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RANDOM MATRIX ENSEMBLES
ASSOCIATED TO COMPACT SYMMETRIC SPACES
EDUARDO DUEN˜EZ
Abstract. We introduce random matrix ensembles that corre-
spond to the infinite families of irreducible Riemannian symmetric
spaces of type I. In particular, we recover the Circular Orthogo-
nal and Symplectic Ensembles of Dyson, and find other families
of (unitary, orthogonal and symplectic) ensembles of Jacobi type.
We discuss the universal and weakly universal features of the global
and local correlations of the levels in the bulk and at the “hard”
edge of the spectrum (i. e., at the “central points” ±1 on the unit
circle). Previously known results are extended, and we find new
simple formulas for the Bessel Kernels that describe the local cor-
relations at a hard edge.
1. Introduction
Local correlations between eigenvalues of various ensembles of ran-
dom unitary, orthogonal or symplectic matrices, in the limit when their
size tends to infinity, are known to exhibit universal behavior in the
bulk of the spectrum. Dyson’s “Threefold Way” [14] predicts that
this behavior is to be expected universally in the bulk of the spec-
trum, depending only on the symmetry type of the ensemble (unitary,
orthogonal or symplectic). Unfortunately, for general ensembles this
conjecture remains open, though in the unitary case (modeled after
the Gaussian Unitary Ensemble) the universality of the local corre-
lations has been proven for some classes of families [9, 7, 3, 2]. In
the orthogonal and symplectic cases the extension of results known for
Gaussian ensembles is technically more complicated but some more re-
cent work deals with families of such ensembles [26]. Most of the focus
has been on non-compact (Gaussian and the like) matrix ensembles.
In the present article we study families of compact (circular) ensem-
bles including, in particular, Dyson’s circular ensembles: the COE,
CUE and CSE [11]. First we fit Dyson’s ensembles into the framework
I wish to thank Prof. Peter Sarnak for his continued encouragement and guidance
as my Ph. D. thesis advisor as well as Brian Conrey for making my stay at AIM
possible. This research has been supported in part by the FRG grant DMS–00–
74028 from the NSF.
1

2 EDUARDO DUEN˜EZ
of the theory of symmetric spaces, and then we proceed to associate
a matrix ensemble to every family of irreducible compact symmetric
space (all of these are known by the work of Cartan [4, 5]). The most
well-known of these are the families of classical orthogonal, unitary and
symplectic groups of matrices, for which questions about universality
have known answers [17]. These are the so-called compact symmetric
spaces of type II. Zirnbauer [30], on the other hand, has constructed
the “infinitesimal” versions of the other (type I) ensembles, namely
their tangent spaces at the identity element, which is enough to derive
their eigenvalue measures. We, however, construct the “global” ensem-
bles associated to the infinite families of compact symmetric spaces of
type I in a very explicit manner analogous to Dyson’s description of
his circular ensembles.
Type G/K Parameters
A I
(COE) U(R)/O(R) β = 1 (not Jacobi)
A II
(CSE) U(2R)/USp(2R) β = 4 (not Jacobi)
A III U(2R + L)/U(R + L)× U(R) β = 2, (a, b) = (L, 0)
BD I O(2R+ L)/O(R+ L)×O(R) β = 1, (a, b) = (L−12 ,−12)
SO(4R)/U(2R) β = 4, (a, b) = (0, 0)
D III SO(4R+ 2)/U(2R + 1) β = 4, (a, b) = (2, 0)
C I USp(2R)/U(R) β = 1, (a, b) = (0, 0)
C II
USp(4R + 2L)/
USp(2R + 2L)× USp(2R) β = 4, (a, b) = (2L+ 1, 1)
Table 1. Parameters of the probability measure of the
eigenvalues for ensembles of type I.
Besides Dyson’s COE and CSE, the other compact matrix ensembles
of type I are Jacobi ensembles in the sense that their joint eigenvalue
measure is given by
(1)
dν(x1, . . . , xR) ∝
∏
1≤j<k≤R
|xj−xk|β
R
∏
j=1
(1−xj)a(1+xj)bdxj on [−1, 1]R
for some parameters a, b > −1 (depending on the ensemble, see ta-
ble 1) and β = 1, 2, 4 (the “symmetry parameter”) in the orthogonal,
unitary and symplectic cases, respectively. Here, the “free” eigenval-
ues are xj ±
√
−1yj —excluding eigenvalues equal to +1 forced by the
symmetry built into the ensemble—. Also, R stands for the rank of the

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 3
corresponding symmetric space, and our interest is in the semiclassical
limit of the eigenvalue statistics as R→ ∞ (L ≥ 0 is a fixed parameter:
different values of L yield different ensembles.) The name “Jacobi en-
sembles” comes from the intimate connection between the measure (1)
and the classical Jacobi polynomials on the interval [−1, 1].
Afterwards, we prove the universality of the local correlations for
general unitary, orthogonal and symplectic Jacobi ensembles (previous
results of Nagao and Forrester [23] are insufficient for our purposes).
We rely on work of Adler et al [1]. At the “hard edges” ±1 of the inter-
val, Dyson’s universality breaks down and we obtain simple formulas
for the Bessel kernel in terms of which the hard edge correlations are
expressed. In a nutshell, for Jacobi ensembles:
• Away from the “hard edge” x = ±1, the local correlations follow
the universal law of the GOE (β = 1), GUE (β = 2) or GSE
(β = 4). Namely, in terms of local parameters ξj around a fixed
zo ∈ (−1, 1) so that xj = cos(αo +(π/R)ξj) (zo = cosαo), these
local correlations are given by
(2) L(n)β (zo; ξ1, . . . , ξn) = DET(K¯β(ξj, ξk))n×n,
where DET stands for either the usual (β = 2) or quaternion
(β = 1, 4) determinant, and Kβ is the (scalar or quaternion)
Sine kernel (cf., equations (74)–(78).)
• At the hard edge zo = +1, the local correlations depend on the
parameter a of the Jacobi ensemble as well as on β. In terms
of local parameters ξj > 0 with xj = cos((π/R)ξj) the same
expression (2) holds except that the kernel K¯β is to be replaced
by a Bessel kernel Kˆ(a)β (ξ, η) given by equations (80)–(86). At
the hard edge zo = −1 the result is obtained by replacing a
by b.
2. Dyson’s Circular Ensembles as Symmetric Spaces
For motivational purposes we start by reviewing the construction of
the circular ensembles of Dyson and their probability measures of the
eigenvalues in a manner in which the theory of Riemannian symmetric
spaces is brought into play.
The Circular Unitary Ensemble (CUE) is the set S = S(N) of allN×
N unitary matrices H , endowed with the unique probability measure
dµ(H) that is invariant under left (also right) multiplication by any
unitary matrix. This requirement makes the measure invariant under
unitary changes of bases, hence the ensemble’s name.

4 EDUARDO DUEN˜EZ
In the study of statistics of eigenvalues, the relevant probability mea-
sure is the one induced by dµ(H) on the torus A = A(N) ⊂ S(N)
consisting of unitary diagonal matrices
(3) A = {diag(λ1 = eiθ1 , . . . , λN = eiθN )},
where Θ = (θ1, . . . , θN ) ∈ [0, 2π)N , say.
To be more precise, let us denote by K = K(N) the unitary group
of N × N matrices (its underlying set is just S(N)). Then we have a
surjective mapping
K ×A ։ S
(k, a) 7→ H = kak−1,(4)
and correspondingly there exists a probability measure dν(a) on A such
that, for any continuous function f ∈ C(S),
(5)
∫
S
f(H)dµ(H) =
∫
K
∫
A
f(kak−1)dν(a)dHaar(k),
where we denote by dHaar(k) the unique translation-invariant proba-
bility measure on K (so here dHaar = dµ). This measure dν(a) can be
pulled back to some measure on the space [0, 2π)N of angles Θ which,
abusing notation, we denote by dν(Θ). The measure dν(Λ) (or dν(Θ))
is the so-called probability measure of the eigenvalues (for the CUE).
We have [21],
(6) dν(Θ) ∝
∣
∣Van(eiΘ)
∣
∣
2 dΘ on [0, 2π)N .
Here the symbol “∝” stands for proportionality up to a constant (de-
pending only on N), dΘ = dθ1 . . . dθN is the usual translation-invariant
measure on the space of angles Θ, eiΘ = (eiθ1 , . . . , eiθN ) and, for a vector
x = (x1, . . . , xN), Van(x) is the Vandermonde determinant
(7) Van(x) = det
N×N
(xk−1j ) =
∏
1≤j<k≤N
(xk − xj).
The construction of the Circular Orthogonal Ensemble (COE) is as
follows. One starts with the set S = S(N) of N×N symmetric unitary
matrices H . However, because S(N) is not a group, the choice of the
probability measure dµ(H) is not as obvious as it was for the CUE.
Let G = G(N) again be the group of N × N unitary matrices g,
and K = K(N) ⊂ G(N) be the group of orthogonal matrices. Let
Ω(g) = (gT )−1 be the involution of G whose fixed-point set is K. Then
we may identify
G/K ≃ S
G ∋ g 7→ H = gΩ(g)−1 =: g1−Ω,(8)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 5
and by general principles the translation-invariant probability measures
on G and K determine a unique G-invariant measure dµ(g¯) = dµ(H)
on G/K ≃ S which satisfies
(9)
∫
G
f(g)dHaar(g) =
∫
G/K
(
∫
K
f(gk)dHaar(k)
)
dµ(g¯),
where on the right-hand side g stands for a choice of an element g ∈ G
such that gK = g¯. The left translation-invariance of dHaar(g) ensures
that dµ(g¯) is invariant under left translations by elements of K, there-
fore the measure dµ(H) is invariant under orthogonal changes of bases,
hence the ensemble’s name.
The probability measure of eigenvalues dν(a) = dν(Θ) is again that
which satisfies (5) (with the same torus A ⊂ S as for the CUE). It is
known that [11]
(10) dν(Θ) ∝ |Van(eiΘ)|dΘ on [0, 2π)N .
The constructions of the Circular Symplectic Ensemble CSE and
of its measure on eigenvalues dν(Θ) are very similar to the case of
the COE. Here S(N) consists of 2N × 2N self-dual unitary matrices.
Namely, letting
(11) J = JN =
(
−IN
IN
)
,
then a matrix H is self-dual if it equals its dual HD := JHTJT . If
we let G = G(N) be the group of 2N × 2N unitary matrices and
K = K(N) be the subgroup of symplectic matrices k (they satisfy
kJkT = J) then K is the fixed-point set of the involution Ω(g) =
(gD)−1. The identification (8) continues to hold and (9) again defines
the probability measure dµ(H) = dµ(g¯) of the ensemble. It is invariant
under symplectic changes of bases.
The torus A consists here of diagonal matrices:
(12) A = {diag(eiθ1 , . . . , eiθN , eiθ1 , . . . , eiθN )}
with twice-repeated eigenvalues. Then the probability measure of the
eigenvalues is characterized by (5), and indeed
(13) dν(Θ) ∝ |Van(eiΘ)|4dΘ on [0, 2π)N .
Summing up, the measure on eigenvalues for the circular ensembles
is given by
(14) dν(Θ) ∝ |Van(eiΘ)|βdΘ,
where β = 1, 2, 4 in the orthogonal, unitary and symplectic cases, re-
spectively.

6 EDUARDO DUEN˜EZ
Remark. It can be appreciated that the parameter β determines the
strength of the repulsion between nearby eigenvalues: this repulsion is
stronger the larger β is. Hence anything that measures the local inter-
actions between eigenvalues is likely to depend on β. This is the case, in
particular, of the “local correlations” between eigenvalues, cf. section 4.
Remark. The apparent dissimilarity in the construction of the measure
dµ(H) in the case of the unitary vs. the orthogonal and symplectic
ensembles is not essential. In fact, the unitary ensemble S(N) is still
a quotient G(N)/K(N) where G(N) = U(N) × U(N) is the direct
product of two copies of the unitary group, andK(N) is the diagonal of
G(N) (isomorphic to the unitary group itself). If we identify S(N) with
the “anti-diagonal” {H = (g, g−1)} ⊂ G(N) and take Ω(g, h) = (h, g)
then the construction of the ensemble and of the measures dµ(H) and
dν(Θ) follows through in essentially the same manner. We omit the
details. The key observation is that the constructions above show that
the circular ensembles are examples of Riemannian globally symmetric
spaces.
3. Compact Symmetric Spaces as Matrix Ensembles
Any Riemannian globally symmetric space X is locally isometric to a
product of irreducible ones (the symbol “≈” means “is locally isometric
to”):
(15) X ≈
∏
i
X(c)i ×
∏
j
X(nc)j ×Eℓ,
where the X(c)i (resp., the X
(nc)
j ) are irreducible symmetric spaces of
compact (resp., non-compact) type, and Eℓ = (E1)ℓ is ℓ-dimensional
Euclidean space (a flat manifold). In the case of the circular ensembles,
we have
CUE = U(N) ≈ SU(N)× S1
COE = U(N)/O(N) ≈ (SU(N)/SO(N))× S1
CSE = U(2N)/USp(2N) ≈ (SU(2N)/USp(2N))× S1(16)
where in each case the first factor is an irreducible symmetric space of
the compact type and the other (Euclidean) factor is a circle S1 ≈ E1
(we write S1 rather than E1 to emphasize that the spaces are compact).
In the language of differential geometry, the probability measure of a
circular ensemble is the one determined by the natural volume ele-
ment of the manifold. Hence the natural question arises as to how to
construct a random matrix ensemble corresponding to each (infinite)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 7
family of irreducible symmetric spaces of compact type. The restric-
tion to infinite families is due to the need to have a large parameter
N such that the number of eigenvalues grows with N , and then we are
interested mainly in limiting statistics.
The presence of the Euclidean factor S1 (which comes from the subset
of scalar multiples of the identity matrix within the ensemble) is rather
convenient and natural. If we were to define “irreducible” circular en-
sembles analogously to Dyson’s circular ensembles, except requiring
that they consist of matrices with unit determinant, then the spaces so
obtained would be irreducible symmetric spaces of the compact type
(i. e., the factors S1 would disappear from (16)). However, the mea-
sure on eigenvalues would no longer be translationally invariant (under
transformations of the form Θ 7→ Θ+(t, . . . , t)). Namely, instead of the
measure (14), we would obtain an asymmetric version given by the same
formula but with Θ replaced by Θ = (θ1, . . . , θN−1,−θ1−· · ·−θN−1) and
with dθN omitted from the volume element dΘ. As may be expected
from such a loss of symmetry, a rigorous analysis of these “irreducible”
ensembles would be more involved.
Since we are considering only compact symmetric spaces, it is pos-
sible to normalize the natural volume element to obtain a probability
measure. This is not the case for symmetric spaces of non-compact
type. To clarify the difference, we analyze the example of the classical
Gaussian matrix ensembles, which also fit within the framework of the
theory of symmetric spaces (the construction is analogous to that of
the circular ensembles):
GUE ≈ SL(N,C)/SU(N)× E1
GOE ≈ SL(N,R)/SO(N)× E1
GSE ≈ SU∗(2N)/USp(2N)× E1.
(17)
Finding the probability measure on eigenvalues also reduces to a factor-
ization of measures dµ(H) = dHaar(k)dν(a) in the sense of (5), where
K is still the group of invariance (orthogonal, unitary, symplectic) of
the ensemble’s measure, but where A ≃ EN is now a Euclidean space,
which in the case of these ensembles consists of real diagonal matrices
which can be parametrized by N -tuples Λ = (λ1, . . . , λN) of real num-
bers. However, the measure dµ(H) is certainly not the one obtained
from the Riemannian volume element dHaar(g) of G through (10) since
the latter is not normalizable. A choice has to be made to make
this measure into a finite one while preserving its left and right K-
invariance. One possibility is provided by a “Gaussian” probability

8 EDUARDO DUEN˜EZ
measure on G proportional to
(18) e−β2 tr g2dHaar(g)
(the symmetry parameter β = 1, 2, 4 corresponds to the orthogonal,
unitary and symplectic cases, respectively, just as in the case of the
Orthogonal ensembles), which in turn yields the measure on eigenval-
ues:
(19) dν(a) ∝ e−β
∑
λ2j |Van(Λ)|βdΛ.
It can be rightfully argued that the choice of the Gaussian normaliza-
tion for the measure on these matrix ensembles is rather arbitrary and
motivated by analytical rather than conceptual considerations. The
point we wish to state here is that making such a choice is unavoid-
able. For the compact spaces, however, no such choice needs to be
made since their volume element already determines a unique proba-
bility measure. We will henceforth restrict our attention to compact
ensembles for that reason.
The general definition of a Riemannian symmetric space of the com-
pact type is as follows. We start with a compact semisimple Lie algebra
g (i. e., exp(ad(g)) ⊂ GL(g) is compact) having an involutive automor-
phism ω. Then g splits into the sum of the (+1)- and (−1)-eigenspaces
of ω as
(20) g = k ⊕ p.
(the subspace p ⊂ g can be identified with the tangent space to G/K at
the identity coset o = K/K). G/K is called a Riemannian symmetric
space of the compact type if
(1) K ⊂ G are Lie groups (G connected). Their Lie algebras are
k, g; and
(2) there is a (necessarily unique) involutive automorphism Ω of
G such that (GΩ)o ⊂ K ⊂ GΩ, where GΩ is the fixed-point
set of Ω in G (a Lie subgroup of G) and (GΩ)o is its identity
component (then dΩe = ω).
The complete list of irreducible symmetric spaces (up to local isom-
etry) is known by the classical work of Cartan. As we will explain
later, it suffices to consider one matrix ensemble in each equivalence
class of locally isometric symmetric spaces, because the measures on
eigenvalues for locally isometric ensembles are the same.
The irreducible symmetric spaces of compact type are classified into
spaces of “Type I” and “Type II”. Of these the latter are simplest
to describe: they are the (connected) simple compact Lie groups G,
provided with a bi-invariant (under both left and right translations)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 9
Riemannian metric. Proving that such a G is a bona fide symmetric
space of the compact type as defined before involves expressing it as
(G×G)/G in a manner analogous to what we did at the end of section 2
for the CUE.
Type G/K Rank R
A I SU(N)/SO(N) N − 1
A II SU(2N)/USp(2N) N − 1
A III SU(M +N)/S(U(M)× U(N)) min(M,N)
BD I SO(M +N)/SO(M)× SO(N) min(M,N)
D III SO(2N)/U(N) ⌊N/2⌋
C I USp(2N)/U(N) N
C II USp(2M + 2N)/USp(2M)× USp(2N) min(M,N)
Table 2. The infinite families of symmetric spaces of type I.
Up to local isometry, the infinite families of Type II spaces are
those of orthogonal SO(N), unitary SU(N) and (compact) symplec-
tic USp(2N) groups. The random matrix theory of these spaces is
well-known [17].
The Type I spaces, on the other hand, are those symmetric spaces
G/K of the compact type with G simple. The bi-invariant Riemannian
metric on G determines that on the quotient G/K. Table 2 lists the
infinite families of Type I spaces, up to local isometry.
Without loss of generality, we assume henceforth that min(M,N) =
N .
Choose a maximal abelian subalgebra a of g contained in p. Then
the subgroup A = exp(a) is a torus that projects onto a totally flat
submanifold AK/K ⊂ G/K (a flat torus). This totally flat manifold
is maximal, and its dimension is the rank R of the symmetric space
G/K. Thus, R = dim(AK/K) = dimA = dim a.
Guided by the exposition in the previous section, it is reasonable to
regard as ensembles the symmetric spaces G/K of type I endowed with
their normalized Riemannian volume elements dµ(g¯), which satisfy (9).
However, the elements of these ensembles are not matrices but rather
cosets g¯ = gK ∈ G/K.
Theorem 1. The infinite families of type I ensembles G/K can be
realized as matrix ensembles S. Indeed, (8) maps G/K bijectively onto
a submanifold S ⊂ G, and G is a classical group of matrices, hence S
is a space of matrices. Under this correspondence, AK/K ⊂ G/K is
mapped onto the torus A. The action of K on G/K by left translation
corresponds to the conjugation H 7→ kHk−1 on matrices H ∈ S, and

10 EDUARDO DUEN˜EZ
any H ∈ S is conjugate to some a ∈ A under this action. Moreover,
two matrices in A are conjugate under K if and only if they have the
same eigenvalues.
The proof of the theorem is a long exercise in elementary linear
algebra. We shall omit most of the details, which can be found in [10].
In what follows we describe the explicit matrix ensembles S which are
the images of the imbedding (8).
In each case, we choose the involution Ω of G so that its fixed-
point set is exactly K. The cases of A I (COE) and A II (CSE) have
been discussed already. We introduce some notation (recall that JN is
defined by equation (11)):
J ′N =
(
IN
IN
)
2N×2N
,(21)
JMN =
(
JM
JN
)
(2M+2N)×(2M+2N)
,(22)
J ′MN =
(
J ′M
J ′N
)
(2M+2N)×(2M+2N)
,(23)
I ′MN =
(
IM
−IN
)
(M+N)×(M+N)
.(24)
The canonical bilinear antisymmetric matrix Jn in the definition of the
compact symplectic group USp(2n) will be taken to be (11) in the case
of ensembles with one parameter N (n = N), and (22) in the case of
ensembles with two parameters M,N (n = M +N).
A III. Take M ≥ N ≥ 1 and G(M,N) = U(M + N). Then
K(M,N) = U(M) × U(N) is the fixed-point set of the involution
(25) g 7→ gΩ := I ′gI ′,
with I ′ = I ′MN as in (24).
The symmetric space U(M+N)/U(M)×U(N) = SU(M+N)/S(U(M)×
U(N)) is realized as the matrix ensemble
(26) S(M,N) := {H = GI ′ such that G ∈ U(M +N)
is Hermitian of signature (M,N)},
under the identification (8). A choice of the abelian torus A is given
by
(27) A =





1M−N
ℜΛN −ℑΛN
ℑΛN ℜΛN






MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 11
where ΛN = diag(λ1, . . . , λN) is an arbitrary diagonal unitary matrix.
Besides the eigenvalue 1 with multiplicity M − N , the eigenvalues of
the matrix in (27) come in R = N pairs λj, λ−1j , |λj| = 1.
BD I. Let M ≥ N ≥ 1, G(M,N) = O(M + N), and K(M,N) =
O(M) × O(N) be the fixed-point set of the involution (25) with I ′ =
I ′MN as in (24). Then G/K = O(M +N)/O(M) × O(N) = SO(M +
N)/S(O(M)× O(N)) ≈ SO(M +N)/SO(M)× SO(N) (the last two
spaces are locally isometric).
The symmetric space O(M + N)/O(M) × O(N) can be realized as
the set of matrices
(28) S(M,N) := {H = gI ′ such that g ∈ O(M +N)
is symmetric of signature (M,N)},
by means of (8). The torus A is just as in (27) and we get the same
description for the eigenvalues.
D III. Let G(N) = SO(2N) and K(N) = SO(2N) ∩ Sp(2N,C) ≃
U(N):
(29) U(N) ∋ g 7→
(
ℜg −ℑg
ℑg ℜg
)
∈ K(N).
Then K(N) is the fixed-point set of the involution
(30) g 7→ gΩ := JT (g−1)TJ = JTgJ
with J = JN as in (11). We can identify G(N)/K(N) with the set
(31) S(N) := {H ∈ SO(2N) s. t. HJ is “dexter” antisymmetric}
using equation (8). We now explain what we mean by a dexter matrix.
Say G is a 2N × 2N orthogonal antisymmetric matrix. Then an or-
thogonal change of basis puts it into the canonical form JN . However,
this may not be possible by means of a proper orthogonal change of
basis (i. e., of determinant +1). Specifically, when N is even, the two
complex structures ±JN are equivalent (under, say, the proper orthog-
onal change of basis J ′N as in (21)), but when N is odd they are not.
We call G dexter if, by a proper orthogonal change of basis, it can be
taken into the canonical form +JN . Thus, for N even, all orthogo-
nal antisymmetric matrices are dexter, whereas for N odd, only half
of them are (in this case, conjugation by J ′N takes “dexter” matrices
into “sinister” ones and vice-versa). Now, for H ∈ S(N), G := HJ is
dexter antisymmetric, so our discussion above proves the surjectivity
of the mapping.

12 EDUARDO DUEN˜EZ
The torus A is
(32)
A =































ℜΛR −ℑΛR
ℑΛR ℜΛR
ℜΛR ℑΛR
−ℑΛR ℜΛR




for N even;







1
ℜΛR −ℑΛR
ℑΛR ℜΛR
1
ℜΛR ℑΛR
−ℑΛR ℜΛR







for N odd.



























,
where ΛR = diag(λ1, . . . , λR) is a diagonal unitary matrix. Besides the
double eigenvalue 1, which occurs for N odd, the matrices in (32) have
R quadruples of eigenvalues λj , λj, λ−1j , λ−1j .
C I. Here G(N) = USp(2N), and K(N) ≃ U(N) is the fixed-point
set of the involution (25) with I ′ = I ′NN as in (24). Explicitly,
(33) U(N) ∋ g 7→
(
g
(gT )−1
)
∈ K(N).
Identify G(N)/K(N) with the set
(34)
S(N) := {H = GI ′ s.t. G ∈ U(2N) is Hermitian and JG = −GJ}
by means of (8). The torus A is
(35) A =
{(
ℜΛN −ℑΛN
ℑΛN ℜΛN
)}
,
with ΛN a unitary diagonal matrix as before. The eigenvalues occur in
pairs just as in the case of (27) with M = N .
C II. Let M ≥ N ≥ 1 and G(M,N) = USp(2M +2N). We take the
complex structure J = JMN as in (22). Then K(M,N) = USp(2M)×
USp(2N) consists exactly of those elements that also stabilize
(36) I ′ =
(
I ′MN
I ′MN
)
,
with I ′MN as in (24), so that K(M,N) is the fixed-point set of the
involution
(37) g 7→ gΩ := I ′gI ′.

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 13
We can realize the symmetric space G(M,N)/K(M,N) as the set of
matrices
(38) S(M,N) := {H = GI ′ such that G ∈ USp(2M + 2N)
is Hermitian of signature (M,N)},
where we mean the quaternionic signature as discussed below. We
recall that any matrix G ∈ Sp(2n,C) which is Hermitian (G = GT ),
has real eigenvalues and can be diagonalized with a symplectic matrix
g ∈ Sp(2n,C), that is,
(39) g−1Gg =
(
∆n
∆−1n
)
for some real diagonal matrix ∆n. The usual signature of G is of
the form (2a, 2b), so we call (a, b) the quaternionic signature. The
identification is, of course, given by (8). The torus A is
(40) A =


















IM−N
ℜΛN −ℑΛN
ℑΛN ℜΛN
IM−N
ℜΛN ℑΛN
−ℑΛN ℜΛN


















with ΛN unitary diagonal. Besides the eigenvalue 1 with multiplicity
2(M −N), the other eigenvalues occur in quadruples like those of the
matrices in (32).
For each of the ensembles, the torus A, which has dimension equal to
the rank R of the symmetric space, is parametrized by diagonal unitary
matrices
(41) ΛR = diag(λ1, . . . , λR), |λj| = 1.
Abusing notation, we will also write ΛR for the vector (λ1, . . . , λR).
The tangent space a to this torus at the identity is identified with the
space of R-tuples iΘ = (iθ1, . . . , iθR), θj ∈ R. Recall that we identify
p with the tangent space to G/K at the base-point o = K/K. The
exponential maps Exp of G/K and exp of G are related by
(42) Exp(X) = exp(X)K ∈ G/K
for X ∈ p = To(G/K). For iΘ ∈ a, exp(iΘ) is given by the matrix on
the right-hand side of equations (27), (32), (35) and (40), respectively,
provided we choose λj = eiθj in (41).

14 EDUARDO DUEN˜EZ
Proposition 1 (KAK decomposition). Let G/K be a symmetric space
of the compact type and A ⊂ G be as above. The mapping
K × A×K ։ G
(k1, a, k2) 7→ k1ak2(43)
is a surjection.
The KAK decomposition has an integral counterpart.
Proposition 2 (Weyl’s integration formula). There is a measure dν¯(a)
on A such that, for any f ∈ C(G),
(44)
∫
G
f(g)dg =
∫
K
∫
K
∫
A
f(k1ak2)dν¯(a)dk2 dk1.
(We have simplified our notation by dropping the name “Haar” of the
respective invariant measures.) Denote by Ξ+ the set of positive roots
of the symmetric Lie algebra (g, ω), and by mα the multiplicity of a
positive root α ∈ Ξ+. Then
(45) dν¯(a) ∝
∏
α∈Ξ+
| sinα(Θ)|mαda = ∆(Θ)da,
say, where Θ is chosen so a = exp(iΘ).
(With the notation above, we write iΘ = log(a). This Θ is well-
defined modulo 2π.)
Now recall that the (positive) roots of (g, ω) are certain non-zero
real-valued linear functionals on a (in fact one should speak about the
roots which are positive with respect to a fixed Weyl chamber in a).
The root systems of the irreducible orthogonal Lie algebras of compact
type are well-known by Cartan’s work.
Proposition 3. The positive roots and multiplicities for the irreducible
orthogonal Lie algebras of type I are as follows (let L = M −N in the
case of ensembles with two parameters).
• A I.
α mα
θk − θj , 1 ≤ j < k ≤ R 1
• A II.
α mα
θk − θj , 1 ≤ j < k ≤ R 4
• A III.
α mα
θk − θj , 1 ≤ j < k ≤ R 4
• BD I.

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 15
α mα
θk ± θj , 1 ≤ j < k ≤ R 1
θj , 1 ≤ j ≤ R L
• D III. N even.
α mα
θk ± θj , 1 ≤ j < k ≤ R 4
2θj, 1 ≤ j ≤ R 1
• D III. N odd.
α mα
θk ± θj , 1 ≤ j < k ≤ R 4
θj , 1 ≤ j ≤ R 4
2θj, 1 ≤ j ≤ R 1
• C I.
α mα
θk ± θj , 1 ≤ j < k ≤ R 1
2θj, 1 ≤ j ≤ R 1
• C II.
α mα
θk ± θj , 1 ≤ j < k ≤ R 4
θj , 1 ≤ j ≤ R 4L
2θj, 1 ≤ j ≤ R 3
We are now ready to derive the measure on eigenvalues for ensembles
of type I.
Theorem 2. The measure on eigenvalues for a symmetric space of
type I is given by
(46) dν(a) ∝ ∆(Θ/2)da =
∏
α∈Ξ+
∣
∣
∣
∣
sin
1
2
α(Θ)
∣
∣
∣
∣
mα
da, iΘ = log a.

16 EDUARDO DUEN˜EZ
Using Weyl’s integration formula, we deduce that, for any f ∈ C(S),
∫
S
f(H)dµ(H) =
∫
G/K
f((gk)1−Ω)dk dµ(g¯) (by (8))
=
∫
G
f(g1−Ω)dg (since k1−Ω = e)
=
∫
K
∫
A
∫
K
f((k1ak2)1−Ω)dk2dν¯(a)dk1
=
∫
K
∫
A
∫
K
f((k1a)1−Ω)dk2dν¯(a)dk1
=
∫
K
∫
A
f((ka)1−Ω)dν¯(a)dk
=
∫
K
∫
A
f(ka2k−1)dν¯(a)dk. (since a1−Ω = a2)
(47)
This ought to be compared with (5), which defines the measure dν(Λ)
on eigenvalues. A key property of the measure dν¯(a) defined by (45) is
reflected in the fact that ∆(Θ) = ∆(Θ′) if Θ ≡ Θ′ mod π (this follows
in general from the fact that the roots take integral values on the “unit
lattice” exp−1(e), and can be verified for ensembles of type I directly
using proposition 3). From that observation, it follows that:
∫
K
∫
A
f(ka2k−1)dν¯(a)dk ∝
∫
K
∫
[0,2π]R
f(k exp(2iΘ)k−1)∆(Θ)dΘ dk
= 2R
∫
K
∫
[0,π]R
f(k exp(2iΘ)k−1)∆(Θ)dΘ dk
=
∫
K
∫
[0,2π]R
f(k exp(iΘ)k−1)∆(Θ/2)dΘ dk
=
∫
K
∫
A
f(kak−1)∆(Θ/2)da dk.
(48)
When put together with (47), this proves (46).
Now we restrict attention to the most interesting case, that of “class
functions” f ∈ C(K\S), that is, those functions on S which depend
only on the eigenvalues of the matrix, viz
(49) f(kak−1) = f(a).
The tori A are parametrized by R-tuples (λj = eiθj ). From the
knowledge of the structure of the set of eigenvalues of the matrices

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 17
in these tori, we see that for all the ensembles of type I except for
Dyson’s A I and A II, changing the sign of any θj does not change
the set of eigenvalues since these always come in pairs {e±iθj} (with
single or double multiplicity), hence any class function f ∈ C(K\S) is
determined by its values on exp([0, π]R) ⊂ A, and correspondingly
∫
S
f(H)dµ(H) =
∫
A
f(a)dν(a) ∝
∫
[−π,π]R
f(exp(iΘ))∆(Θ/2)dΘ
= 2R
∫
[0,π]R
f(exp(iΘ))∆(Θ/2)dΘ.
(50)
Hence, except in the cases of A I and A II, it is convenient to regard
the measure on eigenvalues as one supported on [0, π]R. Noting that
the contribution of a pair of roots θk ± θj to ∆(Θ/2) is
(51)
∣
∣
∣
∣
sin
(θk − θj
2
)
sin
(θk + θj
2
)∣
∣
∣
∣
∝ | cos θk − cos θj |,
it is clear that for all the ensembles of type I, except for the COE and
the CSE, the measure on eigenvalues is proportional to the measure
(52)
∏
1≤j<k≤R
|Van(cosΘ)|β
∏
1≤j≤R
| sin θj |P | sin(θj/2)|QdΘ, on [0, π]R.
(Here β = 1, 2, 4 according to the multiplicity mα of the roots θk ± θj .)
Because | sin θ| = |1− cos θ|1/2|1 + cos θ|1/2 and | sin(θ/2)| = 2−1/2|1−
cos θ|1/2, the above is proportional to the measure
(53)
∏
1≤j<k≤R
|Van(cosΘ)|β
∏
1≤j≤R
|1− cos θj |p|1 + cos θj |qdΘ, on[0, π]R.
We make the change variables Θ 7→ x = cosΘ to obtain
(54)
dν(x) ∝
∏
1≤j<k≤R
|Van(x)|β
∏
1≤j≤R
|1− xj |a|1 + xj|bdx, on[−1, 1]R,
where a = p − 1/2, b = q − 1/2, and dx = dx1 . . . dxR. The weight
function
(55) w(x) = |1− x|a|1 + x|b on [−1, 1]
is that with respect to which the classical Jacobi orthogonal polynomi-
als P (a,b)n (x) are defined, so a matrix ensemble for which the probability
measure of the eigenvalues is given by (54) is called a Jacobi ensem-
ble (with parameters (a, b)). For β = 1, 2, 4 we call such an ensemble
orthogonal, unitary or symplectic, respectively.

18 EDUARDO DUEN˜EZ
Recall that, for the COE and CSE, the probability measure of the
eigenvalues is given by (14). It coincides with that given by Weyl’s
formula (proposition 2) since
(56) |eiθk − eiθj | = 2
∣
∣
∣
∣
sin
(θk − θj
2
)∣
∣
∣
∣
.
For completeness, table 3 is the analogue of table 1 for (the infi-
nite families of) symmetric spaces of type II (compact Lie groups).
The CUE is a circular ensemble with β = 2 and measure on eigen-
values (14), whereas the orthogonal and symplectic groups are unitary
Jacobi ensembles.
Type S(N) Parameters
aN (CUE) U(N) β = 2
bN SO(2N + 1) β = 2, (a, b) = (12 ,−12)
cN USp(2N) β = 2, (a, b) = (12 , 12)
dN SO(2N) β = 2, (a, b) = (−12 ,−12)
Table 3. Parameters of the probability measure of the
eigenvalues for ensembles of type II.
4. Universality of Local Correlations
In this section we analyze the limiting correlation functions for gen-
eral Jacobi ensembles. As we have shown, with the exception of Dyson’s
COE (A I) and CSE (A II), the ensembles of type I are special cases
of (orthogonal, unitary or symplectic) Jacobi ensembles.
We consider the joint probability measure of the R levels (we speak
about levels rather than eigenvalues since the natural variables to use
are xj = ℜλj) given in the general form
(57) dν(xR) = PR(xR) dxR,
where xR = (x1, . . . , xR) is an R-tuple of levels. The n-level correlation
function I(n)R (xn) is defined by
(58) I(n)R (xn) =
R!
(R− n)!
∫
· · ·
∫
PR(xn, xn+1, . . . , xR)dxn+1 · · · dxR.
It is, loosely speaking, the probability that n of the levels, regardless
of order, lie in infinitesimal neighborhoods of x1, . . . , xn (but the total
mass of the measure I(n)R (xn)dxn is now R!/(R− n)! and not 1).

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 19
The semi-classical limit R → ∞ is of great interest. The so-called
“universality conjecture” (which dates back to the work of Dyson [14])
states that the local correlations of the eigenvalues in the bulk of the
spectrum tend to very specific limits that depend only on the symmetry
parameter β. Special cases of the truth of this assertion are known. In
particular, in the unitary case β = 2, the result is proven in certain
generality [8, 7, 2, 3], but for β = 1, 4 it is known only for special
ensembles such as the circular ensembles of Dyson [11, 12, 13] and,
by work of Nagao and Forrester [23], for most Laguerre ensembles and
Jacobi ensembles. However, the latter assumes that the parameters a, b
are strictly positive, hence it is not applicable to ensembles of type I
(cf., table 1).
It is an extremely important fact that for general orthogonal, unitary
and symplectic ensembles the correlation functions can be expressed as
determinants (which discovery goes back, in the unitary case, to the
work of Gaudin and Mehta [15, 19], and in the orthogonal and symplec-
tic cases to Dyson’s study of his circular ensembles, and later extended
by Chadha, Mahoux and Mehta [18, 6, 22] to the general case). In
the case of unitary Jacobi ensembles there exists a scalar-valued kernel
K(a,b)R2 (x, y) defined in terms of the classical Jacobi orthogonal polyno-
mials P (a,b)n (x) (the projector kernel onto the span of the first R Jacobi
polynomials) satisfying [24]
(59) I(n)Rβ (xn) = det(KRβ(xj , xk))j,k=1,...,n.
In the case of the orthogonal (resp., symplectic) Jacobi ensembles, there
exists a matrix-valued kernel [24] (alternatively, a “quaternion” kernel)
(60) K(a,b)Rβ (x, y) =
(
S(a,b)Rβ (x, y) I
(a,b)
Rβ (x, y)− δǫ(x− y)
D(a,b)Rβ (x, y) S
(a,b)T
Rβ (x, y)
)
,
where δ = 1 (resp., δ = 0—the ǫ-term is absent in the symplectic case),
(61) ǫ(z) = 1
2
sgn(z) = 1
2
z
|z| ,
and the scalar kernel S(a,b)Rβ is defined in terms of the skew-orthogonal
polynomials of the second (resp., first) kind depending on the weight (55)

20 EDUARDO DUEN˜EZ
and the other quantities are given by
I(a,b)Rβ (x, y) = −
∫ y
x
S(a,b)β (x, z)dz,(62)
D(a,b)Rβ (x, y) = ∂xS
(a,b)
Rβ (x, y),(63)
S(a,b)TRβ (x, y) = S
(a,b)
Rβ (y, x).(64)
The matrix kernel (60) is self-dual in the sense that K(a,b)Rβ (y, x) =
K(a,b)Rβ (x, y)D (cf., section 2). The correlation functions themselves are
given by
(65) I(n)Rβ (xn) =
√
det(KRβ(xj , xk))n×n.
Indeed, if the matrix (KRβ(xj , xk))n×n is interpreted as a quaternion
self-dual matrix [20], then the right-hand side of (65) is its Dyson’s
“quaternion determinant” qdet [11, 12, 13], so (65) can be rewritten:
(66) I(n)Rβ (xn) = qdet(KRβ(xj, xk))n×n.
Remark. In what follows we will sometimes unify notation by writing
DET (all caps) to signify the usual determinant when β = 2 and the
quaternion determinant when β = 1, 4. Thus, equations (59) and (66)
will be written
(67) I(n)Rβ (xn) = DET(KRβ(xj , xk))n×n.
The first quantity of interest is the (global) level density. Indeed,
since the first correlation function has total mass R, one might expect
that the probability measure R−1I(1)R (x)dx on [−1, 1] tend to a limiting
measure as R→ ∞. We define the level density to be the corresponding
probability density function:
(68) ρ(x) = lim
R→∞
R−1I(1)R (x).
Assuming ρ(x) to be continuous, the bulk of the spectrum is the set
{x : ρ(x) > 0}: points where the level density vanishes or blows up to
infinity are excluded from the bulk of the spectrum.
Theorem 3. For the orthogonal, unitary or Jacobi ensembles asso-
ciated to the weight function (55), the global level density is given by
(69) ρ(x) = 1
π
√
1− x2
on (−1, 1).
The limit in (68) is attained uniformly on compact subsets of (−1, 1).

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 21
This theorem will be proved in the following section.
If we revert to the angular variable θ with x = cos θ, we see that
(70) ρ(x)dx = dθπ = ̺(θ)dθ
so the level density ̺(θ) ≡ 1/π on (0, π) is constant: the eigenvalues
become equidistributed on the unit circle (with respect to its invariant
measure), and uniformly so away from the central eigenvalues ±1, in
the semiclassical limit R→ ∞. The bulk of the spectrum excludes the
edges ±1.
The local n-level correlations are the “local” semi-classical limits of
the n-level correlations I(n)R . When localizing near the neighborhood
of a fixed level zo belonging to the bulk of the spectrum, these local
correlations are universal in the sense that they depend neither on the
specific ensemble nor on the choice of zo but only on the symmetry
parameter β. In particular they coincide with the local correlations
of the Gaussian Orthogonal (β = 1), Unitary (β = 2) or Symplectic
(β = 4) ensemble, respectively. For Jacobi ensembles the bulk of the
spectrum consists of the open interval (−1, 1), whereas the local cor-
relations near the “hard edges” ±1 (which correspond to the “central
eigenvalues” ±1 on the unit circle) have a different behavior which is
sensitive to the parameters (a, b) of the ensemble.
Remark. As we shall see later, the level density vanishes to some or-
der at, say, the hard edge +1 depending on the parameter a (which is
natural since a determines the order to which the weight function (55)
vanishes at xj = +1). The local correlations fail to follow Dyson’s uni-
versal “threefold way”, but rather depend on this parameter. The same
limiting behavior occurs at the hard edge 0 of Laguerre ensembles [23],
so that, at least conjecturally, these “universal” laws—manifestly dif-
ferent from Dyson’s bulk regimes—describe the behavior of the local
correlations at a hard edge for general orthogonal, unitary or symplec-
tic ensembles.
We now fix a level zo ∈ [−1, 1]. Given that the eigenvalue density
is uniform, it is natural to change variables from x to ξ stretching the
angles by a factor R, namely setting
(71) xj = cos
(
αo +
π
Rξj
)
,
where αo = arccos zo (note that the change of variables depends on
R). The semiclassical limit of the correlation functions is obtained by
letting R tend to infinity. What the factor π/R accomplishes is that,

22 EDUARDO DUEN˜EZ
on the bulk of the spectrum, the local level density (i.e., the local limit
of the correlation function I(1)Rβ) will be ρ¯(ξ) ≡ 1.
Theorem 4. For the orthogonal (β = 1), unitary (β = 2) and sym-
plectic (β = 4) Jacobi ensembles associated to the weight function (55),
the local correlations are as follows:
• Bulk local correlations (independent of β and of the choice of a
fixed z0 = cosα0 ∈ (−1, 1)).
– Local level density:
(72) ρ¯(ξ) = lim
R→∞
(Rρ(x))−1I(1)Rβ(x) ≡ 1, ξ ∈ R.
where x depends on ξ as in (71) and ρ(x) is the global level
density (69).
– Local correlations:
(73) L(n)β (zo; ξn) = limR→∞(Rρ(zo))
−nI(n)Rβ (xn) = DET(K¯β(ξj, ξk))n×n,
where xn and ξn are related by (71) (recall that DET stands
for the usual or the quaternion determinant in the cases of
β = 2 and β = 1, 4, respectively). In the case β = 2, K¯2 is
the scalar Sine Kernel
(74) K¯2(ξ, η) =
{
sinπ(ξ−η)
π(ξ−η) , ξ 6= η;
ρ¯(ξ) = 1, ξ = η.
In the case β = 4 the matrix Sine Kernel K¯4 is given by
(75) K¯4(ξ, η) =
(
S¯4(ξ, η) I¯4(ξ, η)
D¯4(ξ, η) S¯T4 (ξ, η)
)
,
where
S¯4(ξ, η) = K¯2(2ξ, 2η),(76)
I¯4(ξ, η) = −
∫ η
ξ
S¯4(ξ, t)dt,
D¯4(ξ, η) = ∂ξS¯4(ξ, η),
S¯T4 (ξ, η) = S¯4(η, ξ).
In the case β = 1 the matrix Sine Kernel K¯1 is given by
(77) K¯1(ξ, η) =
(
S¯1(ξ, η) I¯1(ξ, η)− ǫ(ξ − η)
D¯1(ξ, η) S¯T1 (ξ, η)
)
,

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 23
where
S¯1(ξ, η) = K¯2(ξ, η),(78)
I¯1(ξ, η) = −
∫ η
ξ
S¯1(ξ, t)dt,
D¯1(ξ, η) = ∂ξS¯1(ξ, η),
S¯T1 (ξ, η) = S¯1(η, ξ).
• Hard edge zo = +1 (αo = 0).
– Central point level density. For ξ > 0:
(79) lim
R→∞
(R
π
)−1
I(1)Rβ(x) = ρˆβ(ξ)
(where x depends on ξ by (71)) is given by:
ρˆ(a)2 (ξ) =
π
2
(πξ)[Ja(πξ)2 − Ja−1(πξ)Ja+1(πξ)],(80)
ρˆ(a)1 (ξ) = ρˆ
(2a+1)
2 (ξ) +
π
2
J2a+1(πξ)
∫ ∞
πξ
J2a+1,(81)
ρˆ(a)4 (ξ) = ρˆ
(a)
2 (2ξ)−
π
2
Ja−1(2πξ)
∫ 2πξ
0
Ja+1.(82)
– Local correlations. For ξn > 0 :
(83) L(n)β (+1; ξn) = limR→∞
(R
π
)−n
I(n)Rβ (xn) = DET(Kˆβ(ξj, ξk))n×n,
with xn related to ξn by (71). The scalar “Bessel Kernel”
Kˆ2 = Kˆ(a)2 is given by
(84)
Kˆ(a)2 (ξ, η) =
{ √
ξη
ξ2−η2 [πξJa+1(πξ)Ja(πη)− Ja(πξ)πηJa+1(πη)], ξ 6= η;
ρˆ(a)2 (ξ), ξ = η.
For β = 1, 4 the matrix Bessel Kernels are given by the
same expressions of (75)–(78), except that the bars are to
be replaced by hats and Sˆ1 = Sˆ(a)1 , Sˆ4 = Sˆ
(a)
4 are given by
Sˆ(a)1 (ξ, η) =
√
ξ
η Kˆ
(2a+1)
2 (ξ, η) +
π
2
J2a+1(πη)
∫ ∞
πξ
J2a+1(t)dt,(85)
Sˆ(a)4 (ξ, η) =
√
ξ
η Kˆ
(a−1)
2 (2ξ, 2η)−
π
2
Ja−1(2πη)
∫ 2πξ
0
Ja−1(t)dt.(86)
where the Jν are the Bessel functions of the first kind.

24 EDUARDO DUEN˜EZ
0.5 1 1.5 2 2.5 3
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Figure 1. Graphs of ρˆ(a)2 (ξ) for a = −1/2 (the “even”
Sine Kernel, solid), a = +1/2 (the “odd” Sine Kernel,
dotted), and a = 0 (the Legendre Kernel, dashed).
The next section will be devoted to the proof of this theorem.
Remark. The local limits at the edge z0 = −1 are given by the same
formulae replacing the parameter a by b.
Remark. The integral in (86) diverges for −1 < a < 0. However, in the
next section we provide an alternative version of that equation which
is well-defined for all a > −1.
Remark. In connection with the hard edge correlations for the classical
orthogonal and symplectic groups (table 3), we remark that the unitary
Bessel kernel (84), in the case a = +1/2 (resp., a = −1/2), coincides
with the “odd” (resp., “even”) Sine Kernel [17]:
(87) K¯(±1/2)2 (ξ, η) =
sin(ξ − η)
ξ − η ∓
sin(ξ + η)
ξ + η .
5. Proofs
In this section we prove theorems 3 and 4. First we remark that the
unitary case has been studied in the work of Nagao and Wadati [24, 25],
but we reproduce the proofs here for completeness and also to show
that the hypothesis a > −1 is, in a certain sense, unnecessary. Also
we remark that Forrester and Nagao [23] have studied the hard edge
correlations directly, using skew-orthogonal polynomial expressions for

26 EDUARDO DUEN˜EZ
we take advantage of the more recent work of Adler et al which provides
simple “summation formulas” for the quantities SR1, SR4.
5.1. Some preliminary results and formulas. The various results
we quote on Jacobi polynomials can be found in Szego˝’s book [28]
and in his article on asymptotic properties of Jacobi polynomials [27]
(reproduced in his collected papers [29]). Stirling’s formula and the
Bessel function identities can be found, for instance, in the tables of
Gradshteyn and Ryzhik [16]. We denote by P (A,B)N (x) the classical
Jacobi polynomials defined by
(88)
(1− x)A(1 + x)BP (A,B)N (x) =
(−1)N
2NN !
( d
dx
)N
[
(1− x)N+A(1 + x)N+B
]
.
When A,B > −1, these polynomials are orthogonal on [−1, 1] with
respect to the weight
(89) w(x) = |1− x|A|1 + x|B,
but they are not normalized. However, the formula (88) is meaning-
ful for arbitrary (real or complex) values of the parameters A,B, and
defines a polynomial in A,B, x of degree (at most) N in x. In fact
(90) P (A,B)N (x) =
N
∑
k=0
(A +N
k
)(B +N
N − k
)(x− 1
2
)N−k (x+ 1
2
)k
.
In particular
(91) P (A,B)N (+1) =
(A+N
N
)
.
The derivative of a Jacobi polynomial is related to another Jacobi poly-
nomial by the identity (the apostrophe denotes differentiation with re-
spect to x)
(92) P (A,B)N
′
(x) = 1
2
(N + A+B + 1)P (A+1,B+1)N−1 (x).
Proposition 4 (Darboux’s formula). (With an improved error term
due to Szego˝ [27].) For arbitrary reals A,B,
P (A,B)N (cos θ) = (πN)−1/2
(
sin
θ
2
)−A−1/2(
cos
θ
2
)−B−1/2
cos(N ′θ + γ)
+ E,
(93)
N ′ = N + A +B + 1
2
, γ = −
(
A + 1
2
) π
2
,

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 27
for 0 < θ < π, where the error term E satisfies
(94) E = θ−A−3/2O(N−3/2), uniformly for c/N ≤ θ ≤ π − ǫ,
for any positive constants c, ǫ, and the constant implied by the O symbol
depends only on c, ǫ, A,B.
Proposition 5 (Hilb’s formula). (As generalized by Szego˝ to Jacobi
polynomials [28].) For A > −1 and any real B:
(
sin
θ
2
)A(
cos
θ
2
)B
P (A,B)N (cos θ) = N−A
Γ(N + A+ 1)
N !
√
θ
sin θJA(N
′θ)
+ E,
(95)
where N ′ has the same meaning as in (93) and the error term E is
given by
(96) E =
{
θ1/2O(N−3/2) if c/N ≤ θ ≤ π − ǫ,
θA+2O(NA) if 0 < θ ≤ c/N,
where c, ǫ are arbitrary but fixed positive constants, and the constants
implied by the O symbol depend on A,B, c, ǫ only.
The restriction to A > −1, however, is too strong for some pur-
poses, and we will need the following formula, also due to Szego˝ [27]
(reproduced in [29]):
P (A,B)N (cos θ) =
(
sin
θ
2
)−A(
cos
θ
2
)−B√ θ
sin θ
(
1−
√
tan(θ/2)
2θ
)
×
× JA(N ′θ) +R,
(97)
with N ′ as in (93). Here A,B are arbitrary reals. The error term R
satisfies:
(98) R =
{
θ 12−AO(N−3/2) if c/N ≤ θ ≤ π − ǫ,
O(NA−2) if 0 < θ ≤ c/N,
where c, ǫ are fixed positive numbers, and the constants implied by the
O symbol depend only on A,B, c, ǫ. It must be noted, however, that the
error term R of (98) does not depend on θ on the range 0 < θ < c/N ,
which makes this formula less useful than (95) with the error term (96)
for θ in this range.
Recall Stirling’s asymptotic formula for the Gamma function:
(99) log Γ(x) =
(
x− 1
2
)
log x−x+1
2
log 2π+O(x−1), as x→ ∞.

28 EDUARDO DUEN˜EZ
The Bessel functions of the first kind are defined by the series
(100)
Jν(z) =
(z
2
)ν ∞∑
k=0
(−1)k z
2k
22kk!Γ(ν + k + 1) , z ∈ C\(−∞, 0], ν ∈ R;
they satisfy, among many others, the relations:
J ′ν(z) = Jν−1(z)−
ν
z Jν(z),(101)
J ′ν(z) = −Jν+1(z) +
ν
z Jν(z),(102)
J ′ν(z) =
1
2
[Jν−1(z)− Jν+1(z)],(103)
Jν+1(z) =
2ν
z Jν(z)− Jν−1(z),(104)
d
dz [z
νJν(z)] = zνJν−1(z),(105)
d
dz [z
−νJν(z)] = −z−νJν+1(z).(106)
We also have
∫
Jν = 2
∞
∑
k=0
Jν+2k+1,(107)
∫ ∞
0
Jν(t)dt = 1 for ν > −1.(108)
5.2. Asymptotics of the Unitary Jacobi Kernel. In this section
we recall the proofs of some of the results of Nagao and Wadati [24],
which will be needed later on in the analysis of the orthogonal and
symplectic cases.
Using the Christoffel-Darboux summation formula [28], the scalar
kernel K(A,B)N2 can be written in the form
K(A,B)N2 (x, y) =
2−A−B
2N + A+B
Γ(N + 1)Γ(N + A +B + 1)
Γ(N + A)Γ(N +B)
×
√
w(x)w(y)P
(A,B)
N (x)P
(A,B)
N−1 (y)− P
(A,B)
N−1 (x)P
(A,B)
N (y)
x− y ,
(109)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 29
for x 6= y, and
K(A,B)N2 (x, x) =
2−A−B
2N + A+B
Γ(N + 1)Γ(N + A +B + 1)
Γ(N + A)Γ(N +B)
× w(x)[P (A,B)N
′
(x)P (A,B)N−1 (x)− P
(A,B)
N−1
′
(x)P (A,B)N (x)].
(110)
We observe that the kernelKN2 given by (109) and (110) is well-defined
for A,B > −c for any real constant c provided N is sufficiently large.
First consider the global level density
(111) ρ(x) = lim
N→∞
N−1K(x, x).
Using Darboux’s formula (93) together with the identity (92) in the
expression (110) for the kernel, we find:
(112) K(a,b)N2 (x, x) =
N
π
√
1− x2
+O(1)
where the implied constant depends only on ǫ for −1 + ǫ ≤ x ≤ 1− ǫ.
Equation (112) proves (69) (in the unitary case).
A density functionD = D(x1, . . . , xn) defines a measureDdx1 . . . dxn.
Under a (monotonically increasing or decreasing) differentiable change
of variables xj = X(uj), this density is transformed into the density
(113) D(u1, . . . , un) =
( n
∏
j=1
|X ′(uj)|
)
D(X(u1), . . . , X(un)).
If the density D is given as a determinant with a (scalar) kernelK(x, y),
namely D = det(K(xj , xk))n×n, then the change of variables reflects
itself in the kernel in the following fashion:
Lemma 1. After the (monotonic) differentiable change of variables
u → x = X(u), the correlation functions are given as the determi-
nant (59) defined using the kernel
(114) K(u, v) =
√
|X ′(u)X ′(v)|K(X(u), X(v)).
This is clear since the introduction of the factor
√
|X ′(u)X ′(v)| re-
sults in multiplying the determinant (59) by
∏n
j=1 |X ′(uj)|.
The localization at some −1 < zo = cosαo < 1 given by the change
of variables (71) leads us to consider the limit
K¯(a,b)2 (ξ, η) = limN→∞
(
N
√
ρ(x)ρ(y)
)−1K(a,b)N2 (x, y)
= lim
N→∞
(Nρ(zo))−1K(a,b)N2 (x, y),
(115)

30 EDUARDO DUEN˜EZ
with x, y related to ξ, η by (71), which from Darboux’s formula (93)
can be easily seen to be the Sine Kernel (74), independently of the
value of zo (as long as −1 < zo < 1), for any real a, b, and the limit is
attained uniformly on compacta.
For the localization at zo = +1 (αo = 0) —localization at zo = −1 is
analogous provided a and b are interchanged—, we use the same change
of variables (71) with ξn > 0. To compute the limit
Kˆ(a,b)2 (ξ, η) = limN→∞
(
N
√
ρ(x)ρ(y)
)−1K(a,b)N2 (x, y)
= lim
N→∞
(Nρ(zo))−1K(a,b)N2 (x, y),
(116)
we use Szego˝’s formulas (95), (97), in conjunction with (109) and (110):
(117) Kˆ(a)2 (ξ, η) =
√
ξη
ξ2 − η2 [πξJ
′
a(πξ)Ja(πη)− Ja(πξ)πηJ ′a(πη)].
Using the derivation formula (102) we rewrite this kernel in the form (84).
For the case ξ = η we start with the expression (110) and use the
derivation formula (92) to find:
ρˆ(a)2 (ξ) = Kˆ
(a)
2 (ξ, ξ)
=
π
2
[Ja(πξ)Ja+1(πξ) + πξJ ′a+1(πξ)Ja(πξ)− πξJa(πξ)′Ja+1(πξ)].
(118)
Applying the derivation formula (101) and the recurrence formula (104)
this can be rewritten in the form (80).
5.3. Asymptotics of the Orthogonal Jacobi Kernel. We start
with some general remarks. If a density P = P (x1, . . . , xn) is given
as a quaternion determinant with a self-dual matrix kernel K(y, x) =
K(x, y)D, namely P = qdet(Q(xj , xk))n1 , then under a differentiable
change of variables xj = X(uj) the density is still given as a quaternion
determinant.
Lemma 2. After a (monotonic) differentiable change of variables u→
x = X(u), a density function
(119) P (x1, . . . , xn) = qdet(K(xj , xk))
defined in terms of some self-dual matrix kernel (δ = 0, 1)
(120) K(x, y) =
(
S(x, y) I(x, y)− δǫ(x− y)
D(x, y) ST (x, y)
)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 31
with
I(x, y) = −
∫ y
x
S(x, z)dz,(121)
D(x, y) = ∂xS(x, y),(122)
ST (x, y) = S(y, x).(123)
is transformed into the density
(124) P(u1, . . . , un) = qdet(K(uj, uk)),
where
K(u, v) =
(
S(u, v) I(u, v)− δǫ(u− v)
D(u, v) ST (u, v)
)
(125)
S(u, v) = S(X(u), X(v))|X ′(v)| = ±S(X(u), X(v))X ′(v)(126)
I(u, v) = −
∫ v
u
S(u, w)dw,(127)
D(u, v) = ∂uS(u, v),(128)
ST (u, v) = S(v, u).(129)
For the proof, we need first:
Lemma 3. Let H = HD = JnHTJTn be a 2n × 2n self-dual complex
matrix. Let kj, j = 1, 2, . . . , n be arbitrary complex constants. Set
K = diag(k1, . . . , kn). Then the matrices
(130)
H1 = diag(I,K)H diag(K, I) H2 = diag(−I,K)H diag(−K, I)
(where I = In is the n× n identity matrix) are both self-dual, and
(131) qdet(H1) = det(K) qdet(H) = qdet(H2).
The verification that H1 and H2 are self-dual is trivial. On the other
hand, since (qdetX)2 = detX for any self-dual matrix X, we have that
(qdet(H1))2 = (qdet(H2))2 = (det(K))2 det(H)
= (det(K))2(qdet(H))2.
(132)
Hence equation (131), which is an equality between polynomials in the
entries of the matrices involved, must hold up to a sign. Setting K = In
we see that the first equality in (131) holds, and setting K = −In, so
H2 = −H , the validity of the second equality in (131) is equivalent to
the easy fact that qdet(−H) = (−1)n qdetH = det(−In) qdetH .

32 EDUARDO DUEN˜EZ
Proceeding to the proof of lemma 2, we first observe that, after the
change of variables u → x, the density P (x1, . . . , xn) transforms into
the density
(133) P(u1, . . . , un) = P (X(u1), . . . , X(un))
n
∏
j=1
|X ′(uj)|.
We apply lemma 3 with H = (K(X(uj), X(uk)))n×n and kj = |X ′(uj)|
to conclude that (124) holds with either of the two kernels (we write
X(u, v) for (X(u), X(v)))
(134)
K±(u, v) =
(
S(X(u, v))|X ′(v)| ±(I − δǫ)(X(u, v))
±D(X(u, v))|X ′(u)||X ′(v)| ST (X(u, v))|X ′(u)|
)
.
The plus and minus signs correspond to applying the first and second
of the equalities in (131), respectively. If x → u preserves orientation,
then we observe that ǫ(X(u) − X(v)) = ǫ(u − v) and conclude by
a simple application of the chain rule and a change of variables in
the integral that the kernel K+ coincides with K from (125) for the
choices (126)–(129). If x→ u reverses orientation, we choose the minus
signs, observe that ǫ(X(u) − X(v)) = −ǫ(u − v) and proceed exactly
as before to see that K− coincides with (125) in this case.
Lemma 2 explains the relations (78) between the entries of the lim-
iting kernels K¯β and also of Kˆβ (β = 1, 4). The relations certainly
hold when R is finite after applying the change of variables (71) to
the the matrix kernel KRβ so as to obtain another kernel KRβ . They
can be shown to continue to hold in the limit either by noting that
the sequence of scalar kernels {SRβ(ξ, η)}∞R=0 is a normal sequence of
analytic functions (i.e., it converges uniformly on compacta), or by di-
rect verification that each of the sequences {SRβ}, {IRβ}, {KRβ}, {STRβ}
converges to the correct limit as R→ ∞. In what follows we will only
consider the limit of the quantity SRβ which alone determines the ma-
trix kernel KRβ .
Let A = 2a + 1, B = 2b + 1, where a, b are the parameters of the
orthogonal Jacobi ensemble. Assume also that R is even. Observe that
A,B > −1 if a, b > −1. The summation formula of Adler et al [1]
expresses the orthogonal kernel S(a,b)R1 using the unitary kernel K
(A,B)
R−1,2
and another term. As we shall see, this other term is negligible in the
localized limit (in the bulk of the spectrum), but it does contribute to
the edge limit.

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 33
The summation formula for the quantity S(a,b)R1 (x, y) of (60) is as
follows [1]:
(135) S(a,b)R1 (x, y) =
√
1− x2
1− y2K
(A,B)
R−1,2(x, y) + cR−2ψR−1(y)ǫψR−2(x).
Here ǫ denotes the integral operator (cf., eq. (61))
(136) (ǫf)(x) =
∫ 1
−1
ǫ(x− y)f(y)dy,
and we have set
(137) ψN (t) = ψ(A,B)N (t) = (1− t)(A−1)/2(1 + t)(B−1)/2P
(A,B)
N (t)
and
(138) cN = 2−A−B−1
Γ(N + 2)Γ(N + A+B + 2)
Γ(N + A+ 1)Γ(N +B + 1) .
The quantity S(a,b)R1 determines the entries of the matrix kernel K
(a,b)
R1
as per equations (62)–(64).
From Stirling’s formula (99), the asymptotic behavior of the coeffi-
cient cN is
(139) cN ∼ 2−A−B−1N2, as N → ∞.
Lemma 4. For any real A,B:
(140) lim
N→∞
ψ(A,B)N (cosφ) = 0
for 0 < φ < π, uniformly on compacta.
This follows immediately from Darboux’s formula (93).
This lemma is, however, insufficient to understand the asymptotics
of the function ǫψN as N → ∞ since it says nothing about the behavior
of ψN near the edge. First we note:
Lemma 5. For A > −1 and B arbitrary:
lim
N→∞
N−1ψ(A,B)N (cos(φ/N)) = 2
A+B
2
JA(φ)
φ ,(141)
lim
N→∞
ψ(A,B)N (cos(φ/N)) sin(φ/N) = 2
A+B
2 JA(φ).(142)
The limits hold uniformly on compact subsets of (0,∞).
These follow from Szego˝’s formula (95).

34 EDUARDO DUEN˜EZ
Lemma 6. For A,B real with A > −1 and any 0 < θ < π we have:
lim
N→∞
N
∫ θ
0
ψ(A,B)N (cosφ) sinφ dφ = 2
A+B
2 ,(143)
lim
N→∞
N
∫ θ/N
0
ψ(A,B)N (cosφ) sinφ dφ = 2
A+B
2
∫ θ
0
JA.(144)
These follow again from Szego˝’s formula (95) and equation (108).
When −1 < A < 0, the dependence on θ of the second of the error
terms in (96) is critical to ensure that the contribution of this error
term to the integral is negligible (in particular, this lemma cannot be
proven using the alternate formula (98) unless A > 0.)
Corollary 1. For −1 < A,B and 0 < θ < π:
lim
N→∞
N(ǫψ(A,B)N )(cos θ) = 0,(145)
lim
N→∞
N(ǫψ(A,B)N )(cos(θ/N)) = 2
A+B
2
(
1−
∫ θ
0
JA
)
= 2
A+B
2
∫ ∞
θ
JA.
(146)
This follows from the previous lemma applied to both ψ(A,B)N and
ψ(B,A)N . We also used (108) to obtain the last equality.
We localize at some zo = cosαo ∈ (−1, 1) using the change of variable
x→ ξ of (71). The limit to consider is
(147)
S¯(a,b)1 (ξ, η) = limR→∞(Nρ(y))
−1S(a,b)R1 (x, y) = limR→∞(Nρ(zo))
−1S(a,b)R1 (x, y)
By the lemmas above, the second term on the right-hand side of (135)
is negligible in the limit. Also, the factor
√
1−x2
1−y2 is 1 in the limit. Thus,
the limit (147) is equal to the limiting unitary kernel, namely the Sine
Kernel, whence the expression (78).
As for the central point, let us now localize at z = +1. Using the
summation formula (135), lemma 5 and corollary 1, we readily find:
Sˆa1 (ξ, η) =
√
ξ
η Kˆ
(2a+1)
2 (ξ, η) +
π
2
J2a+1(πη)
[
1−
∫ πξ
0
J2a+1(t)dt
]
=
√
ξ
η Kˆ
(2a+1)
2 (ξ, η) +
π
2
J2a+1(πη)
∫ ∞
πξ
J2a+1(t)dt.
(148)

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 35
As we remarked already, the conditions a > −1 and A > −1 are
equivalent since A = 2a+1. Thus we have derived a weak universality
law for the local correlations at the central points ±1 for any a, b > −1.
Lemma 7. Let κα(x, y) = xJα+1/2(x)Jα−1/2(y)− Jα−1/2(x)yJα+1/2(y).
Then
(149)
√x
y κα±1/2(x, y)−
√
y
xκα∓1/2(x, y)
= ∓
(x2 − y2√xy
)
Jα−1/2∓1/2(x)Jα−1/2±1/2(y).
(This equation stands for two different equations, one with the top signs
and another with the bottom signs.)
We prove the equation with the choice of the top signs (the other
case is analogous). Indeed, expanding the left-hand side we obtain:
(150) x3/2Jα+1(x)y−1/2Jα(y)− x1/2Jα(x)y1/2Jα+1(y)
− x1/2Jα(x)y1/2Jα−1(y) + x−1/2Jα−1(x)y3/2Jα(y).
The central terms can be combined into −2αx1/2Jα(x)y−1/2Jα(y) us-
ing the identity (104) and expanded using this same identity into
−x3/2Jα−1(x)y−1/2Jα(y) − x3/2Jα+1(x)y−1/2Jα(y). Two terms cancel
out, and the remaining two factor to give the right-hand side of (149).
We now have, using lemma 7,
√
ξ
η Kˆ
(A)
2 (ξ, η) =
√
ξη
ξ2 − η2κA+1/2(πξ, πη)
=
η
ξ
√
ξη
ξ2 − η2κA−1/2(πξ, πη) + πJA(πξ)JA−1(πη)
=
√η
ξ Kˆ
(A−1=2a)
2 (ξ, η)− πJA−1(πξ)JA(πη),
(151)
and similarly
(152)
√
ξ
ηKˆ
(A)
2 (ξ, η) =
√η
ξ Kˆ
(A+1)
2 (ξ, η) + πJA+1(πξ)JA(πη).
From (107):
(
∫ πξ
0
JA
)
± 2JA∓1(πξ) =
∫ πξ
0
JA∓2.(153)

36 EDUARDO DUEN˜EZ
The last two equations provide alternative forms of the kernel Sˆ(a)1 ,
namely
Sˆ(a)1 (ξ, η) =
√η
ξ Kˆ
(2a)
2 (ξ, η) +
π
2
J2a+1(πη)
[
1−
∫ πξ
0
J2a−1(t)dt
]
(154)
Sˆ(a)1 (ξ, η) =
√η
ξ Kˆ
(2a+2)
2 (ξ, η) +
π
2
J2a+1(πη)
[
1−
∫ πξ
0
J2a+3(t)dt
]
(155)
As before, the terms in brackets can be replaced by
[
∫∞
πξ
]
.
5.4. Asymptotics of the Symplectic Jacobi Kernel. Here we set
A = a − 1, B = b − 1 where a, b are the parameters of the symplectic
Jacobi ensemble. Note that here a, b > −1 corresponds to A,B > −2.
With cN as in (138) and ψN = ψ(A,B)N as in (137), the summation
formula in this case reads
(156) S(a,b)R4 (x, y) =
1
2
√
1− x2
1− y2K
(A,B)
2R,2 (x, y)−
1
2
c2R−1ψ2R(y)δψ2R−1(x),
where the operator δ acts by
(157) δf(x) =
∫ 1
x
f(t)dt.
The formula (156) only holds verbatim when a > 0 (that is, A,B >
−1), since the integral defining δψ(A,B)N is divergent for A ≤ −1. How-
ever, we note that the skew orthogonal polynomials of the second kind
are analytic functions of the parameters a, b > −1 (corresponding to
A,B > −2), hence the kernel KN4 is an analytic function on a, b > −1.
Thus, we must find a suitable analytic continuation of (156) valid for
A,B > −2. First we remark that, although the original kernel K(A,B)2R,2
of unitary Jacobi ensembles is defined for A,B > −1, equation (109)
is well-defined and analytic for A,B > −2 if R > 1 (which we will

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 37
assume). We write
(158) δψ(A,B)N (x) =
∫ 1
x
(1− t)(A−1)/2(1 + t)(B−1)/2P (A,B)N (t)dt
=
∫ 1
x
(1− t)(A−1)/2(1 + t)(B−1)/2(P (A,B)N (t)− P
(A,B)
N (1))dt
+ P (A,B)N (1)
∫ 1
x
(1− t)(A−1)/2(1 + t)(B−1)/2dt.
The first integral on the right-hand side is well-defined and analytic for
A > −2. The term P (A,B)N (1) =
(A+N
N
)
(cf., equation (91)) vanishes for
A = −1, which is sufficient to extend the second integral on the right-
hand side to a well-defined analytic function on the range A > −2. It
is easy to rewrite that integral as an incomplete Beta function and use
well-known results to achieve the extension, but one can also proceed
elementarily as follows. Integrating the second integral by parts we
obtain, for A > −1:
(159)
(A+N
N
)
∫ 1
x
(1− t)(A−1)/2(1 + t)(B−1)/2dt
=
2
A+ 1
(A+N
N
)
(1− x)(A+1)/2(1 + x)(B−1)/2
+
B − 1
A+ 1
(A+N
N
)∫ 1
x
(1− t)(A+1)/2(1 + t)(B−3)/2dt.
Observe that
(160)
1
A+ 1
(A +N
N
)
=
1
N
(A+N
N − 1
)
,
and the latter is an analytic function of all A. Then both terms on
the right-hand side of (159) are analytic functions of A > −2 for −1 <
x ≤ 1, so this last equation provides the analytic extension of the
integral (158) defining δψN (x), which is sensu stricti undefined for
A ≤ −1, to an analytic function on A > −2.
The rest of the reasoning is analogous to that in the orthogonal
case. The only technical difficulty arises because the error term (98)
in Szego˝’s formula does not depend on θ in the range 0 < θ ≤ c/N ,
effectively making the reasoning of the previous section inapplicable
when −2 < A ≤ −1. This is to be expected since the summation
formula only makes sense after being analytically continued. In what
follows we prove that the various limits of the kernel do in fact depend

38 EDUARDO DUEN˜EZ
analytically on the parameter A, thus allowing the expressions obtained
for A > −1 to be extended to A > −2.
Using Szego’s formula (97) (valid for all A), there is no problem to
obtain this variant of lemma 6:
Lemma 8. For any A,B, θ real and 0 < ψ < π we have:
(161) lim
N→∞
N
∫ φ
θ/N
ψ(A,B)N (cosψ) sinψ dψ = 2
A+B
2
∫ ∞
θ
JA.
Lemma 9. Using equation (159), the expression
(162) N
∫ θ/N
0
ψ(A,B)N (cosφ) sinφ dφ
can be analytically continued to a regular function on A > −2. As
N → ∞, this function tends to a limit which is also analytic for A > −2
and coincides with (144) for A > −1.
We change variables φ → φ/N . As before, we split the integral to
rewrite (162) in the form
(163)
2(A+B)/2
∫ θ
0
(
sin
φ
2N
)A(
cos
φ
2N
)B [
P (A,B)N
(
cos
φ
N
)
− P (A,B)N (1)
]
dφ
+ 2(A+B)/2P (A,B)N (1)
∫ θ
0
(
sin
φ
2N
)A(
cos
φ
2N
)B
P (A,B)N (1) dφ
The first of these terms is analytic for A > −2, the second one has an
analytic continuation given by (159). It is easy to see that this second
term has the asymptotic behavior:
(164) 2(A+B)/2
∫ θ
0
(
sin
φ
2N
)A(
cos
φ
2N
)B
P (A,B)N (1) dφ
∼ 2B−A2 1N
(A +N
N − 1
)( φ
N
)A+1

MATRIX ENSEMBLES ASSOCIATED TO SYMMETRIC SPACES 39
as N → ∞, and from Stirling’s formula (99), the binomial coefficient
(A+N
N−1
)
= Γ(A+N+1)Γ(N)Γ(A+2) = O(NA+1), hence this second terms is asymptot-
ically negligible. As for the first term in (163), we first write
(165)
P (A,B)N
(
cos
φ
N
)
− P (A,B)N (1) = −
1
N
∫ φ
0
P (A,B)N
′
(
cos
ψ
N
)
sin
ψ
N dψ
= −N + A+ B + 1
2N
∫ φ
0
P (A+1,B+1)N−1
(
cos
ψ
N
)
sin
ψ
N dψ,
where we have used the derivation formula (92). We can now use
Szego˝’s formula (95) to estimate P (A+1,B+1)N−1 since A + 1 > −1. The
upshot is that the limit of (162) as N → ∞ can be written as the
following integral, which is an analytic function of A > −2:
(166) −2(A+B)/2
∫ θ
0
∫ φ
0
φAψ−AJA+1(ψ)dψ dφ.
Using the Bessel function identity (106) we can simplify the above
integral, for A > −1:
(167) 2(A+B)/2
∫ θ
0
JA(φ)dφ,
which is in agreement with lemma 6.
We note that the expression (167) can be easily continued to an
analytic function of A > −2 without the need to rewrite it as the
double integral (166). Namely, using (107) we have, for A > −1,
(168)
∫ θ
0
JA(φ)dφ = JA+1(θ) +
∫ θ
0
JA+2(φ)dφ.
The expression on the right-hand side is analytic for A > −2 and
provides the desired analytic continuation.
The global level density is derived identically to the previous sec-
tion. The limiting kernel in the bulk of the spectrum is given by the
sum of two terms: S¯(a)2 (2ξ, 2η) and another term which is negligible in
the limit. For the central point z = +1, the lemmas above yield the
following expression for the limiting kernel:
(169) Sˆ(a)4 (ξ, η) =
√
ξ
ηKˆ
(A)
2 (2ξ, 2η)−
π
2
JA(2πη)
∫ 2πξ
0
JA(t)dt,
whe+re the last integral is to be understood in the sense of equa-
tion (168) for A ≤ −1. Using equations (151) and (152) together

40 EDUARDO DUEN˜EZ
with (153) and the equation above, the kernel can be rewritten in ei-
ther of the forms:
Sˆ(a)4 (ξ, η) =
√η
ξ Kˆ
(a)
2 (2ξ, 2η)−
π
2
Ja−1(2πη)
∫ 2πξ
0
Ja+1(t)dt,(170)
Sˆ(a)4 (ξ, η) =
√η
ξ Kˆ
(a−2)
2 (2ξ, 2η)−
π
2
Ja−1(2πη)
∫ 2πξ
0
Ja−3(t)dt.(171)
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E-mail address : eduenez@math.jhu.edu
American Institute of Mathematics and The Johns Hopkins Univer-
sity, 3400 N. Charles St., Baltimore, MD 21218