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Random Matrices: Theory and Applications
c World Scientific Publishing Company
Computing with Beta Ensembles and Hypergeometric Functions
VESSELIN DRENSKY
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
1113 Sofia, Bulgaria
drensky@math.bas.bg
ALAN EDELMAN
Department of Mathematics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.
edelman@math.mit.edu
TIERNEY GENOAR
Department of Mathematics, San José State University,
1 Washington Square, San Jose, CA 95192, U.S.A.
tgenoar@gmail.com
RAYMOND KAN
Joseph L. Rotman School of Management, University of Toronto
105 St. George Street, Toronto, Ontario, Canada M5S 3E6
kan@chass.utoronto.ca
PLAMEN KOEV
Department of Mathematics, San José State University,
1 Washington Square, San Jose, CA 95192, U.S.A.
koev@math.sjsu.edu
Received (January 25, 2014)
Revised (Day Month Year)
Keywords: Wishart; Jacobi; Laguerre; Hermite; MANOVA; eigenvalue; hypergeometric
function of a matrix argument
Mathematics Subject Classification 2000: 15A52, 60E05, 62H10, 65F15
We survey the empirical models for the Hermite, Wishart, Laguerre, Jacobi, and
MANOVA beta ensembles. The eigenvalue distributions of these ensembles are expressed
in terms of the hypergeometric function of a matrix argument. We present a number of
identities for this function that have been instrumental in computing these distributions
in practice. These identities include a number of new results.
1
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1. Introduction
Multivariate statistical analysis relies heavily on the classical random matrix
ensembles—Hermite, Wishart, Laguerre, Jacobi, and MANOVA—and the distributions of their eigenvalues in order to infer information about multivariate datasets.
Muirhead’s text [29] provides an in-depth analysis of the real (β = 1) case. Most of
the results have complex (β = 2) analogues [31] and the extension to quaternions
(β = 4) is straightforward although lacking on applications thus far.
In the past 10–15 years generalizations of these ensembles to any β > 0 have
been introduced. This suggests that most of the classical results (and perhaps most
of random matrix theory) also generalize to any β > 0 [12]. Advances in computational tools have also made these theoretical results very important in practice.
The theoretical importance of the distributions of the extreme eigenvalues is also
well documented [20,21].
The goal of this paper is to present in one place the empirical models, the eigenvalue distributions, and the identities that have made computing those distributions
possible. We include several new results.
This paper is organized as follows. In section 2 we introduce the random matrix
ensembles, their empirical models as well as the hypergeometric function of a matrix
argument and the various combinatorial objects that we will need later on. In
section 3 we present the identities for the hypergeometric function and the Jack
function. The known distributions and densities of the extreme eigenvalues of the
beta ensembles are in 4. We prove all new results in section 5. The numerical
experiments are in section 6. We finish with open problems in section 7.
2. Preliminaries
In this section we survey the classical definitions from combinatorics and multivariate polynomials. We refer to Stanley [34,35] for a more detailed treatment of the
material in this section.
For an integer k ≥ 0 we say that κ = (κ1 , κ2 , . . .) is a partition of k (denoted
κ ` k) if κ1 ≥ κ2 ≥ · · · ≥ 0 are integers such that κ1 + κ2 + · · · = k. The quantity
|κ| = k is called the size of κ.
All partitions can be partially ordered: we say that µ ⊆ λ if µi ≤ λi for all
i = 1, 2, . . .. Then λ/µ is called a skew shape and consists of those boxes in the
Young diagram of λ that do not belong to µ. Clearly, |λ/µ| = |λ| − |µ|.
The skew shape κ/µ is called a horizontal strip when κ1 ≥ µ1 ≥ κ2 ≥ µ2 ≥
· · · [35, p. 339].
The upper and lower hook lengths at a point (i, j) in a partition κ (i.e., i ≤
κ0j , j ≤ κi ) are, respectively,
h∗κ (i, j) ≡ κ0j − i +
2
(κi − j + 1)
β
and
hκ∗ (i, j) ≡ κ0j − i + 1 +
2
(κi − j).
β
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The products of the upper and lower hook lengths are denoted, respectively, as
Y
Y
Hκ∗ =
h∗κ (i, j)
and
H∗κ =
hκ∗ (i, j).
(i,j)∈κ
(i,j)∈κ
Hκ∗ H∗κ .
Their product is denoted as jκ ≡
For a partition κ = (κ1 , κ2 , . . . , κn ) and β > 0, the Generalized Pochhammer
symbol is defined as
Y i−1
(β)
β+j−1
(2.1)
(a)κ ≡
a−
2
(i,j)∈κ
κi n Y
Y
i−1
a−
=
β+j−1
2
i=1 j=1
n Y
i−1
a−
=
β
,
2
κi
i=1
where (a)k = a(a + 1) · · · (a + k − 1) is the rising factorial.
For a partition κ = (k) in only one part (a)(β)
κ = (a)k is independent of β.
The multivariate Gamma function of parameter β is defined as
m
Y
m(m−1)
i−1
m−1
β
4
Γ(β)
(c)
≡
π
Γ
c
−
β
for <(c) >
β.
(2.2)
m
2
2
i=1
Definition 2.1 (Jack function).
The Jack function Jκ(β) (X) = Jκ(β) (x1 , x2 , . . . , xm ) is a symmetric, homogeneous
polynomial of degree |κ| in the eigenvalues x1 , x2 , . . . , xm of X. It can be defined
recursively [34, Proposition 4.2]:
Jκ(β) (x1 , x2 , . . . , xn ) = 0, if κn+1 > 0;
(β)
J(k)
(x1 ) = xk1 1 + β2 · · · 1 + (k − 1) β2 ;
X
Jκ(β) (x1 , x2 , . . . , xn ) =
Jµ(β) (x1 , x2 , . . . , xn−1 )x|κ/µ|
βκµ ,
n
n ≥ 2,
(2.3)
µ⊆κ
where the summation in (2.3) is over all partitions µ ⊆ κ such that κ/µ is a
horizontal strip, and
Q
∗
κ
hν (i, j), if κ0j = µ0j ;
(i,j)∈κ Bκµ (i, j)
ν
,
where Bκµ (i, j) ≡
βκµ ≡ Q
(2.4)
µ
hν∗ (i, j), otherwise.
(i,j)∈µ Bκµ (i, j)
Definition 2.2 (Hypergeometric function of matrix arguments). Let p ≥ 0
and q ≥ 0 be integers, and let X and Y be m × m complex symmetric matrices
with eigenvalues x1 , x2 , . . . , xm and y1 , y2 , . . . , ym , respectively. The hypergeometric
function of two matrix arguments X and Y , and parameter β > 0 is defined as
(β)
p Fq
(a1 , . . . , ap ; b1 , . . . , bq ; X, Y )
∞ X
(β)
X
1 (a1 )(β)
Cκ(β) (X)Cκ(β) (Y )
κ · · · (ap )κ
≡
·
·
. (2.5)
(β)
k! (b1 )(β)
Cκ(β) (I)
κ · · · (bq )κ
k=0 κ`k
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Beta ensemble (m × m)
Joint eigenvalue density
π
Hermite
m(m−1)β
4
(2π)
m
2
·
(Γ(1 + β/2))m
(β)
e−
tr(Λ2 )
2
dµ(Λ)
Γm (1 + β/2)
n
Wishart (n DOF, cov. Σ)
n
|Σ|− 2 β
|Λ| 2 β−r · 0 F0(β) (− β2 Λ, Σ−1 )dµ(Λ)
(β) n
Km
( 2 β)
Laguerre (with param. a)
tr(Λ)
1
|Λ|a−r e− 2 β dµ(Λ)
Km (a)
Jacobi (with param. a, b)
MANOVA (param. n, p
and covariance Ω)
(β)
(β)
Sm
1
|Λ|a−r |I − Λ|b−r dµ(Λ)
(a, b)
(β)
pβ
p
p
(n + p)
Km
|Ω| 2 |Λ| 2 β−r |I − Λ|− 2 β−r
(β)
(β)
Km (n)Km (p)
−1
× 1 F0(β) ( n+p
, Ω)dµ(Λ)
2 ; Λ(Λ − I)
Table 1. Joint eigenvalue densities of the beta ensembles (where r ≡
m−1
β
2
+ 1).
For one matrix argument X, we define
(β)
p Fq
(a1 , . . . , ap ; b1 , . . . , bq ; X) ≡ p Fq(β) (a1 , . . . , ap ; b1 , . . . , bq ; X, I).
(2.6)
2.1. A new measure—the Vandermonde determinant
The Vandermonde determinant to power β is a part of every multivariate integral
we consider. To avoid it appearing everywhere, we conveniently incorporate it into
a new measure µ
dµ(X) =
Y
|xi − xj |β dx1 dx2 · · · dxm ,
i<j
where X = diag(x1 , x2 , . . . , xm ).
2.2. Beta random matrix ensembles
The Beta random matrices are defined in terms of their joint eigenvalue distributions. A matrix is from a particular beta ensemble if its (unordered)
eigenvalues λ1 , λ2 , . . . , λm have joint density as in Table 2.2. We denote Λ =
diag(λ1 , λ2 , . . . , λm ) and for any matrix A we denote its determinant as |A|.
We require that the parameters above be such that β > 0 and a, b > m−1
2 β. We
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define
m!( β2 )ma
(β) m
Γ(β)
m (a)Γm 2 β
,
Km (a) ≡ m(m−1) ·
m
Γ β2
π 2 β
m!Γ(β) m β
Γ(β) (a) Γ(β)
m (b)
(β)
Sm
(a, b) ≡ m(m−1)m 2 m · m (β)
.
β
Γ
(a
+
b)
β
m
Γ
π 2
(β)
(2.7)
(2.8)
2
The expression (2.8) is the value of the Selberg Integral [32].
We refer to [10] (Hermite), [8,17] (Wishart), [15] (Laguerre), [11] (Jacobi),
and [7] (MANOVA).
2.3. Empirical models for the classical random matrix ensembles
The following are the empirical models for the beta–ensembles above which are
valid for any β > 0.
• Hermite [10]:

N (0, 2) cm−1

 cm−1 N (0, 2)
1 
..
H=√ 
.
2



..
.
..
. c2
c2 N (0, 2) c1
c1 N (0, 2)



,



where ci ∼ χiβ , i = 1, 2, . . . , m − 1, are independent and all elements on
the diagonal are i.i.d. N (0, 2).
• Wishart [8,17]: The procedure is iterative. Let the eigenvalues of the covari(β)
(β)
ance Σ be d21 , d22 , . . . , d2m . The Wishart model is W = β1 (Zm
(n))T Zm
(n),
where


τ1
χβ di
 ..

..


.
.
Zi(β) (n) = 
.


τi−1
χβ di
χ(n−i+1)β di
(β)
Here τ1 , τ2 , . . . , τi−1 are the singular values of Zi−1
(n). The base (m = 1)
(β)
case is Z1 (n) = χnβ d1 .
Note than the number of degrees of freedom n need not be integer.
• Laguerre [10,33]: L ≡ β1 BB T , where

χ2a

 χβ(m−1) χ2a−β

β


B=
 , a > (m − 1).
.
.
..
..
2


χβ χ2a−β(m−1)
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• Jacobi [14] (see also [23,26]):

T
J ≡ Z Z,
where
cm −sm c0m−1


.

cm−1 s0m−1 . .
,

..
0
. −s2 c1 
c1 s01


Z≡


with
ck ∼
q
Beta a −
m−k
2 β, b
−
m−k
2 β
,
q
c0k ∼ Beta k2 β, a + b − 2m−k−1
β ,
2
sk =
s0k =
q
1 − c2k ;
q
1 − c02
k.
• MANOVA [7] Let Λ be m × m beta–Wishart with n degrees of freedom
and covariance Ω and let M be m × m Wishart with p degree of freedom
and covariance Λ−1 . Then
(M −1 + I)−1
is m × m beta–MANOVA with parameters n, p and covariance Ω.
3. Identities involving p Fq(β) (a1:p ; b1:q ; X, Y )
Let
r≡
m−1
2 β
+ 1;
X ≡ diag(x1 , x2 , . . . , xm );
Y ≡ diag(y1 , y2 , . . . , ym );
X̄ ≡ diag(x2 − x1 , x3 − x1 , . . . , xm − x1 );
Ȳ ≡ diag(y2 − y1 , y3 − y1 , . . . , ym − y1 );
Y
∆(X) ≡
(xi − xj );
i<j
∆(Y ) ≡
Y
i<j
(yi − yj ).
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3.1. Identities for p Fq(β)
(β)
0 F0
(β)
0 F0
(X) = etr(X) ;
tr(X)
(3.1)
(β)
0 F0 (X, Y − I)
y1 tr(X)+x1 tr(Y )−mx1 y1
(X, Y ) = e
=e
(β)
1 F0
(a; X) = |I − X|
−a
,
(3.2)
(β)
1 F1
m
( m−1
2 β; 2 β; X̄, Ȳ
);
(X < I);
(3.3)
(β)
−a
· 1 F0(β) (a; X(I − X)−1 , Y − I);
1 F0 (a; X, Y ) = |I − X|
Q
(β) m
−β
2 ;
1 F0 ( 2 β; X, Y ) =
i,j (1 − xi yj )
(β)
1 F1 (a; c; X)
(β) β m
1 F1 ( 2 ; 2 β; X)
(β) β
m
2 F1 ( 2 , n; 2 β; X)
(3.6)
β m
t
1 F1 ( 2 ; 2 β; X − tI) · e ;
(1 − t)n · 2 F1(β) ( β2 , n; m
2 β; (1
(3.7)
=
=
(β)
− t)X + tI), (t 6= 1);
Γm (c)Γm (c − a − b)
;
(β)
Γ(β)
m (c − a)Γm (c − b)
(1)
(see [19] for the definition of A);
2 F1 (a, b; c; xI) = Pf(A)
(β)
(β)
c − a, b; c; −X(I − X)−1 · |I − X|−b
2 F1 (a, b; c; X) = 2 F1
(β)
2 F1
(a, b; c; I) =
c−a−b
= 2 F1 (c − a, c − b; c; X) · |I − X|
(a, b; c; X) = 2 F1(β) a, b; a + b + r − c; I − X
× 2 F1(β) (a, b; c; I),
(β)
p Fq
( β2 , a2:p ; b1:q ; xI)
(2)
p Fq
=
(a1:p ; b1:q ; X) =
(3.8)
(β)
(β)
(β)
(3.5)
= etr(X) · 1 F1(β) (c − a; c; −X);
(β)
2 F1
(3.4)
(3.9)
(3.10)
(3.11)
;
(a or b ∈ Z≤0 );
m
p Fq ( 2 β, a2:p ; b1:q ; x);
(3.12)
(3.13)
m m−j
xi p Fq (a1:p − j + 1; b1:q − j + 1; xi ) i,j=1 ;
(3.14)
∆(X)
Qq
m
Y
(i − 1)! j=1 (bj − m + 1)m−i
(2)
Q
p
p Fq (a1:p ; b1:q ; X, Y ) =
j=1 (aj − m + 1)m−i
i=1
m p Fq (a1:p − m + 1; b1:q − m + 1; xi yj ) i,j=1 ×
;
(3.15)
∆(X)∆(Y )
mt
X
X C (β) (X −1 )
Γ(β)
(β)
m (r + t)
κ
t
F
(r,
−t;
−X)
=
|X|
, (t ∈ Z≥0 ); (3.16)
2 0
k!
Γ(β)
m (r)
k=0 κ`k, κ1 ≤r
(β)
p Fq
(a1:p ; b1:q ; X) = lim
c→∞
(β)
p+1 Fq
(a1:p , c; b1:q ; 1c X)
(β)
= lim p Fq+1
(a1:p ; b1:q , c; cX).
c→∞
(3.17)
The references for the above identities are:
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
[16, (13.3), p. 593] (also [29, p. 262] for β = 1 and [31, p. 444] for β = 2);
[2, section 6] (also [27] for β = 1);
[16, p. 593, (13.4)] (also [29, p. 262] for β = 1 and [31, p. 444] for β = 2);
[27, (6.29)];
[27, (6.10)];
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(3.6) [16, p. 596, (13.16)] (also [29, (6), p. 265] for the case β = 1);
(3.7) New result, see Theorem 5.2. For m = 3, β = 1 this is due to Bingham [3,
Lemma 2.1];
(3.8) New result, see Theorem 5.2;
(3.9) [16, (13.14), p. 594] (also [5, p. 24, (51)] for the case β = 1);
(3.10) [19];
(3.11) [16, Proposition 13.1.6, p. 595]; see also [29, p. 265, (7)] for the case β = 1;
(3.12) [16, Proposition 13.1.7, p. 596]. The condition a or b ∈ Z≤0 implies that
the series expansion for 2 F1(β) terminates.
(3.13) New result, see Theorem 5.3. See also [18, (5.13)] for β = 2;
(3.14) [30, (33), p. 281];
(3.15) [18, Theorem 4.2] (there is a typo in [18, (4.8)]: βn−1 should be βn ), see also
[30, (34), p. 281];
(3.16) [13];
−|κ|
(3.17) [16, (13.5)]. Trivially implied by limx→∞ (x)(β)
= 1.
κ ·x
3.2. Integral identities
Define
−
c(β)
m ≡π
m(m−1)
β
4
m!
m
Y
Γ( i β)
2
i=1
and recall that r =
(β)
p+1 Fq
m−1
2 β
+ 1. Then
(a1:p , a; b1:q ; Y )
Z
1
= (β) (β)
e−tr(X) p Fq(β) (a1:p ; b1:q ; X, Y )|X|a−r dµ(X);
cm Γm (a) Rm
+
(β)
p+1 Fq+1 (a1:p , a; b1:q , b; Y
=
Γ( β2 )
1
(β)
Sm
(a, b − a)
Cκ(β) (Y −1 ) =
)
Z
(β)
[0,1]m
(3.18)
p Fq
(a1:p ; b1:q ; X, Y )|X|a−r |I − X|b−a−r dµ(X);
|Y |a
(β) (β)
cm Γm (a) · (a)(β)
κ
Z
R+
m
(β)
0 F0
(−X, Y )|X|a−r Cκ(β) (X)dµ(X).
(3.19)
(3.20)
The references for the above identities are:
(3.18) [27, (6.20)] (see also [13]);
(3.19) [27, (6.21)] (see also [13]);
(3.20) This is a conjecture of Macdonald [27, Conjecture C, section 8], proven by
Baker and Forrester [2, section 6], with Dubbs and Edelman [7] correcting
a typo in [2].
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Define T ≡ diag(t1 , t2 , . . . , tn ). The following identities are due to Kaneko [22]
Z
m,n
Y
(xi − tk )
[0,1]m i,k=1
m
Y
(β)
xai (1 − xi )b dµ(X) = Sm
(a + r, b + r)
i=1
(4)
× 2 F1 β
Z
m,n
Y
β
(1 − xi tk )− 2
[0,1]m i,k=1
−m, β2 (a + b + n + 1) + m − 1; β2 (a + n); T ;
m
Y
(3.21)
(β)
xai (1 − xi )b dµ(X) = Sm
(a + n + r, b + r)
i=1
×
4
(β
)
2 F1
β
− mβ
2 , 2 (m
− 1) + a + 1; β(m − 1) + a + b + 2; T , (3.22)
(β)
is the value of the Selberg integral (2.8).
where Sm
3.3. Identities involving the Jack function
We will predominantly use the “C” normalization of the Jack function, but there
are other normalizations, Jκ(β) , Pκ(β) , and Q(β)
κ , related as in (3.25). The properties
of these normalizations are that Jκ(β) (X) is such that for |κ| = n the coefficient of
x1 x2 · · · xn in Jκ(β) (X) is n! [34, Theorem 1.1]. The zonal polynomial is Cκ(1) (X) and
Pκ(2) (X) = Q(2)
κ (X) = sλ (X) is the Schur function.
X
Cκ(β) (X) = (trX)k ;
(3.23)
κ`k
Jκ (xIm ) = (x β2 )|κ|
Jκ(β) (X) =
(β)
m
2β κ
jκ
C (β) (X)
2 |κ|
( β ) |κ|! κ
(3.24)
κ (β)
= Hκ∗ Q(β)
κ (X) = H∗ Pκ (X);
(β)
Jκ(β) (X) = |X| · J(κ
(X)
1 −1,κ2 −1,...,κm −1)
m
Y
(m − i + 1 + β2 (κi − 1)),
(3.25)
(3.26)
i=1
where κm > 0;
X κ C (β) (X)
Cκ(β) (I + X)
σ
=
;
σ Cσ(β) (I)
Cκ(β) (I)
σ⊆κ
Pµ̂(β) (X) = |X|N Pµ(β) (X −1 ),
where µ1 ≤ N and µ̂i = N − µm+1−i , i = 1, 2, . . . , m
(i.e., the partition µ̂ is the complement of µ in (N m ));
(3.27)
(3.28)
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For partitions κ = (k) in only one part,
Ck(β) (I) =
β −1
m
2β k
2
k
X
;
(3.29)
k
(β)
k Cs (X)
s Cs(β) (I)
s=0
k
m X
tk−s k
Cs(β) (X);
=
β
m
s
2
k
β
s
s=0 2
Ck(β) (X + tI) = Ck(β) (I)
tk−s
(3.30)
(3.31)
For partitions κ = (1k )
(β)
(β)
2 k m
J(1
k ) (Im ) = ( β ) ( 2 β)(1k ) =
k−1
Y
(m − i);
(3.32)
i=0
(β)
C(1
k ) (Im ) =
k−1
k−1
Y m−i
1 2 k Y
(m − i) =
( β ) k!
;
jκ
iβ + 1
i=0
i=0 2
for β = 2 this is
m
k
. (3.33)
The references for the above identities are:
(3.23) This is the definition of the normalization Cκ(β) (X) [34, Theorem 2.3];
(3.24) [34, Theorem 5.4];
(3.25) [22, (16)], [28, (10.22), p. 381]. Then Pκ(1) = Q(1)
κ = sκ , the Schur function
[34, Proposition 1.2];
(3.26) [34, Propositions 5.1 and 5.5];
(3.27) [22, (33)]. This also serves as a definition for the generalized binomial coefficient σκ ;
(3.28) This is a result of Macdonald [27, (4.5), (6.22)]: Both sides have xµ̂1 1 · · · xµ̂mm
as the leading term and for the scalar product h· , ·i0n defined in [28,
(10.35)]), h|X|N Pµ (X −1 ), |X|N Pν (X −1 )i0n = hPµ (X), Pν (X)i0n , which is 0
for µ 6= ν [28, (10.36)].
(3.29) Directly from (3.24);
(3.31) Directly from (3.27);
(3.32) Directly from (3.24);
(3.33) Directly from (3.25).
3.4. Multivariate Gamma function identities
We found the following identities useful in simplifying expressions involving the
multivariate Gamma function.
Γ(x)
Γ(β)
m (x)
.
=
Γ x− m
Γm x − β2
2β
(β)
Γ(β)
m (x + a)
= 1.
ma
x→∞ Γ(β)
m (x)x
lim
(3.34)
(3.35)
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The first identity (3.34) is due to Pierre-Antoine Absil in the case β = 1. In general,
from the definition (2.2)
m
Y
Γ x − i−1
Γ(x)
Γ(β)
m (x)
2 β
.
=
=
(β)
i
β
Γ x− m
Γ x − 2β
Γm x − 2
2β
i=1
The second identity, (3.35)], follows directly from the definition (2.2). See also [13].
4. Distributions of the extreme eigenvalues
In this section we review the known explicit formulas for the densities and distributions of the extreme eigenvalues of the beta ensembles.
4.1. Wishart
These results are from [13]. Recall again that r = m−1
2 β + 1. Then
x −1 n2 β
βΣ Γ(β) (r)
· 2tr( x βΣ−1 ) 1 F1(β) r; n2 β + r; x2 βΣ−1 ;
P (λmax (A) < x) = (β) mn
Γm 2 β + r
e 2
and when t ≡
n
2β
(4.1)
− r is a nonnegative integer, then
x
P (λmin (A) < x) = 1 − etr(− 2 βΣ
−1
)
X
κ⊆(mt )
1 (β) x −1
C ( βΣ ).
|κ|! κ 2
(4.2)
It is an open problem to obtain an expression that does not have the requirement
for t to be a nonnegative integer—see section 7.
For the trace we have
∞
nβ x X
dens trA (x) = x2 βΣ−1 2 e− 2z β
k=0
×
X
κ`k
(β)
n
2β κ
1
·
Γ( nm
2 β + k)
1
·
· Cκ(β) (I − zΣ−1 ),
k!
β
2z
k−1
x
2z β
(4.3)
where z is arbitrary. Muirhead [29, p. 341] suggests the value z = 2σ1 σm /(σ1 +σm ),
where σ1 ≥ · · · ≥ σm > 0 are the eigenvalues of Σ.
The second sum in (4.3) is the marginal sum of 1 F0(β) ( n2 β; I − zΣ−1 ) thus the
truncation of (4.3) for |κ| ≤ M can be computed using mhg as follows. If s is the
vector of eigenvalues of Σ, these marginal sums (for say k = 0 through M ) are
returned in the variable c by the call:
[f ,c]=mhg(M ,α,n/α,[],1-z./s),
where α = 2/β, making the remaining computation trivial.
Empirically, the trace can be generated off the Wishart model from [8], which
upon closer observation reveals that
1
tr(A) ∼ (χ2nβ σ1 + χ2nβ σ2 + · · · + χ2nβ σm ).
β
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4.2. Laguerre
The eigenvalues of the Laguerre ensemble are Wishart distributed with Σ = I and
n = a β2 .
Interestingly, we have explicit expressions for the densities of the largest and
smallest eigenvalue which are not related in an obvious way to the distributions.
In these expressions the matrix argument is of size m − 1 making them faster to
compute (fewer partitions to sum over) than the straightforward derivatives of the
distributions.
N EW ST U F F
(4.4)
Γm ( m−1
2 β
(β)
dens λmin (L) (x) =
m
2β
·
+ 1)
e−
2
m
2 β + 1, −t; − βx Im−1 .
Γ(β)
m (a)
× 2 F0(β)
tm
·
x
2β
mx
2 β
(4.5)
The density (4.4) is new; it generalizes the β = 1 result of Sugiyama [36]—see
Theorem ??. The result for the smallest eigenvalue (4.5) is a generalization of the
same result of Krishnaiah and Chang [25] for β = 1. Dumitriu [9, Theorem 10.1.1,
p. 147] established (4.5), but did not provide the scaling constant. We give full
derivation in Theorem 5.5.
Forrester [15] used (3.21) to obtain the following expressions for the smallest
eigenvalue
P (λmin < x) = 1 − e−
mxβ
2
4
)
(β
(−m; 2t/β, −xIt );
1 F1
(β)
dens λmin (L) (x) = mxr e−
mxβ
2
Km−1 (a + β)
1 F1
(β)
Km
(a)
4
(β
)
(4.6)
(−m + 1; 2t/β + 2; −xIt ), (4.7)
(β)
with Km
(a) as in (2.7).
(4)
The expression (4.6) is identical (as a function of x) to (4.2)—the 1 F1 β function
in (4.6) and the truncated 0 F0(β) function in (4.2) are the same polynomial in x of
degree mt.
4.3. Jacobi
Let the m × m matrix C be Jacobi distributed with parameters a, b. Define
(β)
Dm
(a, b) ≡
and recall that r =
m−1
2 β
(β)
Γ(β)
m (a + b)Γm (r)
(β)
Γm (a + r)Γ(β)
m (b)
+ 1. Then [11]
(β)
P (λmax (C) < x) = Dm
(a, b) · xma · 2 F1(β) (a, r − b; a + r; xI);
(4.8)
b−r ma−1
(β)
dens λmax (C) (x) = Dm (a, b) · ma(1 − x)
(β)
× 2 F1 (a −
β
2,r
x
− b; a + r; xIm−1 ).
(4.9)
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When t ≡ b − r is a nonnegative integer, the above expressions are finite and (3.12)
yields the following alternatives:
mt
X
(β)
(a)(β)
κ Cκ ((1 − x)I)
;
k!
k=0 κ`k, κ1 ≤t
Γ a + b − m−1
β Γ(b + β2 Γ β2
2
dens λmax (C) (x) =
m
Γ b − m−2
2 β Γ(t + 1)Γ 2 β Γ(a)
P (λmax (C) < x) = xma
X
× (1 − x)t xma−1 2 F1(β) (a − β2 , −t; −t − β; xIm−1 ).
(4.10)
(4.11)
0
For the smallest eigenvalue we use that the matrix C = I − C is Jacobi distributed with parameters b, a:
P (λmin (C) < x) = 1 − P (λmax (C 0 ) < 1 − x)
dens λmin (C) (x) = dens λmax (C 0 ) (1 − x).
(4.12)
(4.13)
The distribution (4.8) is due to Dumitriu and Koev [11]. The density result (4.9)
is new (see section 5.3). For β = 1, the above results are due to Constantine [6,
(61)] (eq. (4.8)), Absil, Edelman, and Koev [1] (eq. (4.9)), and Muirhead [29, (37),
p. 483] (eq. (4.10)).
5. New results
We now prove the identities (3.2), (3.7), (3.8), and (3.13) (section 5.1), and prove
the formulas for the densities of the smallest eigenvalue of the Jacobi ensemble,
(4.9) (section 5.3) and the Laguerre ensemble (4.5) (section 5.4).
5.1. New identities for p Fq(β)
Theorem 5.1. With the notation from Section 3.1 the identity
(β)
0 F0
(X, Y ) = etrX 0 F0(β) (X, Y − I)
m
= ey1 trX+x1 trY −mx1 y1 1 F1(β) ( m−1
2 β; 2 β; X̄, Ȳ )
(5.1)
holds for any β > 0.
Proof. The first part is from Baker and Forrester [2, sec. 6]. Let Ik be the identity
matrix of size k. Since Cκ(β) (X − x1 Im ) = Cκ(β) (X̄) and analogously for Y ,
(β)
0 F0
(X, Y ) = ey1 trX · 0 F0(β) (X, Y − y1 Im )
= ey1 trX+x1 trY −mx1 y1 · 0 F0(β) (X − x1 Im , Y − y1 Im )
∞ X
X
1 Cκ(β) (X − x1 Im )Cκ(β) (Y − y1 Im )
·
= ey1 trX+x1 trY −mx1 y1 ·
k!
Cκ(β) (Im )
= ey1 trX+x1 trY −mx1 y1 ·
k=0 κ`k
∞ X
X
k=0 κ`k
1 Cκ(β) (Im−1 ) Cκ(β) (X̄)Cκ(β) (Ȳ )
·
·
k! Cκ(β) (Im )
Cκ(β) (Im−1 )
m
= ey1 trX+x1 trY −mx1 y1 · 1 F1(β) ( m−1
2 β; 2 β; X̄, Ȳ ).
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The following two theorems use the fact that ( β2 )(β)
κ = 0 for partitions κ in more
than one part.
Theorem 5.2. Let the matrices X and I be m × m. Let t and n be real numbers
and let β > 0. The following identities hold:
β m
2 ; 2 β; X + tI)
(β) β
m
2 F1 ( 2 , n; 2 β; X)
(β)
1 F1
β m
t
2 ; 2 β; X) · e ;
t)n 2 F1(β) ( β2 , n; m
2 β; (1
= 1 F1(β)
= (1 −
(5.2)
− t)X + tI),
t 6= 1.
Proof. We transform the left hand side of (5.2) using (3.29) and (3.31):
β
∞
X
2
1
(β) β m
· m k · Ck(β) (X + tI)
1 F1
2 ; 2 β; X + tI =
k!
2β k
k=0
∞
X
1 Ck(β) (X + tI)
·
k!
Ck(β) (I)
k=0
∞
k
X
1 X k−s
k
C (β) (X)
t
·
· s(β)
=
s
k! s=0
Cs (I)
k=0
∞
∞
(β)
X C (X) X tj−i j i
=
·
·
(β)
j!
i
C
i (I)
i=0
j=i
β
∞
∞
X
X
2
1
tj−i
=
· m k · Ck(β) (X) ·
i!
(j − i)!
2β k
i=0
j=i
=
= 1 F1(β)
β m
2 ; 2 β; X)
· et .
We use (3.31) again to obtain (5.3):
(1 − t)n 2 F1(β) β2 , n; m
2 β; (1 − t)X + tI
β
∞
(n)k
X
2
1
k = (1 − t)n
·
· Ck(β) ((1 − t)X + tI)
m
k!
β
2
k
k=0
(β)
∞
k
X (n)k X
k Cs ((1 − t)X)
= (1 − t)n
tk−s
k! s=0
s
Cs(β) (I)
k=0
∞
∞ j−i X
X
j
Ci(β) (X)
t
n
i
= (1 − t)
(1 − t)
(n)j
(β)
j!
i
C
(I)
i
i=0
j=i
=
∞
∞ l
X
X
1 Ci(β) (X)
t
· (β)
(n)i · (1 − t)n+i
(n + i)l
i! Ci (I)
l!
i=0
l=0
|
{z
}
1
= 2 F1(β) ( β2 , n; m
2 β; X).
(5.3)
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Theorem 5.3.
(β)
p Fq
( β2 , a2:p ; b1:q ; xI) = p Fq ( m
2 β, a2:p ; b1:q ; x).
Proof. Using (3.29),
(β) β
p Fq ( 2 , a2:p ; b1:q ; xI) =
=
∞
β
X
1 ( 2 )k (a2 )k · · · (ap )k (β)
·
Ck (xI)
k!
(b1 )k · · · (bq )k
k=0
∞
X
k=0
β
β)k
1 ( 2 )k (a2 )k · · · (ap )k k ( m
·
x 2β
k!
(b1 )k · · · (bq )k
( 2 )k
= p Fq ( m
2 β, a2:p ; b1:q ; x).
5.2. Density of the largest eigenvalue of the Jacobi ensemble
We repeat the argument in [1] to extend the result for the density of the smallest
eigenvalue of the Jacobi ensemble to any β > 0 (see equations (4.9) and (4.13)). We
start with the joint density of the eigenvalues of the Jacobi ensemble from Table 2.2:
m
Y
Y
1
λa−r
(1 − λi )b−r
|λi − λj |β dλ1 · · · dλm .
(β)
i
Sm (a, b) i=1
i<j
dens(λ1 , λ2 , . . . , λm ) ≡
where, as before, r ≡
m−1
2 β
+ 1. Let x ≡ λ1 ≥ λ2 ≥ · · · λ1 .
Z
dens(x) = m
dens(x, λ2 , . . . , λm )dλ2 · · · dλm
Z
Y
a−r
b−r
|λi − λj |β
x
(1 − x)
[0,x]m−1
=
m
Sm (a, b)
(β)
×
m
Y
[0,x]m−1 1<i<j
|x − λi |β λa−r
(1 − λi )b−r dλ2 · · · dλm .
i
i=2
We change variables λi = xti , i = 2, 3, . . . , m.
dens(x) =
m
xam−1 (1 − x)b−r
(β)
Sm
(a, b)
×
Z
|ti − tj |β
[0,1]m−1
m−1
Y
(1 − xti )b−r ta−r
(1 − ti )β dt2 · · · dtm .
i
i=1
From (3.19) we have for T ≡ diag(t2 , t3 , . . . , tm−1 ), a matrix of size m − 1,
Z
Y
1
(β)
β
m+1
r−b, a+ 2 ; a+ 2 β+1; xIm−1 =
|ti −tj |β
2 F1
(β)
β
β
[0,1]m−1 i<j
Sm−1
+
r,
b
−
2
2
×
m
Y
i=2
(1 − xti )b−r ta−r
(1 − ti )β dt2 · · · dtm−2 .
i
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Qm
(Note that 1 F0(β) (r − a, xIm−1 , T ) = 1 F0(β) (r − a, xT ) = i=2 (1 − xti )a−r . )
Therefore
(β)
β
β
Sm−1
+
r,
b
−
2
2
dens(x) = m
x(a−r)m (1 − x)mb−1
(β)
Sm
(a, b)
× 2 F1(β) −a + r, β2 + r; b + r; x−1
I
,
m−1
x
which using (3.11) gives
(β)
Sm−1
β
2
+ r, b −
β
2
x(a−r)m (1 − x)mb−1
(β)
(a, b)
Sm
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 x−(a−r)(m−1)
(β)
β
β
Sm−1
+
r,
b
−
2
2
xa−r (1 − x)mb−1
=m
(β)
(a, b)
Sm
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 ,
dens(x) = m
(β)
= mb · Dm
(b, a) · xa−r (1 − x)mb−1
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 ,
which yields (4.9) and (4.13).
5.3. Density of the smallest eigenvalue of the Jacobi ensemble
We repeat the argument in [1] to extend the result for the density of the smallest
eigenvalue of the Jacobi ensemble to any β > 0 (see equations (4.9) and (4.13)). We
start with the joint density of the eigenvalues of the Jacobi ensemble from Table 2.2:
dens(λ1 , λ2 , . . . , λm ) ≡
m
Y
Y
1
λa−r
(1 − λi )b−r
|λi − λj |β dλ1 · · · dλm .
(β)
i
Sm (a, b) i=1
i<j
where, as before, r ≡ m−1
2 β + 1. We integrate all but the smallest eigenvalue (call
it x) out of the above density:
Z
dens(x) = m
dens(λ1 , . . . , λm−1 , x)dλ1 · · · dλm−1
[x,1]m−1
Z
Y
m
= (β)
xa−r (1 − x)b−r
|λi − λj |β
Sm (a, b)
[x,1]m−1 i<j≤m−1
×
m−1
Y
i=1
|λi − x|β λa−r
(1 − λi )b−r dλ1 · · · dλm−1 .
i
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We change variables λi = (1 − x)ti + x, i = 1, 2, . . . , m − 1.
Z
m
x(a−r)m (1 − x)mb−1
|ti − tj |β
dens(x) = (β)
Sm (a, b)
[0,1]m−1
a−r
m−1
Y β
x−1
×
ti
(1 − ti )b−r dt1 · · · dtm−1 .
ti 1 −
x
i=1
From (3.19) we have for T ≡ diag(t1 , t2 , . . . , tm−1 ), a matrix of size m − 1,
Z
Y
1
(β)
β
|ti − tj |β
r − a, 2 + r; b + r; zIm−1 =
2 F1
(β)
β
β
[0,1]m−1 i<j
Sm−1
+
r,
b
−
2
2
×
m−1
Y
(1 − zti )a−r tβi (1 − ti )b−r dt1 · · · dtm−1 .
i=1
Qm−1
(Note that 1 F0(β) (−a + r, zIm−1 , T ) = 1 F0(β) (−a + r, zT ) = i=1 (1 − zti )a−r . )
Now, for z = x−1
x
(β)
β
β
Sm−1
+
r,
b
−
2
2
x(a−r)m (1 − x)mb−1
dens(x) = m
(β)
(a, b)
Sm
× 2 F1(β) −a + r, β2 + r; b + r; x−1
x Im−1 ,
which using (3.11) gives
(β)
Sm−1
β
2
+ r, b −
β
2
x(a−r)m (1 − x)mb−1
(β)
(a, b)
Sm
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 x−(a−r)(m−1)
(β)
β
β
Sm−1
2 + r, b − 2
=m
xa−r (1 − x)mb−1
(β)
Sm
(a, b)
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 ,
dens(x) = m
(β)
= mb · Dm
(b, a) · xa−r (1 − x)mb−1
× 2 F1(β) −a + r, b − β2 ; b + r; (1 − x)Im−1 ,
which yields (4.9) and (4.13).
5.4. The density of the smallest eigenvalue of the Laguerre
ensemble
We will obtain this result as a limiting argument from the density of the smallest
eigenvalue of the Jacobi ensemble. The connection is that if λi are the eigenvalues
2bλi
of a Jacobi matrix, then λ̄i ≡ limb→∞ β(1−λ
are the eigenvalues of a Laguerre
i)
matrix.
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Proposition 5.1. Let L(β)
m (a; Λ) be the joint eigenvalue density of the Laguerre
(β)
ensemble and Jm (a, b; Λ) be the joint eigenvalue density of the Jacobi ensemble as
defined in Table 2.2. Then
1
(β)
lim 2 m · Jm
a, b; Λ( β2 bI + Λ)−1 = L(β)
m (a; Λ).
b→∞ ( b)
β
Proof.
(β)
Jm
a, b; Λ( β2 bI + Λ)−1
( β2 b)m
=
1
π
=
1
· (β)
(a, b)
( β2 b)ma Sm
m
Y a− m−1 β−1
×
λi 2
1
i=1
m(m−1)
β
2
Γ
( β2 )ma m!Γ(β)
m
×
m
Y
β
2
+
2−a−b
β
2b λi
Y
|λj − λk |β
j<k
m
Γ(β)
m (a + b)
(β)
m
ma
Γ(β)
m (b)b
2 β Γm (a)
a− m−1
2 β−1
λi
1+
·
2−a−b
λi
2b β
i=1
Y
|λj − λk |β .
j<k
We then take the limit as b → ∞ and use (3.35) to obtain the desired result.
Next, we derive the density of the largest eigenvalue of the Laguerre ensemble.
Theorem 5.4. The density of the largest eigenvalue of the Laguerre ensemble is
nmβ
(β)
2 −1
nmβ 2 Γm (r) − mβx βx
(β) mβ
2
e
+ 1; nβ
+ r; xβ
Im−1 .
(5.4)
1 F1
2
2
2
(β) nβ
2
4Γm
+r
2
Proof. The density of the maximum eigenvalue of the Jacobi ensemble with parameters a and b is given by
(β)
(β)
β
Γm (a + b)Γm (r)
(β)
b−r ma−1
a
−
f (x) = (β)
ma(1
−
x)
x
F
,
r
−
b;
a
+
r;
xI
2 1
m−1
(β)
2
Γm (a + r)Γm (b)
(β)
(β)
Γm (a + b)Γm (r)
β
x
(β)
b−1−a+ β
2 xma−1 F
= (β)
ma(1
−
x)
a
−
,
a
+
b;
a
+
r;
−
I
(5.5)
2 1
m−1
(β)
2
1−x
Γm (a + r)Γm (b)
where r = (m − 1)β/2 + 1. Let y = (2b/β)x/(1 − x), we have
1
βy
= 1+
,
1−x
2b
x=
dx =
1
βy
2b
+ βy
2b
,
β(1 − x)2
dy,
2b
(5.6)
(5.7)
(5.8)
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COMPUTING WITH BETA ENSEMBLES
19
and the density function of y is given by
(β)
(β)
β
βy
Γm (a + b)Γm (r) maβ
(β)
b+1−a− β
2 xma−1 F
(1
−
x)
a
−
,
a
+
b;
a
+
r;
−
I
2 1
m−1
(β)
(β)
2
2b
Γm (a + r)Γm (b) 2b
ma−1 −(m−1)a−b− β2
(β)
(β)
b−ma Γm (a + b)Γm (r) maβ βy
βy
=
1+
(β)
(β)
2
2
2b
Γm (b)Γm (a + r)
βy
β
(β)
× 2 F1
(5.9)
a − , a + b; a + r; − Im−1 .
2
2b
Using the fact that
f (y) =
(β)
lim
b−ma Γm (a + b)
b→∞
(β)
= 1,
(5.10)
Γm (b)
−(m−1)a−b− β2
βy
βy
lim 1 +
= e− 2 ,
(5.11)
b→∞
2b
β
βy
βy
β
(β)
(β)
a − , a + b; a + r; − Im−1 = 1 F1
lim 2 F1
a − ; a + r; − Im−1
b→∞
2
2b
2
2
βy(m−1)
β
βy
(β)
−
2
=e
r + ; a + r;
I(5.12)
1 F1
m−1
2
2
we have
ma−1
(β)
maβ
βy
βy
mβ
Γm (r)
(β)
− mβy
2
+ 1; a + r;
Im−1 ,
e
lim f (y) = (β)
1 F1
b→∞
2
2
2
2
Γm (a + r)
(5.13)
and this is the density of the maximum eigenvalue of Laguerre ensemble with parameter a. Putting a = nβ/2, we prove (5.4).
We now derive the scaling constant in the formula for the density of the Laguerre
ensemble, (4.5). The proportionality result is due to Dumitriu [9, Theorem 10.1.1,
p. 146].
Theorem 5.5. When t ≡ a − m−1
2 β − 1 is a nonnegative integer, the density of
the smallest eigenvalue of a Laguerre matrix is
densλmin (A) (y) =
m−1
my
Γ(β)
m ( 2 β + 1)
· e− 2 β ( y2 β)mt
(β)
Γm (a)
2
× 2 F0(β) ( m
2 β + 1, −t; − βy Im−1 ).
m
2β
·
(5.14)
Proof. Let C be Jacobi distributed. From (4.9), (4.13), and the identity (3.11) we
have
(β)
densλmin (C) (x) = mb · Dm
(b, a) · (1 − x)mb−1 xt
× 2 F1(β) (b − β2 , −t; b +
(β)
m−1
2 β + 1; (1
mb−1 mt
= mb · Dm (b, a) · (1 − x)
(β)
× 2 F1
(m
2β
+ 1, −t; b +
− x)Im−1 )
x
m−1
2 β
+ 1; − 1−x
x Im−1 ).
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DRENSKY, EDELMAN, GENOAR, KAN, and KOEV
β
(with the Jacobian being 2b
(1 − x)2 ) we get:
m−1
−mt
Γ(β)
Γ(β)
m
m (a + b) · b
2 β+1
m
densλmin ( 2b C(I−C)−1 ) (y) = 2 β ·
·
m−1
β
Γ(β)
Γ(β)
m (a)
m b+ 2 β+1
−mb−1 y mt
−mt
× 1 + βy
1 + βy
2b
2β
2b
Substituting y =
2b
β
·
x
1−x
× 2 F1(β) ( m
2 β + 1, −t; b +
m−1
2 β
2
+ 1; b(− βy
)Im−1 ).
We use Proposition 5.1, (3.17), and (3.35) to take the limit as b → ∞ and obtain (5.14).
6. Numerical experiments
All formulas for the distributions and densities in this paper can be approximated
(within the limits of the computational power of modern computers) using the
software mhg [24].
It returns the truncation
M X
(β)
X
C (β) (X)C (β) (Y )
1 (a1 )(β)
κ · · · (ap )κ
· κ (β) κ
·
.
(β)
(β)
k! (b1 )κ · · · (bq )κ
Cκ (I)
k=0 κ`k
of (2.5) as well as the partial sums
X 1 (a1 )(β) · · · (ap )(β) C (β) (X)C (β) (Y )
κ
κ
· κ (β) κ
·
(β)
k! (b1 )(β)
Cκ (I)
κ · · · (bq )κ
κ`k
for k = 0, 1, . . . , M with the additional option to restrict the summation for κ1 ≤ t
(for, e.g., (4.2)). All expressions in this paper are trivially approximated then using
these partial sums.
We performed extensive numerical tests to verify the correctness of the formulas
in this paper and present four examples in Figure 1.
7. Open problems and future work
We finish with two open problems.
Several formulas in this paper, e.g., (4.2), (4.10), are only valid when certain
parameters are integers. It is an open problem to derive formulas that do not have
that restriction. The central problem there is to find an explicit expression (perhaps
in terms of p Fq(β) ) for the integral
Z
(β)
a
b
0 F0 (−X, Y )|X| |I + X| dµ(X).
R+
m
This is the multivariate version of the function Ψ, the confluent hypergeometric
function of the second kind.
The second open problem we pose is to find explicit formulas for the extreme
eigenvalues of the Hermite ensemble, which are analogous to the ones for the other
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Wishart, =4/3, =diag(0.7,1,1.3), n=5
Laguerre, =4/3, m=3, n=5
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
min
min
0.3
min
0.2
max
0.1
max
theoretical CDF
min
empirical CDF
0.3
min
theoretical CDF
0.2
min
empirical CDF
0.1
max
max
0
0
5
10
15
20
21
0
0
25
5
10
x
15
theoretical CDF
empirical CDF
theoretical PDF
empirical PDF
theoretical CDF
empirical CDF
20
25
x
Trace of Wishart, =4, =diag(0.5,0.7,1,1.4), n=7
max
of Jacobi, =3, a=8, b=11, m=4
6
0.1
0.09
theoretical PDF
empirical PDF
0.08
5
0.07
theoretical PDF
empirical PDF
theoretical CDF
empirical CDF
4
0.06
3
0.05
0.04
2
0.03
0.02
1
0.01
0
0
10
20
30
40
50
60
0
0
0.2
0.4
0.6
0.8
1
Fig. 1. Numerical experiments comparing the theoretical predictions of the eigenvalue distributions
for various ensembles vs. the empirical results.
ensembles. This problem boils down to evaluating the integral
Z
e−
tr(X 2 )
2
dµ(X).
[x,∞)m
Bornemann [4] presents algorithms for numerical evaluation of distributions in
Random Matrix Theory that are expressible as Painlevé transcendents or Fredholm
determinants in the β = 1, 2, 4 cases. It is an open problem to devise good numerical
routines for the general β versions of some of these quantities, especially the bulk,
hard edge, and soft edge statistics.
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DRENSKY, EDELMAN, GENOAR, KAN, and KOEV
Acknowledgments
We thank Iain Johnstone, Tom Koornwinder, Richard Stanley, and Brian Sutton
for insightful discussions in the process of preparation of this paper.
This research was supported by the National Science Foundation Grant DMS–
1016086 and by the Woodward Fund for Applied Mathematics at San José State
University. The Woodward Fund is a gift from the estate of Mrs. Marie Woodward
in memory of her son, Henry Teynham Woodward. He was an alumnus of the
Mathematics Department at San José State University and worked with research
groups at NASA Ames.
References
[1] Pierre-Antoine Absil, Alan Edelman, and Plamen Koev, On the largest principal
angle between random subspaces, Linear Algebra Appl. 414 (2006), 288–294.
[2] T. H. Baker and P. J. Forrester, The Calogero-Sutherland model and generalized
classical polynomials, Comm. Math. Phys. 188 (1997), no. 1, 175–216.
[3] Christopher Bingham, An antipodally symmetric distribution on the sphere, Ann.
Statist. 2 (1974), 1201–1225.
[4] F. Bornemann, On the numerical evaluation of distributions in random matrix theory:
a review, Markov Process. Related Fields 16 (2010), no. 4, 803–866.
[5] R. W. Butler and A. T. A. Wood, Laplace approximations for hypergeometric functions with matrix argument, Ann. Statist. 30 (2002), no. 4, 1155–1177.
[6] A. G. Constantine, Some non-central distribution problems in multivariate analysis,
Ann. Math. Statist. 34 (1963), 1270–1285.
[7] Alexander Dubbs and Alan Edelman, The Beta-MANOVA ensemble with general
covariance, arXiv:1309.4328, 2013.
[8] Alexander Dubbs, Alan Edelman, Plamen Koev, and Praveen Venkataramana, The
Beta–Wishart ensemble, J. Math. Phys. 54 (2013), 083507.
[9] Ioana Dumitriu, Eigenvalue statistics for the Beta-ensembles, Ph.D. thesis, Massachusetts Institute of Technology, 2003.
[10] Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys.
43 (2002), no. 11, 5830–5847.
[11] Ioana Dumitriu and Plamen Koev, The distributions of the extreme eigenvalues of
beta–Jacobi random matrices, SIAM J. Matrix Anal. Appl. 1 (2008), 1–6.
[12] Alan Edelman, The random matrix technique of ghosts and shadows, Markov Processes and Related Fields 16 (2010), no. 4, 783–790.
[13] Alan Edelman and Plamen Koev, Eigenvalues distributions of Beta–Wishart matrices, Submitted to Random Matrices: Theory and Applications.
[14] Alan Edelman and Brian D. Sutton, The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems, Found. Comput. Math. 8 (2008),
no. 2, 259–285.
[15] P. J. Forrester, Exact results and universal asymptotics in the Laguerre random matrix ensemble, J. Math. Phys. 35 (1994), no. 5, 2539–2551.
[16] Peter J. Forrester, Log-gases and random matrices, London Mathematical Society
Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010.
, Probability densities and distributions for spiked Wishart β-ensembles,
[17]
arXiv:1101.2261, 2011.
[18] Kenneth I. Gross and Donald St. P. Richards, Total positivity, spherical series, and
January 25, 2014
13:18
WSPC/INSTRUCTION FILE
mvs-paper
COMPUTING WITH BETA ENSEMBLES
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
23
hypergeometric functions of matrix argument, J. Approx. Theory 59 (1989), no. 2,
224–246.
Rameshwar D. Gupta and Donald St. P. Richards, Hypergeometric functions of scalar
matrix argument are expressible in terms of classical hypergeometric functions, SIAM
J. Math. Anal. 16 (1985), no. 4, 852–858.
Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295–327.
, Multivariate analysis and Jacobi ensembles: largest eigenvalue, TracyWidom limits and rates of convergence, Ann. Statist. 36 (2008), no. 6, 2638–2716.
MR 2485010 (2010f:62146)
Jyoichi Kaneko, Selberg integrals and hypergeometric functions associated with Jack
polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086–1110.
Rowan Killip and Irina Nenciu, Matrix Models for Circular Ensembles, Int. Math.
Res. Not. 50 (2004), 2665–2701.
Plamen Koev and Alan Edelman, The efficient evaluation of the hypergeometric function of a matrix argument, Math. Comp. 75 (2006), no. 254, 833–846.
P. R. Krishnaiah and T. C. Chang, On the exact distribution of the smallest root
of the Wishart matrix using zonal polynomials, Ann. Inst. Statist. Math. 23 (1971),
293–295.
Ross A. Lippert, A matrix model for the β-Jacobi ensemble, J. Math. Phys. 44 (2003),
4807–4816.
Ian G. Macdonald, Hypergeometric functions I, arXiv:1309.4568.
, Symmetric functions and Hall polynomials, Second ed., Oxford University
Press, New York, 1995.
R. J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons Inc.,
New York, 1982.
A. Yu. Orlov, New solvable matrix integrals, Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models, vol. 19, 2004, pp. 276–293.
T. Ratnarajah, R. Vaillancourt, and M. Alvo, Eigenvalues and condition numbers of
complex random matrices, SIAM J. Matrix Anal. Appl. 26 (2004/05), no. 2, 441–456.
Atle Selberg, Remarks on a multiple integral, Norsk Mat. Tidsskr. 26 (1944), 71–78.
Jack W. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix,
Ann. Probab. 13 (1985), no. 4, 1364–1368.
Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv.
Math. 77 (1989), no. 1, 76–115.
, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999.
T. Sugiyama, On the distribution of the largest latent root and the corresponding latent
vector for principal component analysis, Ann. Math. Statist. 37 (1966), 995–1001.