0^\infty ds_N \, \prod _{l=1}^N (s + s_l) e^{-2s_l} \prod _{1 \le j < k \le N} (s_k - s_j)^4, \end{aligned}$$

where $C_N$ is such that the LHS equals unity for $s=0$. According to [237, Eq. (13.44) with $a=0,m=1,\beta =4,t_1=-s$], the normalised multiple integral is equal to the hypergeometric polynomial $_1 F_1(-N;1/2;-s)$, which in turn is proportional to the Laguerre polynomial $L_N^{(-1/2)}(-s)$. From the confluent limit of the hypergeometric polynomial, we conclude the simple result

$$\begin{aligned} \lim _{N \rightarrow \infty } E_N^\textrm{q W}(0,(0,x/N)) =e^{-2x} _0F_1(1/2;x) = e^{-2x} \cosh (2 \sqrt{x}); \end{aligned}$$

(10.72)

this is equivalent to [233, Eq. (2.15a)]. Consequently, the scaled condition number $\kappa _N /(2N)$ has the limiting PDF

$$\begin{aligned} - {2 \over y^3} {d \over dx} \Big ( e^{-2x} \cosh (2 \sqrt{x}) \Big ) \Big |_{x=1/y^2}. \end{aligned}$$

In keeping with our previous discussion relating to singular values, we record here too the explicit form of the distribution of $|\det X|^2$ for X a square GinSE matrix [269, Proposition 2 with $\beta = 4, \sigma _l^2 = 1/4$],

$$\begin{aligned} |\det X|^2 \mathop {=}\limits ^\textrm{d} \prod _{j=1}^N {1 \over 4} \chi _{4j}^2. \end{aligned}$$

(10.73)

Here we have defined $|\det X|^2 = \prod _{l=1}^Ns_l$, even though the eigenvalues of X are doubly degenerate. One approach to the derivation of (10.73) is to make use of knowledge of the joint PDF (10.71) and the Laguerre weight version of Selberg’s integral to compute first the Mellin transform of the distribution; recall (6.12).

8 GinSE Eigenvectors

Consider an eigenvalue $z_j$ of a GinSE matrix, and let $\boldsymbol{\ell }_j, \boldsymbol{r}_j$ denote the corresponding left and right eigenvectors. As for GinUE and GinOE eigenvectors, of interest is the matrix of overlaps (6.19). In particular, the considerations of the paragraph above Proposition 6.5 are again valid, to allow for the quaternion structure. Thus while $\boldsymbol{r}_j$ can be chosen as the first standard basis vector (now in $\mathbb C^{2N}$), we now require $\boldsymbol{\ell }_1 = (b_1,c_1, \dots , b_N,c_N)$ where $b_1 :=1$, $c_1:=0$ so that

$$\begin{aligned} \mathcal O_{11} = \langle \boldsymbol{\ell }_1, \boldsymbol{\ell }_1 \rangle = \sum _{k=1}^N ( |b_k|^2 + |c_k|^2). \end{aligned}$$

Moreover, with $Z_{pq} = \begin{bmatrix} z_{pq} & w_{pq} \\ - \bar{w}_{pq} & \bar{z}_{pq} \end{bmatrix}$ now a $2 \times 2$ block element of the strictly upper triangular portion of the triangular matrix Z in the block Schur decomposition $\{b_k, c_k \}$, and regarding now $\ell _1$ as a left eigenvector of Z with eigenvalue $z_1$ (this does not effect $\mathcal O_{11}$), it follows that $\{b_k, c_k\}$ satisfy the coupled recurrence

$$\begin{aligned} b_p = {1 \over z_1 - z_p} \sum _{k=1}^{p-1} ( b_k z_{k,p} - c_k \bar{w}_{k,p} ), \quad c_p = {1 \over z_1 - \overline{z}_p} \sum _{k=1}^{p-1} ( b_k w_{k,p} + c_k \bar{z}_{p,q} ), \end{aligned}$$

subject to the initial conditions $b_1 = 1$ and $c_1 = 0$.

Writing now $\boldsymbol{\ell }_1 = \boldsymbol{\ell }_1^{(N)}$ and denoting by $\boldsymbol{\ell }_1^{(n)}$ the truncation of this vector ending with $b_n, c_n$, it follows that

$$\begin{aligned} & || \boldsymbol{\ell }_1^{(n+1)} ||^2 = || \boldsymbol{\ell }_1^{(n)} ||^2 \bigg ( 1 + {1 \over |z_1 - z_{n+1}|^2} \Big | \sum _{q=1}^N (\tilde{b}_k z_{k,p} - \tilde{c}_k w_{k,p}) \Big |^2 \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + {1 \over |z_1 - \overline{z}_{n+1}|^2} \Big | \sum _{q=1}^N (\tilde{b}_k w_{k,p} + \tilde{c}_k \overline{z}_{k,p}) \Big |^2 \bigg ), \end{aligned}$$

where

$$\begin{aligned} \tilde{b}_k = b_k \Big /\Big ( \sum _{q=1}^n ( |b_q|^2 + |c_q|^2) \Big )^{1/2}, \qquad \tilde{c}_k = c_k \Big /\Big ( \sum _{q=1}^n ( |b_q|^2 + |c_q|^2) \Big )^{1/2}. \end{aligned}$$

Taking into consideration that $(\tilde{b}_1, \tilde{c}_1, \dots , \tilde{b}_N, \tilde{c}_N)$ is a unit vector and each $z_{k,p}, w_{k,p}$ is an independent standard complex Gaussian, the analogue of (6.22) now follows [27, 199]:

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

(10.74)

Here $X_n, Y_n$ are independent standard complex Gaussians.

Remark 10.4

1. A Pfaffian formula for the average of (10.74) over $\{z_j\}_{j=2}^N$ is given in [27]. To obtain from (10.74) the analogue of (6.22) one requires knowledge of the distribution of $\{ | z_n |^2 \}_{n=2}^N$ conditioned on $z_1 = 0$. The first result of this type, which is an easy consequence of (10.52) and the normalisation formula in Proposition 10.1 gives that [469] $\{|z_n|^2 \}_{n=1}^N \mathop {=}\limits ^\textrm{d} \{ \Gamma [2n;1] \}_{n=1}^N$. As shown in [199], this same result holds true conditioned on $z_1 = 0$ with the sets now beginning at $n=2$. From this, conditioning on $z_1 = 0$ and averaging over $\{ | z_n |^2 \}_{n=2}^N$ in (10.74) gives [199] $\langle \mathcal O_{11} \rangle |_{z_1 = 0} \mathop {=}\limits ^\textrm{d} 1/\textrm{B}[4,2N]$ and hence ${1 \over 2N} \langle \mathcal O_{11} \rangle |_{z_1 = 0} \mathop {\rightarrow }\limits ^\textrm{d} 1/\Gamma [4;1]$.

2. In [27], evidence is given for the validity of the analogue of the large N forms (6.28) (with $M = 1$) and (6.22) (with respect to the leading-order term at least).