Probability densities and distributions for spiked Wishart $β$-ensembles

0.22 MB·English

by  Peter J. Forrester

Probability densities and distributions for spiked Wishart
β-ensembles
Peter J. Forrester
1
1
0
2 Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

n email: [email protected]
a
J
Abstract

2
1
A Wishart matrix is said to be spiked when the underlying covariance matrix has a

] single eigenvalue b different from unity. As b increases through b = 2, a gap forms from
h
the largest eigenvalue to the rest of the spectrum, and with b 2 of order N−1/3 the
p −
- scaled largest eigenvalues form a well defined parameter dependent state. Recent works
h
by Bloemendal and Vir´ag [BV], and Mo, have quantified this parameter dependent state
t
a
for real Wishart matrices from different viewpoints, and the former authors have done
m
similarly for the spiked Wishart β-ensemble. We use a recursive structure to give an
[
alternative construction of the spiked Wishart β-ensemble, and we give the exact form

1
of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real
v
quaternion Wishart matrices (β = 4) the latter is recognised as having appeared in earlier
1
6 studies on symmetrized last passage percolation, allowing the exact form of the scaled
2
distribution of the largest eigenvalue to be given. This extends and simplifies earlier work
2
. of Wang, and is an alternative derivation to a result in [BV]. We also use the construction
1
of the spiked Wishart β-ensemble from [BV] to give a simple derivation of the explicit
0
1 form of the eigenvalue PDF.
1
:
v
i 1 Introduction
X
r
a
1.1 Background
Very recently, an outstanding problem in random matrix theory has been solved from two
different perspectives [8, 20]. The problem relates to real Wishart matrices, specified as the
ensemble of random matrices of the form XTX, where X is an n N real Gaussian matrix
×
with distribution proportional to
N
exp Tr(XTXΣ−1) . (1.1)
− 2
(cid:16) (cid:17)
The case of interest is when the N N covariance matrix Σ is of the spiked form
×
Σ = diag(b,1N−1). (1.2)
1
Here the notation 1N−1 denotes the eigenvalue 1 repeated N 1 times, and the eigenvalue b
−
corresponds to the spike.
With γ = n/N 1 and n,N large it was shown by Baik and Silverstein [4] that for
≥
b > 1 + √γ the largest eigenvalue separates from the remainder of the spectrum, which is
otherwise supported on ((1 √γ)2,(1+√γ)2). A simplified derivation of this fact can be found
−
in [6, Prop. 2.4], and some recent generalisations are given in [7]. This same effect holds for
spiked Wishart matrices with complex entries [2]. There it has been explicitly demonstrated
that for large N, and with b (1+1/√γ) of order N−1/3, the scaled largest eigenvalues form
−
a parameter dependent state at the onset of the eigenvalue separation. It was conjectured in
[2] that for these same scaling parameters, spiked real Wishart matrices similarly exhibit a
parameter dependent state. The outstanding problem has been to quantify this state.
Before discussing the two recent works which solve this problem, let us say some more
about the complex case. For this the parameter dependent state was shown in [16] (in the
case n = N 1), in [2] for general γ > 1, and in [21] for γ < 1, to be a determinantal point
−
process with correlation kernel independent of γ. Technically, the correlation kernel was shown
to be a rank one perturbation of the familiar Airy kernel [11]. This explicit form was used in
turn [1] to express the distribution of the largest eigenvalue in terms of a member the Lax pair
for the Hasting-McLeod solution of the Painlev´e II equation, the latter being specified as the
transcendent q(s) satisfying
q′′ = sq +2q3, q(s) Ai(s). (1.3)
s→∼∞
Asanapplication,aknowncorrespondence[18]betweentheeigenvaluesofcomplexWishartma-
trices and last passage times for a directed percolation model based onthe Robinson-Schensted-
Knuth correspondence allowed for an interpretation of these results to be given within a sta-
tistical mechanics setting.
The most prominent application of Wishart matrices is to principal component analysis
in multivariate statistics. There X in (1.1) corresponds to a data matrix for n distinct mea-
surements of N different quantities, and thus has real entries. We know from explicit results
obtained in the null case for γ > 1 [19] (the null case refers to Σ = I ) that the scaled largest
N
eigenvalues now form a Pfaffian rather than determinantal point process. The distribution of
the scaled largest eigenvalue again involves the Hasting-McLeod solution of the Painlev´e II
equation (1.3), but is distinct from that in the complex case (see e.g. [14, 9.7]).
§
More recently this so called soft edge state, for both the real (β = 1) and complex (β = 2)
cases, hasfurtherbeencharacterised[10,22]intermsofthesmallesteigenvaluesofthestochastic
Airy operator
d2 2
+x+ B′(x), x 0. (1.4)
−dx2 √β ≥
Here B(x) denotes standard Brownian motion and the eigenfunctions are subject to a Dirich-
let boundary condition at x = 0. This in turn allows for a diffusion characterisation of the
distribution of the largest eigenvalue.
In two recent works — by Bloemendal and Vir´ag [8] and Mo [20] — the problem of quantify-
ing the soft edge, parameter dependent state for spiked real Wishart matrices has been solved.
The characterisations are very different, in keeping with the two distinct characterisations of
the scaled largest eigenvalues revised above in the null case.
2
Consider first the work [8]. With
√γ
b (1+1/√γ) w
−(n−1/2 +N−1/2)2/3 − →
(cid:16) (cid:17)
as n,N , and the scaling of the large eigenvalues λ > λ > of XTX
1 2
→ ∞ ···
1 1
λ (√n+√N)2 =: Y ,
k k
√nN (n−1/2 +N−1/2)4/3(cid:16) − (cid:17)
it is proved that Y form a well defined parameter dependent state. The latter is again
k
{ }
specified by the smallest eigenvalues of the stochastic Airy operator, but now with the eigen-
functions satisfying the boundary condition ψ′(0) = wψ(0). Furthermore, it is shown that the
distribution function for the largest eigenvalue, F (x) say, is the unique bounded solution to
β,w
the boundary value problem
∂F 2 ∂2F ∂F
+ +(x w2) = 0
∂x β ∂w2 − ∂w
F(x,w) 1 as x,w together
→ → ∞
F(x,w) 0 as w with x x < , (1.5)
0
→ → −∞ ≤ ∞
where x is fixed. An essential step, following [23, 9] is to use Householder transformations to
0
reduce XT to the N N bidiagonal form
×
√bχ
βn
χ χ 
β(N−1) β(n−1)
BβT :=  χβ(N−2) χβ(n−2)  (1.6)
 ... ... 
 
 χ χ 
β β(n−N+1)
 
with β = 1. Here χ2 refers to the particular gamma distribution Γ[n/2,2] (in general Γ[s,σ] is
n
specified by the PDF. proportional to xs−1e−s/σ, x > 0), it has been assumed for definiteness
that n N and some zero columns which do not effect the non-zero eigenvalues of XTX have
≥
been removed.
It is the Pfaffian point process characterisation of the null case that is generalised in [20].
Here knowledge of the joint eigenvalue PDF in the finite system is essential. With N even, this
is shown to be proportional to
N N
b 1 −1/2
λ(n−N−1)/2e−λj/2 (λ λ ) et t − λ dt, (1.7)
Yj=1 j 1≤jY<k≤N j − k ZΓ Yj=1(cid:16) − 2b j(cid:17)
for Γ a simple closed contour enclosing the branch points of the integrand. Next, using integra-
tion methods based on skew orthogonal polynomials (see [14, Ch. 6]) a Pfaffian formula is given
for the correlations, and the distribution of the largest eigenvalue is expressed in terms of the
corresponding Fredholm determinant. (Actually, the working of [20] applies to the finite system
only; in the abstract of that work we are told the asymptotic analysis will follow shortly.)
3
The starting point of [20] is the expression
N
1
λ(n−N−1)/2e−λj/2 e−21Tr(OXTXOTΣ−1)(OTdO), (1.8)
C j Z
Yj=1 O(N)
where C is the normalization, for the eigenvalue PDF of real Wishart matrices. In the case
of complex Wishart matrices, the corresponding formula involves an average over U(N) rather
thanO(N). AccordingtothewellknownHarish-Chandra/Itzykson-Zuberformula(seee.g.[14,
Prop. 11.6.1]) an evaluation in terms of determinants is possible. However, until [20], it was
not known that the O(N) matrix integral admitted a tractable evaluation.
1.2 An alternative viewpoint and outline
We have seen that two seemingly distinct viewpoints have led to the quantification of the pa-
rameter dependent state formed at the spectrum edge for spiked real Wishart matrices in the
critical regime. In this paper we will emphasize a third viewpoint. The idea, initiated in [16], is
to consider the spiking as a perturbation, and to focus attention on the joint eigenvalue distri-
bution of the perturbed and unperturbed matrices. This follows naturally from the recurrence
XXT = X˜X˜T +b~x~xT (1.9)
for n N matrices X specified by (1.1) and (1.2), where X˜ is an n (N 1) standard Gaussian
× × −
obtained from X by deleting its first column.
We will use this formalism to give an alternative construction of the spiked Wishart β-
ensemble, specified as the random matrices BTB , with B the N N bidiagonal matrix (1.6).
β β β ×
This in turn relies on knowledge of the eigenvalue PDF for the Wishart β-ensemble as specified
in terms of (1.6). We begin in Section 2 by showing how to deduce the eigenvalue PDF (1.7)
from the bidiagonal matrix (1.6) with β = 1. Our derivation applies for all β > 0, so we are
able to give the β generalization of (1.7). This is given in (2.1) below.
In Section 3 we use known results from [16] to compute the joint eigenvalue PDF of the
non-zero eigenvalues of the random matrix pair (X˜X˜T,XXT), as related by (1.9) but with
X˜X˜T replaced by diagX˜X˜T , in the case that the non-zero eigenvalues of X˜X˜T have PDF
proportional to
N−1
yβ(n−N+2)/2−1e−yl/2 (y y )β. (1.10)
l j − k
Y Y
l=1 1≤j<k≤N−1
Results from [16] tell us that this joint eigenvalue PDF can be realized as the zeros of two
polynomials generated recursively from a three term recurrence. We give the explicit form of
the matrix eigenvalue problem implied by the recurrences. Although different to the tridiagonal
matrix eigenvalue problem for BTB , it similarly involves only 2N 1 independent entries.
β β −
We also take up the problem of integrating over the eigenvalues of X˜X˜T, with the aim of
showing that the non-zero eigenvalues of XXT have the same PDF as found in Section 2 for
the spiked Wishart β-ensemble, thus providing an alternative construction of this ensemble.
We remark that this construction also allows for a β-generalisation of the general variance
Wishart ensemble, and the corresponding eigenvalue PDF. From the latter, for the special
4
value β(n N +1)/2 1 = 0, it is possible to show that the probability of no eigenvalues in
− −
(0,s) has a simple exponential distribution.
With n < N, X˜X˜T no longer has any zero eigenvalues. This setting is studied in Section 4.
In the case β = 4, and for a special n we obtain a joint eigenvalue PDF proportional to
N
e−PNj=1(yj+(xj−yj)/b)/2 (xi xj)(yi yj) xi yj , (1.11)
− − | − |
1≤iY<j≤N iY,j=1
subject to the interlacing
x > y > x > y > > x > y 0. (1.12)
1 1 2 2 N N
··· ≥
The corresponding parameter dependent soft edge correlations were calculated as a Pfaffian in
[15]. By the universality results of [8] these same correlations must hold for all cases of the
β = 4 spiked Wishart matrices (i.e. for all choices of n and N in (1.6) provided they both
go to infinity). Moreover, results from [3] tell us that the scaled distribution of the largest
eigenvalue with PDF (1.12) can be written in terms of the same member of the Lax pair for
the Hastings-McLeod solution of the Painlev´e II equation as known for the complex case [1].
And universality tells us that this result must persist for all cases of the β = 4 spiked Wishart
matrices. An alternative derivation of this fact was given by Bloemendal and Vir´ag [8], who
showed that the distribution satisfies (1.5). Our results of this section extend the results of
Wang [25], who considered a particular value of the parameter only.
2 Eigenvalue PDF for the spiked Wishart β-ensemble
BythespikedWishartβ-ensemblewerefertothetridiagonalmatricesBTB , withB theN N
β β β ×
bidiagonal matrix (1.6). For β = 1,2 and 4 we know that this tridiagonal matrix corresponds
to a unitary similarity transformation of the spiked real, complex and real quaternion Wishart
matrices, and so shares the same eigenvalue PDF. Here we seek the eigenvalue PDF of BTB
β β
for general β > 0. In the case b = 1, this has been done in [9]. We can adapt the workings of
that calculation to the general b > 0 case.
Proposition 2.1 The tridiagonal matrix BTB has eigenvalue PDF proportional to
β β
N ∞ N b 1 −β/2
λβ(n−N+1)/2−1e−λj/2 (λ λ )β eit it − λ dt. (2.1)
j j − k Z − 2b j
Yj=1 1≤jY<k≤N −∞ Yj=1(cid:16) (cid:17)
Proof. Let us write
x
n
 y x 
N−1 n−1
BβT :=  ... ... . (2.2)
 
 y x 
1 n−N+1
 
Then, according to the definition (1.6), the probability measure P(B )(dB ) has, up to pro-
β β
portionality, the factorization
P(B )(dB ) e(1−1/b)x2n/2. (2.3)
β β
(cid:16) (cid:17)(cid:12)b=1
(cid:12)
(cid:12)
5
Letusdenoteby λ the(ordered)eigenvaluesofB ,andby q thefirstcomponent
j j=1,...,N β j
{ } { }
of the corresponding (normalized) eigenvector. The working of [9] (see also [14, proof of Prop.
3.10.1]) tells us that in terms of these variables P(B )(dB ) is proportional to
β β b=1
|
N N N
λβ(n−N+1)/2−1e−λj/2 (λ λ )β qβ−1δ q2 1 (d~λ)(d~q), (2.4)
j j − k i j −
Yj=1 1≤jY<k≤N Yi=1 (cid:16)Xj=1 (cid:17)
where δ( ) denotes the Dirac delta function. Furthermore, if we write
·
a b
N N−1
 b a b 
N−1 N−1 N−2
BβTBβ =  ... ... ... ,
 
 b2 a2 b1 
 
 b a 
1 1
 
then we see from (2.2) that a = x2. But we also know [14, proof of Prop. 1.9.3] that
N n
a = N q2λ . Hence, substituting x2 = N q2λ in (2.3) we see from (2.4) that our
N j=1 j j n j=1 j j
remainiPng task is to compute P
N N
qβ−1δ q2 1 e(1−1/b)PNj=1qj2λj/2(d~q). (2.5)
ZRN Yi=1 i (cid:16)Xj=1 j − (cid:17)
Introducing the integral form of the delta function
N 1 ∞
δ q2 1 = lim eit(PNj=11−qj2)e−ǫt2 dt
(cid:16)Xj=1 j − (cid:17) ǫ→0+ 2π Z−∞
and supposing temporarily that b < 1 so the coefficient on the exponential in (1.6) is negative,
we see that (2.5) is equal to
1 ∞ ∞
eit qβ−1e−q2(it+(1/2)(1/b−1)λj)dq dt.
2π Z−∞ (cid:16)Z−∞ (cid:17)
Evaluating the integral, up to proportionality this reduces to
∞ N b 1 −β/2
eit it − λ dt, (2.6)
j
Z−∞ Yj=1(cid:16) − 2b (cid:17)
and we see furthermore that the restriction to b < 1 can now be relaxed. Multiplying the
(cid:3)
eigenvalue dependent factors of (2.4) with (2.6) gives (2.1).
3 An alternative construction of the spiked Wishart β-
ensemble
3.1 Joint eigenvalue PDF for (X˜X˜T,XXT)
We begin by giving the derivation of the recurrence (1.9). With X distributed as in (1.1),
set X = YΣ1/2. We see that YTY is then distributed as a real Wishart matrix with variance
6
I
matrix equal to the identity (Σ = ). With Σ as in (1.2) it then follows that each element
N
in the first column of X has distribution N[0,√b] (i.e. is a zero mean, standard deviation √b
Gaussian), and all other elements are distributed independently as N[0,1]. Hence the matrix
product XXT can be factorized according to the RHS of (1.9). An analogous factorization
holds for X having complex elements (β = 2) or real quaternion elements (β = 4). Moreover
in each case, by the invariance of the distribution of a Gaussian vector under conjugation by a
unitary matrix, we have that
ePDFXXT = ePDF diag(X˜X˜T)+b~x~xT (3.1)
(cid:16) (cid:17)
where with q := x2 (the square of the entries of ~x), we have that each q is distributed
i i { i}
according to the gamma distribution Γ[β/2,2]. The notation ePDF in (3.1) refers to the
eigenvalue PDF.
Fordefiniteness, let us suppose that n N. Then X˜X˜T will have n N+1 zero eigenvalues,
≥ −
with the PDF of the remaining N 1 eigenvalues given by (1.10) [14, eq.(3.16)] (in the case
−
β = 4 of real quaternion entries, all eigenvalues are furthermore doubly degenerate). We take
up the the problem of computing the joint distribution of the eigenvalues of X˜X˜T, and the
eigenvalues of XXT, with the two matrices related by (3.1).
Let the non-zero eigenvalues of X˜X˜T be denoted by y . It is a simple exercise to
i i=1,...,N−1
{ }
show that the secular equation for the eigenvalue problem implied by (3.1) is
N−1
q q
0 j
0 = 1+b + , (3.2)
− λ y λ
(cid:16) Xj=1 j − (cid:17)
d d
where q =Γ[β/2,2] (j = 1,...,N 1) and q =Γ[β(n N +1)/2,1/2]. Furthermore, we know
j 0
− −
from [16, Cor. 3] that the PDF of the roots of this equation, and thus the conditional PDF of
the non-zero eigenvalues x of X˜X˜T, is proportional to
i i=1,...,N
{ }
N N−1
xβ(n−N+1)/2−1e−xj/(2b) y−β(n−N+2)/2+1e−yl/2b
j l
Yj=1 Yl=1
(x x ) N N−1
1≤j<k≤N j − k x y β/2−1, (3.3)
× Q (y y )β−1 | i − j|
1≤j<k≤N−1 j − k Yi=1 Yj=1
Q
subject to the interlacing (1.12) with y := 0. Our sort result can now be deduced.
N
Proposition 3.1 Let XXT and X˜X˜T be related by (3.1), and suppose that the non-zero eigen-
values of X˜X˜T are denoted y and have PDF given by (1.10). With x
i i=1,...,N−1 i i=1,...,N
{ } { }
denoting the non-zero eigenvalues of XXT, we have that the joint eigenvalue PDF of both sets
of non-zero eigenvalues is proportional to
N N−1
xβ(n−N+1)/2−1e−xj/2b e−(1−1/b)yl/2
j
Yj=1 Yl=1
N N−1
(y y ) (x x ) x y β/2−1, (3.4)
j k j k i j
× − − | − |
1≤j<Yk≤N−1 1≤jY<k≤N Yi=1 Yj=1
subject to the interlacing (1.12) with y := 0.
N
7
Proof. The joint PDF is given by the product of the conditional PDF for x given
i i=1,...,N
{ }
y , times the PDF of y . Thus we need to multiply together (3.3) and
i i=1,...,N−1 i i=1,...,N−1
{ } { }
(cid:3)
(1.10), and (3.4) results.
What is the marginal distribution of x ? In the cases β = 1, 2 and 4, the construc-
i i=1,...,N
{ }
tion (3.1) is equivalent the distribution of XTX being given by the spiked Wishart distribution
(1.1) and (1.2). Furthermore the non-zero eigenvalues of XXT are the same as the non-zero
eigenvalues of XTX. Hence it must be in these cases at least, x has PDF (2.1). We
i i=1,...,N
{ }
would like to show that this remains true for general β > 0. Our task then is to integrate over
y in (3.4), and show that (2.1) results. This can be accomplished by the use of Jack
i i=1,...,N−1
{ }
polynomial theory [14, Ch. 12&13]. But before taking on this task, we will make note of a
realization of (3.4) in terms of a generalised eigenvalue problem.
3.2 Relationship to a bidiagonal generalised eigenvalue problem
Let
a =d Γ[(N j)β/2+α +1,2] (j = 1,...,N 1) a =d √bΓ[α +1,2]
j 0 N 0
− −
b =d Γ[jβ/2,2] (j = 1,...,N 2) b =d √bΓ[(N 1)β/2,2] (3.5)
j N−1
− −
where α := β(n N +1)/2 1, and set too b := 0. Then results from [16, Section 5.2] tell us
0 0
− −
that with the monic random polynomials B (x) defined by the three term recurrence
j j=0,...,N
{ }
B (x) = (x a )B (x) b xB (x) (j = 1,...,N), (3.6)
j j j−1 j−1 j−1
− −
we have that the joint PDF of the zeros of (B (x),B (x)) is given by (3.4).
N N−1
In general (see e.g. [24]) the recurrence (3.6) is satisfied by the characteristic polynomials
B (x) = det(xM L ) where L and M are the top j j blocks of the bidiagonal matrices
j j j j j
− ×
a 1 1
1
 a 1   b 1 
2 1
−
L :=  ... ... , M :=  −b2 1 
 aN−1 1   ... ... 
   
 a   b 1
N N−1
   − 
Wesee fromthe specification of the entries of (3.5) that the spike b effects only the single entries
a and b in L and M respectively, which is analogous to how b enters (1.6).
N N−1
An open problem is to obtain the stochastic characterisation of the soft edge spiked Wishart
β-ensemble starting from the generalised eigenvalue problem L~v = λM~v.
3.3 Jack polynomials and hypergeometric functions
The conditional PDF (3.3) is a special case of the Dixon-Anderson density [14, eq. (4.11)].
Another special case is the conditional PDF for y given x
i i=1,...,N−1 i i=1,...,N
{ } { }
Γ(Nβ/2) (y y ) N−1 N
1≤j<k≤N−1 j − k y x β/2−1 (3.7)
(Γ(β/2))N Q (x x )β−1 | i − j|
1≤j<k≤N j − k Yi=1 Yj=1
Q
8
subject to the interlacing (1.12) with y := 0. Let this be referred to as DA (β/2).
N N
Intimatelyrelatedto(3.7)arethesymmetricJackpolynomialsP (z;α),wherez = (z ,...,z ),
κ 1 N
κ denotes a partition of length less than or equal to N (we write ℓ(κ) N), and α is a parame-
≤
ter. The Jack polynomials can be specified as the polynomial eigenfunctions of the differential
operator
N
∂ 2 2 z +z ∂ ∂
j k
z + ,
j
∂z α z z ∂z − ∂z
Xj=1 (cid:16) j(cid:17) 1≤jX<k≤N j − k(cid:16) j k(cid:17)
with leading term given by the monomial symmetric function m (z) (see [14, 12.6] for more
κ
§
details). Thus with the generalised Pochhammer symbol specified by
N
Γ(u (j 1)/α+κ )
[u](α) = − − j , (3.8)
κ Γ(u (j 1)/α)
Yj=1 − −
we have [14, eq. (12.209)]
2/β
[βN/2]
κ
P (x;2/β) = P (y;2/β) , (3.9)
κ [β(N 1)/2](2/β)h κ iDAN(β/2)
κ
−
valid for ℓ(κ) N 1.
≤ −
Let us define the quantity d′ as in [14, eq. (12.60)] (it’s precise value plays no explicit role
κ
in the following), and use this in the definition of the renormalized Jack polynomials
α|κ| κ !
C (x;α) = | | P (x;α).
κ d′ κ
κ
The generalized hypergeometric functions based on Jack polynomials are then specified by
(α) (α)
1 [a ] [a ]
F(α)(a ,...,a ;b ,...,b ;x) = 1 κ ··· p κ C (x;α). (3.10)
p q 1 p 1 q κ ! [b ](α) [b ](α) κ
Xκ | | 1 κ ··· q κ
Important for our present purposes is the fact that [14, eq. (13.3)]
F(α)(x) = ePNj=1xj. (3.11)
0 0
The use of relevance of the generalized hypergeometric functions is see upon multiplying
both sides of (3.9) by
[β(N 1)/2](2/β)(2/β)(|κ| 1 1 |κ|
κ
− 1
[βN/2]2κ/β d′κ (cid:16)2(cid:16)b − (cid:17)(cid:17)
and making use of (3.10) with p = q = 1 on the LHS. On the RHS we first use the fact that the
Jackpolynomials arehomogeneous ofdegree κ , andso forc a scalar P (xc;α) = c|κ|P (x;2/β),
κ κ
| |
then use (3.11). We thus obtain the following corollary of (3.9).
Proposition 3.2 For ℓ(κ) N 1 and x := (x ,...,x ) we have
1 N
≤ −
De(1/b−1)PNj=−11yj/2EDAN(β/2) = 1F1(2/β)(β(N −1)/2;βN/2;(1/b−1)x/2)
= e(1/b−1)PNj=1xj/2 F(2/β)(β/2;βN/2;(1 1/b)x/2). (3.12)
1 1 −
9
Proof. It remains to explain the second line. This follows from a generalisation of the second
Kummer identity [14, (13.16)], which states that
F(α)(a;c;x) = ePNj=1xj F(α)(c a;c; x).
1 1 1 1 − −
(cid:3)
Comparing the explicit form of DA (β/2) (3.7) with the joint PDF (3.4), it follows that
N
the marginal distribution of x is proportional to
i i=1,...,N
{ }
N
xjβ(n−N+1)/2−1e−PNj=1xj/2 (xj −xk)β1F1(2/β)(β/2;βN/2;(1−1/b)x/2).(3.13)
Yj=1 1≤jY<k≤N
Comparison of (3.13) with (2.1) shows that our remaining task is to show that for c a scalar
∞ N −β/2
F(2/β)(β/2;βN/2;cx) eit it cx dt. (3.14)
1 1 ∝ Z − j
−∞ Yj=1(cid:16) (cid:17)
(2/β)
Forthispurpose, webeginbyobservingfrom(3.10)and(3.8)thatingeneral F (β/2;b;x)
1 1
is very special. Thus the only partitions giving a non-zero contribution to the sum (3.10) are
of the form (k,0N−1), and so the summation is one-dimensional. In the case b = βN/2, as
is the case in (3.14) there is a further special feature, relating to the particular generalized
hypergeometric function based on two sets of variables [14, eq. (13.20)]
(α) (α)
C (x)C (y)
(2/β) κ κ
(x;y) := , (3.15)
0F0 κ !C(α)(1N)
Xκ | | κ
where x := (x ,...,x ) and y := (y ,...,y ). To see the relation, note that for κ = (k,0N−1)
1 n 1 n
we have
(α)
[N/α]
C(α)(y,0N−1) = yk, C(α)(1N) = κ
κ κ (α)
[1/α]
κ
(for the second formula see e.g. [26, eq. (243)]), while for κ with two or more non-zero parts,
C(α)(y,0N−1) = 0. Thus the summation over κ in (3.15) is also one-dimensional, and moreover
κ
we have that
F(2/β)(β/2;βN/2;cx) = (2/β)(x;(c,0N−1)). (3.16)
1 1 0F0
We remark that an alternative derivation of the marginal distribution being given by (3.13)
with the substitution (3.16) can be given by using the recursive integration formula [17] (see
also [16, Appendix C])
(2/β)
( x ; z )
0F0 { }i=1,...,N { }i=1,...,N
= ezNPNj=1xjhe−zNPNj=−11yj0F0(2/β)({yi}i=1,...,N−1;{zi}i=1,...,N−1)iDAN(2/β). (3.17)
Also, as noted in [26], there is a further alternative derivation in the cases β = 1, 2 and 4.
Thus with (U†dU) denoting the normalized Haar volume form for unitary matrices with real
(β = 1), complex (β = 2) and real quaternion (β = 4) entries, and H, H(0) Hermitian matrices
10