Anthony Mays
email: a.mays@ms.unimelb.edu.au

A real quaternion spherical ensemble of random matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form Y = A -1 B, where A and B are independent N × N matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β = 1, 2, 4.

We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure Y = A -1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.

Introduction and main results

This paper follows on from [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF] where the author described a so-called geometrical triumvirate of random matrix ensembles; that is, classes of matrices that (in some sense) live naturally on the three surfaces of constant curvature: the plane, the sphere and the antisphere (this categorization was also used in [START_REF] Krishnapur | Zeros of random analytic functions[END_REF] in a similar context). Through the circular law [START_REF] Girko | Circular law" (trans. Durri-Hamdani)[END_REF][START_REF] Bai | Circular law[END_REF][START_REF] Götze | The circular law for random matrices[END_REF][START_REF] Tao | Random matrices: universality of ESDs and the circular law[END_REF], we identify the Ginibre ensembles [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF] (and other iid ensembles) with the plane, since in the large N limit the eigenvalue density is uniform in some planar disk and zero outside. In fact, in [START_REF] Tao | Random matrices: universality of local spectral statistics of non-Hermitian matrices[END_REF] it is shown that the general n-point correlation functions are universal in the class of iid matrices, assuming the first four moments are finite. We identify the sphere with matrices Y = A -1 B, where A and B are independent iid matrices. The spherical law (conjectured in [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF] and [START_REF] Rogers | Universal sum and product rules for random matrices[END_REF], and proven in [START_REF] Bordenave | On the spectrum of sum and products of non-Hermitian random matrices[END_REF]) establishes that the eigenvalues of these matrices (under stereographic projection) are uniformly distributed on the sphere in the limit of large matrix dimension. However, as yet nothing is known about the higher order correlations for the spherical ensembles beyond the Gaussian case. Lastly, the anti-sphere we may identify with truncated ensembles, since in the two studied cases (truncated unitary [START_REF] Życzkowski | Truncations of random unitary matrices[END_REF][START_REF] Petz | On asymptotics of large Haar distributed unitary matrices[END_REF][START_REF] Forrester | Quantum conductance problems and the Jacobi ensemble[END_REF] and truncated orthogonal [START_REF] Khoruzhenko | Truncations of random orthogonal matrices[END_REF]) the eigenvalues are uniformly distributed in a hyperbolic space. There is, to date, no 'anti-spherical law'.

This paper addresses the outstanding spherical ensemble in the traditional β = 1, 2, 4 collection: the real quaternion (β = 4) spherical ensemble. Previously, the complex (β = 2) and real (β = 1) cases have been studied in [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF][START_REF] Krishnapur | Zeros of random analytic functions[END_REF][START_REF] Krishnapur | From random matrices to random analytic functions[END_REF][START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF][START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]. To quickly gain some insight into the behaviour of these eigenvalues we can simulate examples (see Figure 1). Note the Figure 1: Plots of (stereographically projected) eigenvalues for 120 independent 100 × 100 spherical matrices with real (left), complex (right) entries.

ring of eigenvalues in the real spherical ensemble, which is absent in the complex ensemble; this ring corresponds to real eigenvalues, which have a finite probability of appearing in real ensembles. The remaining eigenvalues come in complex conjugate pairs, meaning that the real spherical ensemble has reflective symmetry across the ring (generally the complex eigenvalues of real matrices occur in complex conjugate pairs) and rotational symmetry about the axis through the centre of the ring. The complex ensemble has unbroken rotational symmetry. For the real quaternion spherical ensemble in the present study we will find that there is a depletion of eigenvalues along the ring corresponding to the real axis. Such an effect in the real quaternion Ginibre ensemble follows from the joint eigenvalue probability density function computed in Ginibre's original paper [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF].

We note that the eigenvalue problem for matrices Y = A -1 B is equivalent to the generalized eigenvalue problem, that is solutions λ j to det(B -λA) = 0, where A is invertible. The real spherical ensemble was discussed in [START_REF] Edelman | How many eigenvalues of a random matrix are real?[END_REF] in the context of these generalized eigenvalues, where the authors establish that the arctan of the real eigenvalues are uniformly distributed on the unit circle. (See [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF] for more discussion on the statistics of the real eigenvalues.)

To approach the real quaternion ensemble, we follow the skew-orthogonal polynomial method used in [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF], which includes an iterated integral technique (see Section 3) used successfully for the β = 2 spherical ensemble [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] (a similar technique was also used in [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF]). For a more detailed description of the techniques used in this paper and the underlying theory, see [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF].

Since quaternions are crucial to this study, we first provide a quick overview. A quaternion is analogous to a complex number, except that it has four basis elements instead of two. Typically they are written in the form q = q 0 + iq 1 + jq 2 + kq 3 , with the relations i 2 = j 2 = k 2 = ijk = -1, and the q l are in general complex. We will also use an alternative representation as 2 × 2 matrices:

q = w x y z , (1)
where w = q 0 + iq 1 , x = q 2 + iq 3 , y = -q 2 + iq 3 , z = q 0 -iq 1 . The analogue of complex conjugation for quaternions we denote q * = q 0 -iq 1 -jq 2 -kq 3 , or in the matrix representation

q * = z -x -y w .
In the case that q 0 , q 1 , q 2 , q 3 ∈ R we say that q is a real quaternion and from (1), with α = q 0 + iq 1 and β = q 2 + iq 3 , we have

q = α β -β ᾱ , (2)
with conjugate

q * = ᾱ -β β α . With Q = [q j,k ] we denote by Q D the matrix [q * k,j ]
, and we call it the dual of Q. If Q = Q D then Q is said to be self-dual. We will regularly use quaternion analogues of the usual matrix trace and determinant [START_REF] Dyson | Correlations between eigenvalues of a random matrix[END_REF].

Definition 1.1. For an N × N matrix Q with real quaternion entries the quaternion trace is defined as the sum of the scalar parts of the diagonal entries

qTr Q := N j=1 (q 0 ) j,j . (3)
The quaternion determinant is defined by

qdet Q := P ∈S N (-1) N -l l 1 qTr(q ab q bc • • • q sa ), ( 4
)
where P is any permutation of (1, ..., N ) that consists of l disjoint cycles of the form

(a → b → c → • • • → s → a).
Note that the definition (3) gives

qTr Q N ×N = 1 2 Tr Q2N×2N , ( 5
)
where Q2N×2N is the matrix corresponding to Q N ×N with the quaternions replaced by their 2 × 2 representatives (2). Furthermore, it is shown in [START_REF] Dyson | Correlations between eigenvalues of a random matrix[END_REF] that with the definition (4) and with Q N ×N a self-dual real quaternion matrix

qdet Q N ×N = det Q2N×2N 1/2 . ( 6
)
Since we will be mostly using the 2 × 2 representation for the quaternions we will most often make use of ( 5) and ( 6) instead of Definition 1.1, although we will suppress the 'hat' notation for ease of reading. An element of the real quaternion spherical ensemble is a product of two real quaternion Ginibre matrices. These matrices, as defined in [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF], are N × N matrices with independent Gaussian real quaternion entries. For ease of reference, we make an explicit definition. Definition 1.2. Define standard Gaussian random real quaternion variables as q = q 0 + iq 1 + jq 2 + kq 3 , where q 0 , q 1 , q 2 , q 3 are independent real standard Gaussian variables.

Let A, B be independent N × N matrices with standard Gaussian real quaternion entries. Then the real quaternion spherical ensemble consists of N × N matrices of the form Y = A -1 B.

Equivalently, choose A and B to be independent 2N × 2N matrices containing the 2 × 2 blocks (2), where α, β are complex numbers with real and imaginary parts being independent standard Gaussian variables. In this case Y = A -1 B is a 2N × 2N complex matrix.

In Section 2 we calculate the matrix probability density function (pdf) for these matrices. It turns out that we can combine the results in [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF] for the complex and real cases (also later derived independently in [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] and [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF] respectively) to state the matrix pdfs for the three spherical ensembles (real, complex and real quaternion) in the one form. Note that in [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF] the author shows that a matrix Cauchy distribution is obtained for all spherical matrices, independent of the elemental distribution.

Proposition 1.3. With Y a real (β = 1), complex (β = 2) or real quaternion (β = 4) spherical matrix, the matrix pdf is

P β (Y) = π -βN 2 /2 N -1 j=0 Γ N +1+j 2 β Γ j+1 2 β det(1 + YY † ) -βN , (7)
where the 'dagger' should be interpreted as 'transpose', 'Hermitian conjugate' or 'quaternion dual' respectively. Note that for β = 4, the determinant is to be understood as a quaternion determinant.

We shall be interested in the eigenvalue distribution; in Figure 2 we present a simulation like those in Figure 1, showing that (on stereographic projection) the eigenvalue distribution splits into two symmetrical hemispheres indicating repulsion from the ring corresponding to the real axis, with the complex eigenvalues coming in conjugate pairs (the latter is a general property of self-dual real quaternion matrices). Using a Schur decomposition, in Section 3 we change variables in [START_REF] Edelman | How many eigenvalues of a random matrix are real?[END_REF] to extract the eigenvalue joint probability density function (jpdf), by applying the iterated integration technique used in [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] and [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]. Then, taking advantage of the rotational symmetry about one axis in Figure 2, we apply the fractional linear transformation

λ = 1 i w -1 w + 1 , (8)
which takes the upper half plane to the unit disk D, and the lower half-plane to its complement, and we find the eigenvalue jpdf

Q(w) = C N N j=1 1 i |w j | 2 τ (w j )τ 1 w j ∆ w, 1 w , (9)
with w = {w 1 , . . . , w N }, w j ∈ D,

1 w = 1 w 1 , . . . , 1 w N , 1 w j ∈ C\D, C N = (-1) N (N -1)/2 (π) -N Γ(N + 1) N j=1 Γ(2N + 2) (Γ(2j)) 2 , τ (x) = 1 x N -1/2 1 |x| -|x| 1/2 |x| + 1 |x| N +1 , ∆(x 1 , ..., x n ) = 1≤j<k≤n (x k -x j ).
Note that the factors of (1/|x| -|x|) signify the expected repulsion from the real axis. The average over this jpdf is calculated in terms of a Pfaffian expression in Section 4. By finding appropriate skew-orthogonal polynomials (Section 5), and using techniques similar to those in [START_REF] Borodin | The Ginibre ensemble of real random matrices and its scaling limits[END_REF] we calculate the eigenvalue correlation functions in Section 6, finding the same structure as in the real quaternion Ginibre ensemble [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF][START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF].

In Section 6.1 we find the eigenvalue density to be

ρ (1) (w) = 1 r 2 1 r -r r + 1 r 2N +2 (2N + 1)(2N ) π N -1 j=0 2N -1 j r 2j+1-2N -r 2N -1-2j 2N -2j -1 = 1 r 2 1 r -r r + 1 r 2N +2 2 2N -1 (2N + 1)N π 1-r 2 0 (1 -γ) -N -1/2 (1 -γ/2) 2N -1 dγ, (10)
where w = r e iθ , which for large N tends to

ρ (1) (w) ∼ N →∞ 2N π(1 + r 2 ) 2 , (11)
which is the expected analogue of [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]Eqn. (59)]. This limiting density tells us that, on stereographic projection, we have a uniform density of eigenvalues on the sphere in the limit of large matrix dimension. This is a consequence of the spherical law, which was discussed at the beginning of this introduction. With λ j → λ j / √ 2N in ( 22), then we note that (for large

N ) Q( λ) approaches CN N j=1 e -|λ j | 2 |λ j -λ j | 2 1≤j<k≤N |λ k -λ j | 2 |λ k -λ j | 2 , ( 12
)
where CN is independent of the λ j . This expression [START_REF] Ginibre | Statistical ensembles of complex, quaternion and real matrices[END_REF] is the same as the eigenvalue jpdf for the Ginibre ensemble of real quaternion matrices [12, Eq. (2.28)], and so we may expect to recover the same bulk correlation functions (see [START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF]) from our correlation functions here by a suitable scaled limit. In Section 6.2 we apply the rescaling (39) and obtain Proposition 6.2, which matches the result for the β = 4 Ginibre case.

Matrix probability density function

With A and B each being N ×N matrices from the real quaternion Ginibre ensemble (that is, having iid standard normal real quaternion entries) the joint distribution of

Y = A -1 B is (2π) -4N 2 e -qTr(AA D +BB D )/2 (dA)(dB) = (2π) -4N 2 e -Tr( Â Â † + B B † )/4 (d Â)(d B), (13)
where we have changed to the 2 × 2 representation for the RHS. (For the remainder of this paper we will suppress the 'hat' notation, however we will endeavour to preserve the distinction between D for quaternion dual and † for Hermitian conjugate, which, along with 'qTr' and 'qdet', will serve to indicate which representation we are using.) The plan now is to change variables to Y and integrate out the remaining degrees of freedom. To achieve this we will follow the real quaternion analogue of the procedure used in [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] and [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]. First we list some required lemmata specific to real quaternion matrices.

Lemma 2.1 ([26]

). Let X be an n × m (n ≥ m) matrix with real quaternion entries, and

A = X D X. Then (dX) = c (qdet A) 2(n-m)+1 (dA),
where c is independent of X.

Lemma 2.2 ([21]

). With A and M both n × n real quaternion matrices (with A nonsingular) and X = A D MA we have

(dX) = (qdet A D A) 2n-1 (dM).

Lemma 2.3 ([10]

). A corollary of the Selberg integral [START_REF] Selberg | Bemerkninger om et multipelt integral[END_REF] is the evaluation

∞ 0 dx 1 • • • ∞ 0 dx N N l=1 x 2a l e -2x l 1≤j<l≤N |x l -x j | 4 = 2 -2N (N +a) N -1 j=0 Γ (3 + 2j) Γ(2a + 1 + 2j).
Lemma 2.4. Let B m×n = x m×m A m×n y n×n , where each of the matrices consist of 2 × 2 real quaternions (2), then

(dB m×n ) = |qdet x m×m | 4n |qdet y n×n | 4m (dA m×n ).
This last lemma is the real quaternion analogue of the statement about real matrices in Theorem 2.1.5 of [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF]. From the working there we can verify this lemma by noting that x acts on the columns of A (of which there are n) and y on the rows (of which there are m), and then each quaternion has 4 independent components.

Letting C := AA D (which is positive definite) we see from Lemma 2.1 that

(dA) = c qdet C (dC), (14)
and also, letting B = AY, by Lemma 2.4 we have (dB) = |qdet A| 4N (dY) = (det C) N (dY), and so ( 13) becomes

(2π) -4N 2 e -Tr(C(1+YY † )) (det C) N +1/2 (dC)(dY). ( 15
)
To calculate c we multiply both sides of ( 14) by e -(TrAA † )/4 and integrate over the whole domain

e -(TrAA † )/4 (dA) = c C>0 e -(TrC)/4 (det C) 1/2 (dC),
where the positive definiteness of C is indicated by C > 0 in the integral terminal. We note that the LHS is just an integral over 4N 2 independent standard Gaussians and so

c = (2π) 2N 2 C>0 e -(TrC)/4 (det C) 1/2 (dC)
. [START_REF] Gradshteyn | Table of integrals, series and products, Corrected and enlarged edition[END_REF] and then integrate over D to obtain

Let D := (1 + YY † ) 1/2 C(1 + YY † ) 1/2 in
(2π) -2N 2 det(1 + YY † ) -2N D>0 e -(TrD)/4 (det D) N +1/2 (dD) C>0 e -(TrC)/4 (det C) 1/2 (dC) (dY),
where we used Lemma 2.2 to obtain the Jacobian for changing variables from C to D.

Since both C and D are self-dual real quaternion matrices we can make use of the well known Jacobians for changing variables from the matrix entries to the matrix eigenvalues (see for example [START_REF] Forrester | Log-gases and random matrices[END_REF]Chapter 1.3]) so that the ratio of integrals becomes

∞ 0 dλ 1 . . . ∞ 0 dλ N N j=1 λ 2N +1 j e -λ j /2 j<k |λ k -λ j | 4 ∞ 0 dλ 1 . . . ∞ 0 dλ N N j=1 λ j e -λ j /2 j<k |λ k -λ j | 4
.

By changing variables λ j → 4λ j we can apply Lemma 2.3 to evaluate the integrals, giving the final expression

π -2N 2 N -1 j=0 Γ (2(N + 1 + j)) Γ (2(j + 1)) qdet(1 + Y N Y D N ) -4N (dY).
By comparing this to the analogous results in the real and complex cases we can write them in the single equation [START_REF] Edelman | How many eigenvalues of a random matrix are real?[END_REF].

Eigenvalue jpdf

Here we will change variables in the matrix pdf [START_REF] Edelman | How many eigenvalues of a random matrix are real?[END_REF] from the elements of Y to the eigenvalues of Y. Using the 2 × 2 representation of quaternions (2) apply the Schur decomposition [START_REF] Akemann | The complex Laguerre symplectic ensemble of non-Hermitian matrices[END_REF] for a detailed description of this in the real quaternion case), with R 2N an upper triangular real quaternion matrix

Y = Q † R 2N Q (see
R 2N =            0 λ 1 λ 1 0 r 1,2 s 1,2 -s 1,2 r 1,2 . . . r 1,N s 1,N -s 1,N r 1,N 0 0 0 0 0 λ 2 λ 2 0 . . . r 2,N s 2,N -s 2,N r 2,N . . . . . . . . . . . . 0 0 0 0 0 0 0 0 . . . 0 λ N λ N 0           
, where λ j and λ j are the N complex conjugate-paired eigenvalues of Y, r j,k and s j,k are complex numbers, and Q ∈ Sp(N )/(U (1)) N is a symplectic (ie. unitary real quaternion) matrix (using the 2 × 2 representation). Since the decomposition is only unique up to ordering on the eigenvalues, we divide through by 2 N Γ[N + 1]. The Jacobian for this change of variables is [START_REF] Mehta | Random matrices[END_REF][START_REF] Akemann | The complex Laguerre symplectic ensemble of non-Hermitian matrices[END_REF] (dY

) = (Q † dQ)(dR 2N )| ∆( λ)| 2 ,
where ∆( λ) := N j=1 (λ j -λ j ) 1≤j<k≤N (λ j -λ k )(λ j -λ k ) and (dX) represents the wedge product of all the independent elements of a matrix of differentials dX. Then

P 4 (Y)(dY) = (Q † dQ)(dR 2N ) π 2N 2 2 N Γ(N + 1) N -1 j=0 Γ (2(N + 1 + j)) Γ (2(j + 1)) | ∆( λ)| 2 det(1 2N + R 2N R † 2N ) -2N . (16)
We will now apply the iterated integration technique used in [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] and [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF] to integrate over the strictly upper triangular components of R 2N . First we separate out the last column

R 2N = R 2N -2 u 2N -2 0 z N ,
where z N is the bottom right 2 × 2 block on the diagonal, which contains the eigenvalues λ N , λ N , and u 2N -2 is of size (2N -2) × 2; explicitly

u 2N -2 =        r 1,N -1 s 1,N -s 1,N r 1,N . . . . . . r N -1,N -1 s N -1,N -s N -1,N r N -1,N       
.

Then the determinant in [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] becomes

det(12N + R2N R † 2N ) -2N = det 12N-2 + R2N-2R † 2N -2 + u2N-2u † 2N -2 u2N-2z † N zN u † 2N -2 12 + zN z † N -2N = det 12N-2 + R2N-2R † 2N -2 + u2N-2u † 2N -2 -uz † N (1 + zN z † N ) -1 zN u † 2N -2 0 zN u † 2N -2 12 + zN z † N -2N = det 12 + zN z † N -2N det 12N-2 + R2N-2R † 2N -2 + u2N-2u † 2N -2 -uz † N (1 + zN z † N ) -1 zN u † 2N -2 .
We use the identity

z † (1 + zz † )z = 1 -(1 + z † z) -1
to write the second determinant above as

det 1 2N -2 + R 2N -2 R † 2N -2 + u 2N -2 (1 2 + z † N z N ) -1 u † 2N -2 = det 1 2N -2 + R 2N -2 R † 2N -2 det 1 2 + v † 2N -2 v 2N -2 ,
where

v 2N -2 := (1 2N -2 + R 2N -2 R † 2N -2 ) -1/2 u 2N -2 (1 2 + z † N z N ) -1/2 . (17)
So [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] becomes

P 4 (Y)(dY) = (Q † dQ)(dR 2N ) π 2N 2 2 N Γ(N + 1) N -1 j=0 Γ (2(N + 1 + j)) Γ (2(j + 1)) | ∆( λ)| 2 det 1 2 + z N z † N -2N × det 1 2N -2 + R 2N -2 R † 2N -2 -2N det 1 2 + v † 2N -2 v 2N -2 -2N . ( 18
)
Note that

(dR 2N ) = N j=1 dλ j N -1 s=1 (du 2N -2s ) 10
and we want to integrate over the independent elements of u 2N -2 , ie. we want to calculate

(du 2N -2 ) det 1 2 + v † 2N -2 v 2N -2 -2N . (19)
Applying Lemma 2.4 to [START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF] we have

(du 2N -2 ) = qdet(1 N -1 + R N -1 R D N -1 ) 2 qdet(1 + z N z D N ) 2N -2 (dv 2N -2 ) = det(1 2N -2 + R 2N -2 R † 2N -2 ) det(1 2 + z N z † N ) N -1 (dv 2N -2 ). ( 20
)
For ease of notation, let a 2N -2

:= v † 2N -2 v 2N -2 , so a 2N -2 = N -1 j=1 |r j,N | 2 + |s j,N | 2 0 0 N -1 j=1 |r j,N | 2 + |s j,N | 2 .
We can again make use of Lemma 2.1 by applying it to a 2N -2 giving

(dv 2N -2 ) = c (qdet a 2N -2 ) 2N -3 (da 2N -2 ) = c (det a 2N -2 ) N -3/2 (da 2N -2 ). ( 21
)
Multiplying both sides of ( 21) by e -Tr(v † 2N -2 v 2N -2 ) and integrating we have for the LHS

e -Tr(v † 2N -2 v 2N -2 ) (dv 2N -2 ) = (-∞,∞) 4N -4 e -2 N -1 j=1 (|sj,N | 2 +|r j,N | 2 ) N -1 j=1
dRe(s j,N )dIm(s j,N )dRe(r j,N )dIm(r j,N )

= 4N -4 j=1 ∞ -∞ e -2x 2 dx = π 2 2N -2
, and for the RHS

c e -Tr a 2N -2 (det a 2N -2 ) N -3/2 (da 2N -2 ) = c ∞ 0 e -2x x 2N -3 dx = c Γ(2N -2) 4 N -1 . because a 2N -2 = diag(x, x), where x ∈ (0, ∞). We deduce that c = π 2N -2 Γ(2N -2) . Now to calculate (19) (du 2N -2 ) det 1 2 + v † 2N -2 v 2N -2 -2N = det(1 2N -2 + R 2N -2 R † 2N -2 ) det(1 2 + z N z † N ) N -1 (dv 2N -2 ) det 1 2 + v † 2N -2 v 2N -2 -2N = K (1) 2N -2 π 2N -2 Γ(2N -2) det(a 2N -2 ) N -3/2 det(1 + a 2N -2 ) -2N (da 2N -2 ) = K (2) 2N -2 ∞ 0 x 2N -3 (1 + x) -4N dx = K (2) 2N -2 ∞ 0 x 1 + x 2N -3 1 (1 + x) 2N +3 dx, where K (1)
2N -2 stands for the determinants in front of the integral after the first equality, and

K (2) 2N -2 = K (1) 2N -2 × π 2N -2 /Γ(2N -2)
. Letting y = x/(1 + x), and so dx = (1 + x) 2 dy, we find a Selberg integral [START_REF] Selberg | Bemerkninger om et multipelt integral[END_REF] resulting in

K (2) 2N -2 ∞ 0 x 1 + x 2N -3 1 (1 + x) 2N +3 dx = K (2) 2N -2 1 0 y 2N -3 (1 -y) 2N +1 dy = K (2) 2N -2 Γ(2N -2)Γ(2N + 2) Γ(4N ) = π 2N -2 Γ(2N + 2) Γ(4N ) det(1 2N -2 + R 2N -2 R † 2N -2 ) det(1 2 + z N z † N ) N -1 .
Substituting this into (18) we have

* P 4 (Y)(dY) = π -2(N (N -1)+1) 2 N Γ(N + 1) Γ(2N + 2) Γ(4N ) N -1 j=0 Γ (2(N + 1 + j)) Γ (2(j + 1)) | ∆( λ)| 2 (1 + |λ N | 2 ) 2(N +1) × det 1 2N -2 + R 2N -2 R † 2N -2 -(2N -1) N j=1 dλ j N -1 s=2 (du 2N -2s ) (Q † dQ),
where the * on the integral indicates that we have only done some of the integrals required. We then iterate this procedure over each of the N -2 remaining columns. Lastly, we need the integral over (Q † dQ). As mentioned above, Q ∈ Sp(N )/(U (1)) N , and using a result from [START_REF] Nachbin | The Haar integral[END_REF] (see [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF]Ch. 7.1] for more details on this point) we have

(Q † dQ) = Vol(Sp(2N )) (Vol(U (1))) N = π N 2 N j=1 Γ(2j)
, where we have used the result from [5, Eq. ( 108)]

Vol Sp(N ) = 2 N π N (N +1) N j=1 Γ(2j)
.

So having integrated out all independent variables in [START_REF] Khoruzhenko | Truncations of random orthogonal matrices[END_REF], except for those corresponding to the eigenvalues λ j = x j + iy j , we are left with the eigenvalue jpdf

Q( λ)(d λ) = π -N Γ(N + 1) N j=1 Γ(2N + 2) (Γ(2j)) 2 × N j=1 |λ j -λ j | 2 (1 + |λ j | 2 ) 2(N +1) 1≤j<k≤N |λ k -λ j | 2 |λ k -λ j | 2 N j=1 dx j dy j , (22)
where, for later convenience, we restrict λ to the upper half plane (in other words we order the complex conjugate pairs Im(λ) > Im(λ)) multiplying through by 2 N .

Fractional linear transformation

As in [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF] we can take advantage of the expected rotational symmetry by applying the fractional linear transformation [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF]. Then with λ j = x j + iy j and w j = a j + ib j we have

y j = 1 -|w j | 2 |1 + w j | 2 and dx j dy j = 4 |1 + w j | 4 da j db j , in which case 1≤j<k≤N |λ k -λ j | 2 |λ k -λ j | 2 = 1≤j<k≤N 16 |w j + 1| 4 |w k + 1| 4 |w k -w j | 2 |w j w k -1| 2 = N j=1 2 w j |w j + 1| 2 2(N -1) 1≤j<k≤N (w k -w j ) 1 w j - 1 w k w k - 1 w j w j - 1 w k . Also (λ j -λ j ) = 2 i w j |w j + 1| 2 w j - 1 w j .
Substituting into ( 22) and simplifying we have

Q(w)(dw) = (-1) N (N -1)/2 (π) -N Γ(N + 1) N j=1 Γ(2N + 2) (Γ(2j)) 2 × N j=1 1 |w j | 2 w j w j N -1/2 1 |w j | -|w j | |w j | + 1 |w j | 2N +2 ∆ w, 1 w (dw), ( 23
)
from which (9) follows, where ∆(x) is the standard Vandermonde product. (Note that the factor of (i) -1 in ( 9) is an artifact of splitting the product 1/|w j | -|w j | into the factors of τ (w j ) and τ (1/w j ). It was for this splitting that we ordered the complex conjugate pairs in [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF].)

Generalized partition function

We will now find a Pfaffian expression for the generalized partition function

Z N [v] := N j=1 v(w j ) Q = D v(w 1 ) (dw 1 ) • • • D v(w N ) (dw N ) Q(w), ( 24
)
where D is the unit disk, with which we shall calculate the correlation functions in Section 6. Using the Vandermonde identity we have ∆ w,

1 w = det   [p k-1 (w j )] j=1,...,N p k-1 ( 1 w j ) j=1,...,N   k=1,...,2N = (-1) N (N -1)/2 det p k-1 (w j ) p k-1 ( 1 w j ) j=1,...,N k=1,...,2N
and so [START_REF] Forrester | Quantum conductance problems and the Jacobi ensemble[END_REF] becomes

Q(w) = (-1) N (N -1)/2 C N N j=1 1 i |w j | 2 τ (w j )τ 1 w j det p k-1 (w j ) p k-1 ( 1 w j ) j=1,...,N k=1,...,2N . (25)
Before proceeding we will reorder the columns of the determinant in [START_REF] Nachbin | The Haar integral[END_REF], as in [11, eqn. (30)], as follows

p 0 , p N -1 , p 2 , p N -3 , • • •, p N -2 , p 1 , (26)
introducing another factor of (-1) (N/2)(N/2-1)/2 . For labeling purposes we define

q 2j (x) = p 2j (x), q 2j+1 (x) = p N -1-2j (x).
This reordering will make finding the relevant skew-orthogonal polynomials straightforward (see Section 5). With the relabeling we have

Q(w) = C N N j=1 1 i |w j | 2 τ (w j )τ 1 w j det q k-1 (w j ) q k-1 ( 1 w j ) j=1,...,N k=1,...,2N . ( 27
)
Substituting this into [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF], expanding out the determinant using the Leibniz formula, and then ordering the terms to give a Pfaffian we obtain the following result (see [START_REF] Forrester | Log-gases and random matrices[END_REF]Chapter 6.1.2] for explicit details).

Proposition 4.1. The generalized partition function for the real quaternion spherical ensemble can be written as

Z N [v] = Γ[N + 1] C N Pf[γ j,k [v]] j,k=1,...,2N , (28)
where

γ j,k [v] := 1 i D v(w) |w| 2 τ (w)τ 1 w q j-1 (w)q k-1 1 w -q j-1 1 w q k-1 (w) (dw).

Skew-orthogonal polynomials

In this section we find the monic polynomials that will reduce the Pfaffian in (28) (with v = 1) to block diagonal form, where the diagonal blocks are the 2 × 2 matrices 0 h j -h j 0 , in which case

Z N [1] = Γ[N + 1] C N N j=1 h j . (29)
In other words, define the skew-symmetric inner product q j , q k := γ j+1,k+1 [START_REF] Akemann | The complex Laguerre symplectic ensemble of non-Hermitian matrices[END_REF] and look for polynomials to satisfy the skew-orthogonality conditions

p 2j , p 2k = p 2j+1 , p 2k+1 = 0, p 2j , p 2k+1 = -p 2k+1 , p 2j = δ j,k h j . (30)
By writing out γ j,k [START_REF] Akemann | The complex Laguerre symplectic ensemble of non-Hermitian matrices[END_REF] explicitly, we see that the only time it is non-zero is when the angular dependence cancels. With this fact, and the convenient reordering in [START_REF] Olkin | The 70th anniversary of the distribution of random matrices: a survey[END_REF], it can be seen that the skew-orthogonal polynomials are the monomials

q 2j (w) = w 2j , q 2j+1 (w) = w 2N -1-2j . (31)
With w = re iθ we substitute these polynomials into γ j,k [START_REF] Akemann | The complex Laguerre symplectic ensemble of non-Hermitian matrices[END_REF] to obtain

h j = 2π 1 0 1 -r 2 (1 + r 2 ) 2N +2 r 4j+1 -r 4N -4j-1 dr.
By changing variables c = r 2 we can make use of the identity in [15, Equation 3.216 (1)]: for general a, b such that Re b > 0, Re (a -b) > 0,

1 0 (t b-1 + t a-b-1 )(1 + t) -a dt = Γ(b)Γ(a -b) Γ(a) ,
and so, with b = y, a -b = x, we have a non-standard form of the beta integral

1 0 t x-1 (1 -t) y-1 dt = Γ(x)Γ(y) Γ(x + y) .
This gives

h j = π(2N -4j -1) (2N + 1)(2N ) 2N -1 2j -1 , j = 0, ..., N -1,
and, from [START_REF] Rains | Correlation functions for symmetrized increasing subsequences[END_REF],

N -1 j=0 h j = 1 C N Γ(N + 1) . ( 32
)

Correlation functions

To calculate the n-point correlation functions we will use the functional differentiation formula

ρ (n) (w) = 1 Z N [v] δ n δv(w 1 ) • • • δv(w n ) Z N [v] v=1 . (33)
Indeed, by the skew-orthogonality [START_REF] Selberg | Bemerkninger om et multipelt integral[END_REF], it is straightforward to calculate the 1-pt correlation function (the eigenvalue density) by applying ( 33) to [START_REF] Rains | Correlation functions for symmetrized increasing subsequences[END_REF],

ρ (1) (w) = 1 Z N [1]    N j=1 ∂ ∂v(w) γ 2j-1,2j [v] N -1 k=0 k =j-1 h k    ,
which reduces to [START_REF] Forrester | Log-gases and random matrices[END_REF] by making use of ( 29) and [START_REF] Tao | Random matrices: universality of ESDs and the circular law[END_REF].

For the general correlations, we make use of the formula det(1+AB) = det(1+BA), or its quaternion determinant analogue qdet(1 + AB) = qdet(1 + BA) (for BA self-dual)or an equivalent Pfaffian formula, see [START_REF] Rains | Correlation functions for symmetrized increasing subsequences[END_REF] -in conjunction with the Fredholm quaternion determinant

qdet[1 + λK] := 1 + ∞ s=1 λ s s! ∞ -∞ dx 1 • • • ∞ -∞ dx s qdet[K(x j , x k )] j,k=1,...,s ,
where K is an integral operator with 2×2 kernel K(x, y), and the matrix [K(x j , x k )] j,k=1,...,s is self-dual. The details are lengthy and tedious, however it is directly analogous to the same approach for the real spherical and real Ginibre ensembles, see [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF] for a detailed exposition (the techniques therein were inspired by those in [START_REF] Borodin | The Ginibre ensemble of real random matrices and its scaling limits[END_REF] and [START_REF] Forrester | Log-gases and random matrices[END_REF]). We find that the correlation functions are of the same structure as those for the real spherical ensemble [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]. Proposition 6.1. With the eigenvalues transformed according to [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF], then using the skeworthogonal polynomials [START_REF] Tao | Random matrices: universality of local spectral statistics of non-Hermitian matrices[END_REF] the n-point eigenvalue correlation functions for the real quaternion spherical ensemble are

ρ (n) (w 1 , ..., w n ) = Pf [K N (w l , w m )] , w i ∈ D, (34)
K N (x, y) = D(x, y) S(x, y) -S(y, x) I(x, y) ,

where

D(x, y) = 1 i N -1 j=0 1 h j a 2j (x)a 2j+1 (y) -a 2j+1 (x)a 2j (y) , S(x, y) = 1 i N -1 j=0 1 h j a 2j (x)b 2j+1 (y) -a 2j+1 (x)b 2j (y) , I(x, y) = 1 i N -1 j=0 1 h j b 2j (x)b 2j+1 (y) -b 2j+1 (x)b 2j (y) , and
a j (x) = |x| -1 τ (x) q j (x), b j (x) = |x| -1 τ 1 x q j 1 x .
Substituting in the various factors we have

S(w, z) = A wz (2N + 1)(2N ) π N -1 j=0 2N -1 2j (wz) 2j+1/2-N -(wz) N -1/2-2j 2N -4j -1 = A wz (2N + 1)(2N ) π N -1 j=0 2N -1 j (wz) j+1/2-N -(wz) N -1/2-j 2N -2j -1 , (36)
where

A wz = 1 r w r z 1 rw -r w 1/2 1 rz -r z 1/2 r w + 1 rw N +1 r z + 1 rz N +1 ,
with |w| = r w and |z| = r z . The other kernel elements satisfy the relations

I(w, z) = 1 |w| 2 S 1 w , z , D(w, z) = 1 |w| 2 S w, 1 z . ( 37
)
In the following, we analyze the eigenvalue density, which includes several checks of our working, however we first rewrite S(w, z). Let

σ(γ) := N -1 j=0 2N -1 j γ j+1/2-N -γ N -1/2-j 2N -2j -1 , then γ dσ(γ) dγ = - (1 + γ) 2N -1 2γ N -1/2 ,
by anti-differentiating we have

σ(1) -σ(v) = 1 v dσ(γ) dγ dγ = - 1 2 1 v (1 + γ) 2N -1 γ N +1/2 dγ.
From the definition of σ(γ) we see that σ(1) = 0 and therefore

σ(v) = 2 2N -2 1-v 0 (1 -γ) -N -1/2 (1 -γ/2) 2N -1 dγ,
where we have changed variables γ → 1 -γ, thus giving the integral expression

S(w, z) = A wz 2 2N -1 (2N + 1)N π 1-wz 0 (1 -γ) -N -1/2 (1 -γ/2) 2N -1 dγ, (38)
which we will make use of in analyzing the asymptotic behaviour of the correlation functions.

Eigenvalue density

With n = 1 (34) reduces to ρ (1) (w) = S(w, w). We see from ( 36) and (38) that this immediately gives us the two expressions for the density in [START_REF] Forrester | Log-gases and random matrices[END_REF]. We can check that The points are the bin mid-points for a simulated radial (normalized) density of eigenvalues of 2000 instances of 500 × 500 matrices, where the eigenvalues have been transformed according to [START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF]. The noisiness of the data near r = 0 is due to the relatively small number of eigenvalues near the origin, which is a consequence of the relatively smaller area one is sampling from. The solid line is a plot of [START_REF] Forrester | Log-gases and random matrices[END_REF], scaled by 1/N . as required. For another check, in Figure 3 we find good agreement between [START_REF] Forrester | Log-gases and random matrices[END_REF] and a simulated eigenvalue density. Note the elevated density near r = 1, which we recall corresponds to the real line. We interpret this as the effect of repulsion from the real line, as discussed in the Introduction.

To analyze the behaviour for large N we first exponentiate the logarithm of the integrand in [START_REF] Forrester | Log-gases and random matrices[END_REF] giving

1-r 2 0 exp [(2N -1) log(1 -γ/2) -(N + 1/2) log(1 -γ)] dγ.
For large N this integral will be dominated by points close to the upper terminal, so we Taylor expand the exponent about 1 -r

2 (2N -1) log(1 -γ/2) -(N + 1/2) log(1 -γ) ≈ log (1 + r 2 ) 2N -1 2 2N -1 r 2N +1 + (γ -1 + r 2 ) 2N + 1 2r 2 - 2N -1 1 + r 2
and so for large N the integral in ( 10) is approximated by

(1 + r 2 ) 2N -1 2 2N -1 r 2N +1 1-r 2 0 exp (γ -1 + r 2 ) 2N + 1 2r 2 - 2N -1 1 + r 2 dγ.
Substituting this into [START_REF] Forrester | Log-gases and random matrices[END_REF] and taking N to infinity (noting that r 2 -1 < 0) yields (11).

Scaled limit

Here we aim to recover the bulk real quaternion Ginibre correlation functions of [START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF], by zooming in on the origin. Under the transformation (8) the origin in the original coordinates is mapped to the point (1, 0) and so we make the replacements

w → w = 1 + 2iW π N , W = X + iY, z → z = 1 + 2iZ π N , Z = A + iB, (39)
in (38) and look for the large N limit of

4π N S(w , z ) = 2 2N +1 (2N + 1)A w z 1-w z 0 (1 -γ) -N -1/2 (1 -γ/2) 2N -1 dγ. (40)
We have chosen the transformations (39) to make a correspondence with the results in [START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF], however we note that the density will be twice as large as in that work since here we have restricted the eigenvalues to be in the upper half plane. First rewrite A w z as

A w z = (r w r z ) N -1/2 (1 -r 2 w ) 1/2 (1 -r 2 z ) 1/2 (1 + r 2 w ) N +1 (1 + r 2 z ) N +1 ,
where

r w = 1 - 4 √ πY √ N + 4π |W | 2 N 1/2
.

Using the asymptotic behaviour

1 + x √ N N ∼ e x √ N -x 2 /2 , (41)
which holds up to terms of order 1/ √ N in the exponent, we find that r

N -1/2 w ∼ exp -2Y √ πN + 2π|W | 2 -4πY 2 (1 + r 2 w ) N +1 ∼ 2 N +1 exp 2π|W | 2 -2 √ πN Y -2πY 2 , (1 -r 2 w ) 1/2 ∼ 4 √ π Y √ N 1/2
, and so

A w z ∼ √ π (Y B) 1/2 2 2N √ N e -2π(Y 2 +B 2 ) .
Ignoring terms of order 1/N we take

v = 1 + 2iW π N -2iZ π N ≈ w z ,
and the integral in (40) becomes

-2i √ π N (W -Z) 0 (1 -γ) -N -1/2 (1 -γ/2) 2N -1 dγ ∼ -2i √ π N (W -Z) 0 1 + γ 2 4 N dγ,
since, for large N , γ is restricted to lie close to zero. Then, by changing variables γ → -2iγ/ √ N the integral approaches

√ π i √ N erf √ π(W -Z) .
Combining the previous we have the large N behaviour of (40).

Proposition 6.2. Changing to the variables w and z from (39) we find (38) has the limiting behaviour 4π

N S(w , z ) ∼ N →∞ 4π i (Y B) 1/2 e -2π(Y 2 +B 2 ) erf √ π(W -Z) , (42)
which, with W = Z, gives us the scaled limit of the density 16πY 2 e -4πY 2 1 0 e 4πY 2 u 2 du.

The scaled limit of the remaining kernel elements (D and I) can be obtained similarly, or from (42) by using formulae analogous to (37) that apply to the real quaternion Ginibre ensemble

I(w, z) = iS(w, z), D(w, z) = -iS(w, z).
The result from [START_REF] Kanzieper | Eigenvalue correlations in non-Hermitean symplectic random matrices[END_REF]

corresponding to (42) is 2π i (Y B) 1/2 e -π(|W | 2 +|Z| 2 )+π(Z 2 +W 2 ) erf √ π(W -Z) . (43)
Ignoring the imaginary parts of (43), which always cancel in Kanzieper's formulation, we can use the properties of the complex error function to see that the correlation functions for the real quaternion spherical ensemble agree with those of the corresponding Ginibre ensemble under the scaling limit implied by (39) (up to the aforementioned factor of 2). Further, from the recent results in [START_REF] Tao | Random matrices: universality of local spectral statistics of non-Hermitian matrices[END_REF], we see that in this limit, we recover the universal correlation functions for planar (iid, with first four moments finite) ensembles.

Discussion

As discussed in the introduction, a tripartite categorization of matrix ensembles into geometrically-themed universality classes was suggested in [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF] (which was also used in a similar context in [START_REF] Krishnapur | Zeros of random analytic functions[END_REF]). Each of these classes corresponds to a surface of uniform curvature: the plane, the sphere and the anti-or pseudo-sphere. The Ginibre ensembles, being ensembles of iid matrices, obey the circular law, which states that the eigenvalues are uniformly distributed on a planar disk in the limit of large matrix dimension, and so we may associate the Ginibre ensembles with the plane. Taking two of these iid matrices and forming the product Y = A -1 B, which is a matrix generalization of a Cauchy random variable, gives us spherical ensembles. The spherical law (analogously to the circular law) states that the eigenvalues have uniform density on a sphere (under stereographic projection) in the limit of large matrix dimension and so these ensembles are naturally associated with the sphere. In this work we have applied the (skew-)orthogonal polynomial method to calculate the correlation functions for the real quaternion spherical ensemble, which is the β = 4 analogue of the ensembles considered in [START_REF] Krishnapur | Zeros of random analytic functions[END_REF][START_REF] Krishnapur | From random matrices to random analytic functions[END_REF][START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] and [START_REF] Forrester | Pfaffian point process for the Gaussian real generalised eigenvalue problem[END_REF]. This completes the exploration of the matrices in the spherical universality class using the Dyson indices β = 1, 2, 4. As the spherical law was established in [START_REF] Bordenave | On the spectrum of sum and products of non-Hermitian random matrices[END_REF] by adapting the proof of the circular law in [START_REF] Tao | Random matrices: universality of ESDs and the circular law[END_REF], we suspect it is possible to adapt the proof in [START_REF] Tao | Random matrices: universality of local spectral statistics of non-Hermitian matrices[END_REF] to show that the general n-point correlation functions for the complex spherical ensemble are universal for the spherical class of matrices.

The third universality class in this categorization corresponds to the anti-sphere, which is a surface of constant negative curvature. The ensembles in this class are given by sub-blocks (or truncations) of unitary matrices. To date, truncations of complex unitary [START_REF] Życzkowski | Truncations of random unitary matrices[END_REF][START_REF] Petz | On asymptotics of large Haar distributed unitary matrices[END_REF][START_REF] Forrester | Quantum conductance problems and the Jacobi ensemble[END_REF] and real orthogonal [START_REF] Khoruzhenko | Truncations of random orthogonal matrices[END_REF] matrices have been analyzed (for a discussion of the real orthogonal case also see [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF]). Naturally enough, the next step would seem to be an analysis of a β = 4 real quaternion analogue of truncated unitary and orthogonal ensembles. Further, to conclude the program laid down in [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF], perhaps by again adapting the proof of the circular law in [START_REF] Tao | Random matrices: universality of ESDs and the circular law[END_REF], then a proof of a so-called 'anti-spherical' law may be achieved (see [START_REF] Mays | A geometrical triumvirate of real random matrices[END_REF]Chapter 7.6.2] for more on this), and, even further, perhaps using the methods of [START_REF] Tao | Random matrices: universality of local spectral statistics of non-Hermitian matrices[END_REF] find that the general correlations are universal.

1

  arXiv:1209.0888v2 [math-ph] 13 Sep 2012

Figure 2 :

 2 Figure 2: (Stereographic) eigenvalue plot for 120 independent 100 × 100 spherical matrices with real quaternion entries.

Figure 3 :

 3 Figure3: The points are the bin mid-points for a simulated radial (normalized) density of eigenvalues of 2000 instances of 500 × 500 matrices, where the eigenvalues have been transformed according to[START_REF] Feinberg | On the universality of the probability distribution of the product B -1 X of random matrices[END_REF]. The noisiness of the data near r = 0 is due to the relatively small number of eigenvalues near the origin, which is a consequence of the relatively smaller area one is sampling from. The solid line is a plot of[START_REF] Forrester | Log-gases and random matrices[END_REF], scaled by 1/N .

Acknowledgements

AM was partly supported by the Australian Mathematical Society (AustMS) LiftOff Fellowship during the work leading to this paper. Special thanks to Peter Forrester for reading an early draft of this work and providing many useful comments. The author would also like to thank Gernot Akemann, Tilo Wettig and Anita Ponsaing for several interesting discussions, and the following for their kind hospitality: Department of Mathematics, Vanderbilt University; Department of Mathematics, University of Geneva; Faculty of Physics, University of Bielefeld; and Faculty of Physics, University of Regensburg. Thanks to Joshua Feinberg for pointing me to an earlier work of his which is relevant to this paper.


Philippe Woloszyn
email: philippe.woloszyn@univ-rennes2.fr

-TOWARDS A VIABLE INFERENCE MODEL OF RESILIENCE DYNAMICS

Keywords: Territorial resilience, sustainable capital, inductive model, viability kernel, sustainability convergence

In this paper, we will present the last evolution of the Territorial Intelligence (TI) networking vulnerability model. To introduce it, we'll first describe a well-known late 80's model of socioeconomic crack-up, known as "Silent Weapons for Quiet Wars", constituted by three passive components as potential energy, kinetic energy, and energy dissipation. To extend this model to social and ecological sustainability pillars, we propose to present the E(Economic)-S(Social)-O(Organic) IT-collaborative model, based on the three sustainability capitals. Goal of this model is the developement system viability computation, which related "Viability theory" computational framework is able to define sustainability constraints of action politics, in order to propose requirements for resilience processing of a territorial challenge.

Résumé

Dans cet article, nous proposons une lecture de la résilience territoriale à travers le prisme du "modèle inductif de la vulnérabilité", dernière mouture du modèle collaboratif E.S.O. (Economic Social and Organic) produit au sein du réseau Intelligence Territoriale (IT). Inspiré du modèle dit du "Silent Weapon", encore appelé "modèle E" (E-model), il propose une analogie simple des éléments conceptuels utilisés dans la mécanique et l'électronique, à savoir l'énergie potentielle, l'énergie cinétique et l'énergie dissipative. En faisant l'hypothèse que ces théories mathématiques développées dans le cadre de l'étude des systèmes énergétiques ont la faculté d'être applicables à d'autres systèmes, à savoir les systèmes sociaux, économiques et écologiques du développement humain, ce modèle associe les trois capitals correspondants aux aux trois piliers de la durabilité. La théorie de la viabilité permet alors de définir l'ensemble des politiques d'action qui permettent au système de rester dans l'ensemble des contraintes de durabilité, et donc de proposer des solutions de résilience pour une situation territoriale donnée.

1.

Introduction

"Resilience" is a physical term which achieved significance in psychology [START_REF] Cyrulnik | Un merveilleux malheur[END_REF]) and then in a wide variety of fields, namely ecology, sociology and economy. As a physical analogy, resilience refers to the property of a system for its state variables to return to their equilibrium values after a disturbance. From an ecological perspective, resilience is defined as: "... the capacity of a system to absorb disturbance and still retain its basic function and structure." (Walter & Salt, 2006). With regard to social systems, concepts and characteristics of resilient communities have been developped within different perspectives: "... the intentional action to enhance the personal and collective capacity of its citizens and institutions to respond to and influence the course of social and economic change." (Centre for Community Enterprise, 2000), or: "The capacity of a system, community or society potentially exposed to hazards to adapt, by resisting or changing in order to reach and maintain an acceptable level of functioning and structure." (United Nations, 2004). Through developping a recovering system where the combination of social, ecological and economical experiencing increases stress as a result of unpredictable change in environment, resilience describes individual and social systems property to face a catastrophic change by resisting, adapting or transforming [START_REF] Walker | Resilience, adaptability and transformability in social-ecological systems[END_REF][START_REF] Holling | Understanding the complexity of economic, ecological and social systems[END_REF][START_REF] Scheffer | Catastrophic regime shifts in ecosystems: linking theory to observation[END_REF]. In other words, community resilience describes the ability and capacity to deal with and adapt to changing conditions and continue to develop after a shock. Thus, resilience unveils the ability of the social system or community to increase its capacity for learning from past disasters for better future protection. For this to happen, governments, private industries and organisations, formal and informal structures, and individuals all have a role to play in the process, by identifying, developing and retaining the essential ingredients of resilience, in terms of sustainable development capitals.

Resilience and development capitals

Commonly, the three kinds of development capitals can be declined as:

-social-cultural capital. Pierre Bourdieu defines Social Capital as « the sum of resources, actual or virtual, that accrue to an individual or a group by virtue of possessing a durable network of more or less institutionalized relationships of mutual acquaintance and recognition » [START_REF] Bourdieu | Invitation to Reflexive Sociology[END_REF]. In this acceptance, social capital shares a productive dimension with the economical system : « Social capital is defined by its function. Like other forms of capital, social capital is productive, making possible the achievement of certain ends that would not be attainable in its absence » [START_REF]Foundations of Social Theory[END_REF]. In this way, social capital manifestation includes norms and values which facilitate exchanges, reduce information and transaction cost, permits trade in the absence of contracts, encourage citizenship responsibility, and, therefore, the collective management of resources [START_REF] Fukuyama | Trust: the social virtues and the creation of prosperity[END_REF]. This notion incorporates social shared values as knowledge, skills, attachments, cultural knowledge, education, training, co-operation networks, shared trusts and values of the population. Social capital is thus treated as a mediating variable, shaped by public and private institutions, which strategy combination has important impacts of development outcomes. It can also be a powertrade for cooperative action, through policies participation or collaborative intelligence at a territorial level: « social capital refers to features of social organization, such as trust, norms, and networks, that can improve the efficiency of society by facilitating coordinated actions » [START_REF] Putnam | Making Democracy Work[END_REF]).

-economical capital. It is sometimes separately considered as financial capital, which refers to the funds that are available to individuals and groups in a community. This form of capital can generate a flow of income to support the holder's immediate wellbeing, such as money, machinery and infrastructure providing physical assets of businesses and households, as well as public or community physical infrastructures. Community wellbeing is then produced by converting stocks ("inputs") to "outputs" such as flows of services; economical capital can also be converted into some other form of capital contributing to community wellbeing.

-ecological (organic) capital. This recovers goods and services items such as exploitation or transformation of natural resources, specific land-use and settlement patterns, biodiversity management, nature valuation, landscape preservation, sustainable farming and food production. Hence, the ecological stock represents objectified and accumulated labour [START_REF] Bourdieu | Reproduction in Education[END_REF] and context-related knowledge describes the interrelations between natural resources and amenity production: in one word, effective adjustments in land-use to reconnect nature with society [START_REF] Swagemakers | Ecologisch kapitaal. Over het belang van aanpassingsvermogen, flexibiliteit en oor-deelkundigheid (Ecological capital. On the importance of adaptability, flexibility and power of judgment)[END_REF]. As neo-classical economics stated a separate organisation from environment and a total freedom from biophysical constraints, economical think-tank interested with natural resources and environment introduced ecological (organic) capital notion to refeer to the limited natural resources stocks, through considering economy as an open-growing totally dependant subsystem of a closed, non-growing finite ecosphere system. Thus, ecological capital dimensioning introduce a highly-ordered dynamic system governed by the second law of thermodynamics, its entropy being directly in-and out-putting to ecosphere energy/matter equilibrium [START_REF] Rees | Economic Development and Environmental Protection : an Ecological Economics perspective[END_REF].

Stocks and flows modelling

Those three kinds of capital, when combined, generate a wide range of "outputs", or wellbeing attributes, that are important in terms of community valuation. As a complex combination between economic, environmental and social life values, outputs that people may need or seek in order to maintain their wellbeing involve complex forms of capital that are the result of joined eco/socio/environmental inputs. So, communities can be considered as complex systems that provide economic, social-cultural, and environmental goods and services that contribute to societal amenities, produced by converting stocks of various forms of capital ("inputs") into flows of services, or common amenity attributes ("outputs").

Therefore, a community resilience hypothesis can be defined as a function of the capability to extent its available capitals in the three (social/ economical/ environmental) dimensions of sustainable development. A decline or absence in stocks of one type of capital may signal a resilience decrease. Alternatively, large or growing stocks of capital may act as a buffer against forces that test a community's ability to cope with change. Some forms of capital might be emerging while others become obsolete, so that equibrium vector between those social, economic, environmental capital dimensions is important for helping to determine practices and policies for the decision making and political directions to follow sustainability laws.

To enable good decision-making practices, sustainablility modelling should therefore evaluate these previous capitals circulation structure, through stock to flow transformation characterization during production/ consuption/ exchange processes between economy, society and environment.

In the study of energy systems, the two first elementary concepts, potential energy and energy dissipation constitute the physical analogical counterparts of stock and flow generation into development process. Moreover, stocks of a community's capital ability to generate flows of community wellbeing is therefore modelled as an induction process of the community system, which physical counterpart corresponds to the inertial concept of energy systems, kinetic energy. Those physical properties are known as "passive components" of the corresponding mechanical/electronical systems. This analogy was first inspired by a well-known late 80's model of socio-economic crack-up, known as "Silent Weapons for Quiet Wars", which presents economy as a social extension of natural energy systems. This last, also named "E-model", is constituted by the three passive components, potential energy, kinetic energy, and energy dissipation, thus allowing economical data to be treated as a thermodynamical system [START_REF] Cooper | Behold a pale horse[END_REF]. To extend this model to social and ecological sustainability pillars, we propose to built an extended E(Economic)-S(Social)-O(Organic) model, based on those three components of sustainable development (Woloszyn & alii., 2012), as seen figure 1:

The E S O local/global transition paradigmatic model

In the science of physical mechanics, potential energy is associated with a physical property called elasticity or stiffness, and can be therefore represented by a stretched spring. In electronic science, potential energy is stored in a capacitor instead of a spring, which property is called capacitance. Second passive component, kinetic energy, is associated with a physical property called inertia or mass, and can be represented by a flywheel in motion. In electronic science, kinetic energy is stored in an inductor, a magnetic field, which property is called inductance. Endly, energy dissipation is associated with a physical property called friction or resistance, thus converting energy into heat.

As E-Model assumes that economics can be considered as social extension of a natural energy system, also constituted by its three passive components, Social and Ecological pillars could therefore be defined as subsystems based on those three previous passive components understood as capitals dynamics, thus constituting the global open sustainability model.

By converting capital stocks into service flows in the 3 dimensions of sustainable development, stocks are understood as constitutive of the "capacitance" property of the system, as "flows" constitute its conductance, leading to capital stocks growing or declining. This system evolution describes the "inductive" effect of the (stock/flow) conversion process. As a consequence, association of potential, dissipative and kinetic energy concepts with the three pillars of sustainable development leads us to define the three dynamical notions of a general theory of social-eco-environmental entropy, also called "generalized social energy": capacitance, conductance and inductance. Thus, sustainable co-evolution of environmentalized systems answer to those complementary potential, dissipative and cinetic processes constitute the ESO IT collaborative model [START_REF] Dumas | Modèle participatif d'évaluation de la résilience[END_REF], involving economic inductance, social inductance and ecologic inductance as illustrated figure 2: Economic inductance, constitutive of the " E-Model Silent Weapon", can be understood as an analogy with an electrical inductor: this last has an electric current as its primary phenomenon and a magnetic field as its secondary phenomenon. Analogic view of this process describes an economic inductor as a complex flow driving, constituted by economic value as its primary phenomenon and population behavior as its secondary field phenomenon of inertia. Social Inductance (S-Model) considers social implying cooperation action as primary phenomenon and environmental integration process as secondary field phenomenon. Last but not least, ecologic/organic Inductance (O-Model) implies ecological management as primary phenomenon and ecosystemic interactions between air (athmosphere), water (hydrosphere), ground (litosphere) and life (biosphere), or environmental benefits/losts, as secondary inertial field phenomenon.

A further description of those passive components is given following figure 3: Thus, this analogy identifies the "stock" as the capacitive property of the system to maintain or develop its capital, the "flow" as the conductance processes of capital production, and the "benefit" (or loss) of the system driving as the inductance effect of stock to flow capital recovering. During the "flows" motions, inductance supplies to both capacitance and conductance, with taking into account the systems transformation potential through their temporal activities. This paradigmatic change of the process leads us to consider a thermodynamic approach of open-living systems, instead of mechanical-closed systems: stability key of those lasts closed systems, retro-action, is here considered as the open-systems evolution source.

When a community's capital stocks is growing, its capacity to generate flows of goods and services will also grow, thus enabling the community's ability to improve its wellbeing in the future, or consolidate its resilience capability. As a consequence, induction of the socio-ecoenvironmental system depends on the community's capital stocks capacity to generate flows of community wellbeing. This can be designed as a non-linear retro-active effect of community flow generation. As an illustration of this retroaction process, [START_REF] Hicks | The foundations of welfare economics[END_REF] stated that maintenance of the wellbeing of a community was a function of income flow maintenance, the conductance process, from a done stock of capital, the community capacitance. The question of defining the stocks of a community's capital that generate enough flows for community wellbeing provides here an econologic view of resilience: a rise (or fall) in a community's wellbeing can be explained by increases (or decreases) in its stocks of capital. In case of disfunction between capacitance level and conductance process, inductance should exceed capacitance, and a leading power factor is produced, driving community to uncontrolled economic/social flows driving. This induced effect generared the 2008's crisis, due tu excessive financial flows resulting from american subprimes system, which has led to a "savage" inductance process, traduced by financial crash & social crisis. Harmonic problems of eco/socio/environmental inductive interactions have then emerged, so that the dissipative system disrupted its equilibrium between the sustainable dimensions of the development process. Induction process has been also illustrated in 2012 U.S. drought event [START_REF] Woloszyn | Territorial intelligence of vulnerability systems 2-Sustainable Modelling of Globalization Challenge[END_REF] as well as for indian population (Woloszyn & al. 2013) or french Coast natural hazard [START_REF] Woloszyn | Vulnérabilité territoriale et résiliences : résistances et capacités adaptatives face aux aléas climatiques[END_REF]) resiliences.

Routes to viability

To achieve economical/sociocultural/ecological coefficients valuation, approach of this "Triple bottom line" structure of sustainability, understood here as three inter-related systems, imply informational dimensioning of: (1): Economical macro and microeconomic dynamics (Akerlof 2002, Stiglitz and[START_REF] Rees | Ecological footprints and appropriated carrying capacity: what urban economics leaves out[END_REF]), (2): Population-wealth distribution (Wolff 2007, Davies andalii., 2007), and (3): Ecological human footprints [START_REF] Ayres | Commentary on the utility of the Ecological Footprint concept[END_REF][START_REF] Costanza | The dynamics of the Ecological Footprint concept[END_REF][START_REF] Rees | Ecological footprints and appropriated carrying capacity: what urban economics leaves out[END_REF]. Each of those described subsystems, feeded (inputs) and feeded by (outputs) the two complementary sustainability pillars should enable to solve a viability criticity matrix of socio-ecological transition process. As System State measurement provides the mentioned coefficients in terms of vulnerability levels in the multidimensional (economical/sociocultural/ecological) evaluation system. Thus, action driving maintaining the system into sustainability limits, responding to E.S.O. criticity conditions, can be provided through a specific "Viability kernel" computational framework. This framework fosters operational definitions of resilience, vulnerability and adaptive capacity, thus helping to understand which response has to be brought to face environmental changes.

Assessing resilience within an economical-ecological-social system is becoming a challenge in the context of sustainable development. In this context, the mathematical theory of viability enable to study the compatibility of sustainability dynamical systems with "desirable" constraints (Aubin et al., 2011), defined by viability studies to provide efficient territorial control policies. When the system evolves outside the desirable constraint set, viability theory can assess whether and how the system can be driven back to desirable states.

Conclusion

The ability to preserve territorial identity in a changing environment is a indication of viability. Also a resilient system must have the ability to anticipate, perceive and respond in an environment of scarcity and pressure.

As a diagnosis key of resilience process, inductive requirements could be finded into politics and action skills. (Hollnagel, Woods & Leveson, 2006) claimed that this approach defines: "Resilience as a form of control: ... In order to be in control it is necessary to know what has happened (the past), what happens (the present) and what may happen (the future), as well as knowing what to do and having the required resources to do it...". Processes control, including ability to absorb perturbations and adaptation appears to be a way to understand resilience and viability. Such a resilient and viable system, able to resume a steady state after it has been disturbed in a way not envisioned by its designer, have been described by [START_REF] Ashby | An introduction to cybernetics[END_REF] as an ultra-stable homeostatic system.

Thus, this viability framework gives indication on the possible consequences of a hazard or an environmental change, but also on the policies that can cope with the situation. Adaptive capacity as the reduction in a given vulnerability statistic due to the introduction of new policy options should then be achieved by the acknowledgement of the inductive process within the viability kernel decision space. Thus, Achieving sustainability goals through effective economic, social and ecological reforms as a route to the sustainability transition may propose societal anticipation and socio-cultural response to environmental constraints and shocks. To respond to this requirement, mathematical framework of viability theory fosters operational definitions of resilience, vulnerability and adaptive capacity, to help understanding which response one should bring to environmental changes. But this wil be part of another paper.

Fig 1 :

 1 Fig 1: Articulation and convergences between sustainable development capitals and ESO model extension.

Fig 2 :

 2 Fig 2: Open-analogical model of sustainability and inertial inductance behaviour

Fig 3 :

 3 Fig 3: The three passive components of the sustainability pillars

Chargé de Recherche CNRS, ESO -UMR 6590 CNRS/Université Rennes2, Laboratoire Espaces et Sociétés Rennes, Université de Haute Bretagne, Maison de la Recherche en Sciences Sociales, 35043 Rennes, France.