On spectral measures of beta ensembles
Trinh Khanh Duy
Institute of Mathematics for Industry,
Kyushu University
Random matrix theory and strongly correlated systems
21–24 March 2016, University of Warwick
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
1 / 32
Outline
1
Motivation
2
Jacobi matrices
3
Gaussian beta ensembles
4
Wishart beta ensembles
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Random Jacobi matrices
• Random Jacobi matrix

a1
b1

J=


b1
a2
..
.
b2
..
.
bN−1
(
 {a }N : real random variables,

i i=1
,
.. 

.
{bj }N−1
j=1 : positive random variables.
aN
• Empirical distributions LN
N
1 X
LN =
δλj ,
N
{λ1 , . . . , λN }: (distint) eigenvalues of J .
j=1
w
• Asymptotic behaviour of LN as N → ∞, i.e., LN → ∃ µ∞ , or
hLN , f i =
N
1 X
f (λj ) → h∃ µ∞ , f i (f : bounded continuous function).
N
j=1
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Spectral measures of beta ensembles
2016/03/21
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Gaussian beta ensembles
• Gaussian beta ensembles.

(β)
HN = √

N (0,2) χ(N−1)β
χ
1 
 (N−1)β

βN
N (0,2) χ(N−2)β
..
.
..
.
χβ
..
.

.

N (0,2)
• Joint probability density function of the eigenvalues
(λ1 , . . . , λN ) ∝
Y
|λi − λj |β exp(−
i<j
• Semi-circle law. µ∞ = sc(x) =
N
βN X 2
λj ).
4
j=1
√
4 − x 2 /(2π), |x| ≤ 2,
√
Z 2
N
1 X
4 − x2
dx
f (λj ) →
f (x)
N
2π
−2
almost surely.
j=1
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Spectral measures of beta ensembles
2016/03/21
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Empirical distributions
Example
• Gaussian beta ensembles (GβE). µ∞ = sc: semi-circle law;
• Wishart beta ensembles (WβE). µ∞ = mp: Marchenko-Pastur law.
• Fluctuations of eigenvalues around the limit:
• GβE. (Johansson 98), for a ‘nice’ function f ,
N X
d
f (λj ) − hsc, f i → N (0, af2 ).
j=1
• WβE. (Dumitriu & Edelman 2006), for polynomial p,
N X
d
p(λj ) − hmp, pi → N (0, abp2 ).
j=1
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Spectral measures of beta ensembles
2016/03/21
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Spectral measures
• µN : spectral measure of (J, e1 ) if
hµN , x k i = (J k e1 , e1 ) = J k (1, 1), k = 0, 1, . . . .
• µN =
N
X
qj2 δλj . 1-1 correspondence with Jacobi matrix.
j=1
Main resutls: Law of large numbers and central limit theorem.
• GβE.
LLN. hµN , f i → hsc, f i, (f : bounded continuous function);
√
d
CLT .
N(hµN , pi − E[hµN , pi]) → N (0, σp2 ), (p: polynomial).
• WβE.
LLN. hµN , f i → hmp, f i, (f : bounded continuous function);
√
d
CLT .
N(hµN , pi − E[hµN , pi]) → N (0, σ
bp2 ), (p: polynomial).
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
6 / 32
Outline
1
Motivation
2
Jacobi matrices
3
Gaussian beta ensembles
4
Wishart beta ensembles
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
7 / 32
Spectral measures of Jacobi matrices
• Given a Jacobi matrix J, finite or infinite

a1
b1
J=
b1
a2
..
.

b2
..
.
..

,
ai ∈ R, bi > 0.
.
• There is a measure µ on R, called spectral measure of (J, e1 ), s.t.
hµ, x k i = (J k e1 , e1 ) = J k (1, 1), k = 0, 1, . . .
• Uniqueness is equivalent to the essential seft-adjointness of J on
`2 (N).
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Jacobi matrices
µ: nontrivial prob. meas. on R s.t.
R
|x|k dµ(x) < ∞, k = 0, 1, . . . .
• {1, x, x 2 , . . . } are independent in L2 (R, µ).
• Define
{Pn (x)}∞
n=0
(
Pn (x) = x n + lower order,
as
Pn ⊥ x j , j = 0, . . . , n − 1.
• pn := Pn /kPn kL2 .
Theorem
(i) xpn (x) = bn+1 pn+1 (x) + an+1 pn (x) + bn pn−1 (x),
where bn+1 =
n = 0, 1, . . . ,
kPn k
kPn+1 k , an+1
=
hPn ,xPn i
kPn k2 , P−1
≡ 0.
(ii) Multiplication by x in the orthonormal set {pj } has the matrix


a1 b1


J = b1 a2 b2

.. ..
..
.
.
.
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Jacobi matrices
• µ: trivial prob. meas.,i.e.,
µ=
N
X
qj2 δλj ,
j=1
•
N−1
{x j }j=0
: independent
pn := Pn /kPn k;
(
{λj } : distinct,
P 2
qj = 1, qj > 0.
in L2 (R, µ). Define P0 , . . . , PN−1 .
•

a1
b1

J=

b1
a2
..
.

b2
..
.
bN−1


.. 
.
aN
N
• {λj }N
j=1 : the eigenvalues of J, {vj }j=1 : the corresponding normalized
eigenvectors. Then
µ=
N
X
j=1
Trinh Khanh Duy (IMI, Kyushu University)
|vj (1)|2 δλj =
N
X
qj2 δλj .
j=1
Spectral measures of beta ensembles
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Jacobi matrix of semicircle distribution
• Semicircle distribution sc with density
sc(x) =
•
Trinh Khanh Duy (IMI, Kyushu University)
1 p
4 − x 2,
2π

0
1
J=
1
0
..
.
|x| ≤ 2.

1
..
.
..
Spectral measures of beta ensembles

.
.
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Outline
1
Motivation
2
Jacobi matrices
3
Gaussian beta ensembles
4
Wishart beta ensembles
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Gaussian orthogonal ensemble
GOE(N)

(1)
GN
N (0, 2) N (0, 1) N (0, 1) . . .
 ∗
N (0, 2) N (0, 1) . . .

 ∗
∗
N (0, 2) . . .
=
 ..
..
..
..
 .
.
.
.
∗
∗
∗
...

N (0, 1)
N (0, 1)

N (0, 1)

.. 
. 
N (0, 2)
• Invariant under orthogonal conjugation,i.e.,
(1)
d
(1)
HGN H t = GN ,
H : (deterministic) orthogonal matrix.
• Joint distribution of the eigenvalues:
N
(λ1 , . . . , λN ) ∝
Y
|λi − λj | exp(−
i<j
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Spectral measures of beta ensembles
1X 2
λj ).
4
j=1
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Gaussian unitary ensemble
GUE(N)
(2)
GN

N (0, 1)

 ∗
=
 .
 ..
∗
N (0,1)+i N (0,1)
√
2
...
N (0, 1)
..
.
∗
...
..
.
...
N (0,1)+i N (0,1)
√
2
N (0,1)+i N (0,1) 

√

2

..
.
N (0, 1)


• Invariant under unitary conjugation,i.e.,
(2)
d
(2)
UGN U ∗ = GN ,
U : (deterministic) unitary matrix.
• Joint distribution of the eigenvalues:
N
(λ1 , . . . , λN ) ∝
Y
i<j
Trinh Khanh Duy (IMI, Kyushu University)
2X 2
λj ).
|λi − λj | exp(−
4
2
Spectral measures of beta ensembles
j=1
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Gaussian beta ensembles
GβE (β > 0)
•
(λ1 , . . . , λN ) ∝
Y
|λi − λj |β exp(−
i<j
N
βX 2
λj ).
4
j=1
• β = 1, GOE.
• β = 2, GUE.
• β = 4, GSE (Gaussian symplectic ensemble).
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Spectral measures of beta ensembles
2016/03/21
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GOE, Jacobi/tridiagonal matrix model (Dumitriu & Edelman 2002)
•
(1)
GN
a1 x t
=
,
x B


a1 ∼ N (0, 2),
x t ∼ (N (0, 1) . . . N (0, 1)),

 d (1)
B = GN−1 .
• H: (N − 1) × (N − 1) orthogonal matrix (depending only on x) s.t.
Hx = (kxk2 0 . . . 0)t = kxk2 e1 .
•
1 0
a1 x t
1 0
a1
kxk2 e1t
=
0 H
x B
0 Ht
kxk2 e1 HBH t

a1 kxk2
0
kxk2

d 
= 0
 ..
(1)
 .
G
...
0







N−1
0
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Spectral measures of beta ensembles
2016/03/21
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GOE, Jacobi/tridiagonal matrix model (Dumitriu & Edelman 2002)
• a1 ∼ N (0, 2).
• kxk2 =
N (0, 1)2 + · · · + N (0, 1)2
|
{z
}
1/2
∼ χN−1 : chi distribution.
N−1
•

(1) 1st step
GN
N (0, 2) χN−1
 χ
 N−1
 ..
 .
0
(1)
GN−1
...

0




0

finally
Trinh Khanh Duy (IMI, Kyushu University)
N (0, 2)
χN−1
 χ
N
(0, 2) χN−2
 N−1

..
..

.
.
χ1
Spectral measures of beta ensembles



.. 
. 
N (0, 2)
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GOE, Jacobi/tridiagonal matrix model (Dumitriu & Edelman 2002)
• ∃ H: (random) orthogonal matrix s.t.
1 0 (1) 1 0
GN
0 Ht
0 H

N (0, 2)
χN−1
 χ
N
(0, 2) χN−2
N−1
(1) d 
=: JN = 
..
..

.
.
χ1
(1)
• The eigenvalues of GN
•
Trinh Khanh Duy (IMI, Kyushu University)


.. 
. 
N (0, 2)
(1)
are the same as those of JN .
(1) k
GN

(1) k
(1, 1) = JN
(1, 1).
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GβE, Jacobi/tridiagonal matrix model (Dumitriu & Edelman 2002)

(β)
• HN
:=
√1
βN

N (0,2) χ(N−1)β
 χ(N−1)β


N (0,2) χ(N−2)β
..
.
..
.
χβ
(β)
• The eigenvalues of HN
..
.



N (0,2)
have (scaled) GβE distribution,i.e.,
(λ1 , . . . , λN ) ∝
Y
|λi − λj |β exp(−
i<j
N
βN X 2
λj ).
4
j=1
• (q1 , . . . , qN ) is distributed as (χβ , . . . , χβ ) normalized to unit length,
independent of (λ1 , . . . , λN ).
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Spectral measures of beta ensembles
2016/03/21
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Empirical distributions: semi circle law & CLT (Johansson ’98)
• Empirical distributions
(β)
LN
N
X
1
δλ ,
:=
N j
(λ1 , . . . , λN ) : (scaled) GβE.
j=1
• f : R → R: bounded continuous function,
(β)
hLN , f i =
N
1 X
f (λj ) → hsc, f i a.s. as N → ∞.
N
j=1
• f : ‘nice’ function
N
X
d
(f (λj ) − hsc, f i) → N (0, af2 ),
j=1
where af2 is a quadratic functional of f .
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Spectral measures of beta ensembles
2016/03/21
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Spectral measures of GβE
• Empirical distributions
(β)
LN :=
N
X
1
δλ .
N j
j=1
• Spectral measures
(β)
µN =
N
X
qj2 δλj .
j=1
• E[qj2 ] =
1
N.
(q1 , . . . , qN ) independent of (λ1 , . . . , λN ).
Consequently, L̄N = µ̄N , namely,
E[hµN , f i] =
N
X
j=1
Trinh Khanh Duy (IMI, Kyushu University)
E[qj2 ]E[f (λj )]
N
X
1
E[f (λj )] = E[hLN , f i].
=
N
j=1
Spectral measures of beta ensembles
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Semicircle law and CLT (Dette & Nagel (2012), D. 2015–)
Theorem
(i) The spectral measures µN converges weakly to the semicircle
distribution, almost surely, i.e.,
hµN , f i → hsc, f i a.s. as N → ∞.
(ii) For any polynomial p of positive degree,
p d
βN hµN , pi − E[] −→ N (0, σp2 ) as N → ∞,
where σp2 > 0 (does not depend on β).
• Recall that for a ‘nice’ function f ,
(β)
a.s.
hLN , f i −→ hsc, f i as N → ∞.
d
(β)
N hLN , f i − hsc, f i −→ N (0, af2 ) as N → ∞.
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Spectral measures of beta ensembles
2016/03/21
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Related results: spectral measures of Wigner matrices
• {ξii }i : i.i.d. E[ξ11 ] = 0; E[|ξ11 |k ] < ∞, k = 2, 3, . . . .
2 ] = 1; E[|ξ |k ] < ∞, k = 3, 4, . . . .
• {ξij }i<j : i.i.d. E[ξ12 ] = 0, E[ξ12
11
• XN : symmetric matrix

ξ11 ξ12 ξ13
 ∗ ξ22 ξ23
1 

∗ ξ33
XN := √  ∗
..
..
..
N
 .
.
.
∗
∗
∗
...
...
...
..
.

ξ1N
ξ2N 

ξ3N 

.. 
. 
...
ξNN
called a (real) Wigner matrix.
• νN : spectral measure, i.e., hνN , x k i = XNk (1, 1).
• νN converges weakly to the semicircle law.
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Spectral measures of beta ensembles
2016/03/21
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Central limit theorems
(i) k = 1, hνN , xi = XN (1, 1) =
√
ξ11
√
.
N
Therefore
d
N (hνN , xi − E[hνN , xi]) = ξ11 −→ ξ11 .
In general, the limit distribution is not Gaussian.
P
1 2
(ii) k = 2, hνN , x 2 i = XN2 (1, 1) = N
i=1 N ξ1i . We consider
√
N hνN , x 2 i − E[hνN , x 2 i]
√
2 − 1) + (ξ 2 − 1) + · · · + (ξ 2 − 1)
2 − E[ξ 2 ]
ξ11
N − 1 (ξ12
13
11
1N
√
√
=
+ √
N
−
1
N
N
d
2
−→ N (0, E[(ξ12
− 1)2 ]).
The limit distribution is Gaussian!!
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
24 / 32
Spectral measures of Wigner matrices
Theorem (Pizzo et al. (J. Stat. Phys. 2012), D. (Osaka J. Math. 2016))
Let
SN,k :=
√
N hνN , x k i − E[] .
(i) k = 3, 5, . . . ,
d
SN,k −→ ck Y + Zk ,
where ck is a constant, Y ∼ ξ11 , Zk ∼ N (0, ak2 ) and Y and Zk are
independent.
(ii) k = 2, 4, . . . ,
d
SN,k −→ Zk ∼ N (0, ak2 ).
(iii) Multidimensional version. There are jointly Gaussian random variables
{Zk } independent of Y ∼ ξ11 such that
d
{SN,k }K
k=1 −→ (Y , Z2 , c3 + Z3 , Z4 , . . . ).
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Chi distribution
What is χn , n > 0?
• n = 1, 2, . . . ,
N (0, 1)2 + · · · + N (0, 1)2
{z
}
|
1/2
∼ χn .
n
• For n > 0,
p.d.f . =
2 n−1 −x 2
x
e ,
Γ( n2 )
•
χn =
√
n
χn −
Trinh Khanh Duy (IMI, Kyushu University)
√
,1
1/2
,
2
where Γ(k, θ) denotes the gamma distribution.
• As n → ∞,
2Γ
x > 0.
1
d
n → N (0, ).
2
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Decomposition of GβE matrix

√

N (0,2) χ(N−1)β
χ
1 
 (N−1)β

βN
N (0,2) χ(N−2)β
..
.
..
..
.
N (0,2)
χβ
0
≈
1
1 0
1
.. .. ..
. . .
.




+√
1
βN
N (0,2) N (0, 12 )
 N (0, 12 ) N (0,2) N (0, 21 )
..
.
..
.

..

.
This implies the semi-circle law and the CLT for hµN , pi with polynomial p.
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Spectral measures of GβE
• Variance formula:
Var[hµN , f i] =
βN
Var[hLN , f i]
βN + 2
2
+
E[hµN , f 2 i] − E[hµN , f i]2 .
Nβ + 2
• Consequently
Nβ Var[hµN , f i] → hsc, f 2 i − hsc, f i2 ,
provided that Var[hLN , f i] = o(N).
Theorem (D. 2016)
For a ‘nice’ function f ,
√
d
Nβ
√
hµN , f i − E[hµN , f i] →N (0, σf2 ),
2
where σf2 = hsc, f 2 i − hsc, f i2 .
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Outline
1
Motivation
2
Jacobi matrices
3
Gaussian beta ensembles
4
Wishart beta ensembles
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
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Jacobi matrices for WβE
• m ∈ N, n > m − 1,


χβn
χβ(m−1) χβ(n−1)

Bm = 
..

.
..
.
χβ χβ(n−m+1)


t
.
 , Wm = Bm Bm

• Eigenvalues of Wm , (a = βn/2, p = 1 + β(m − 1)/2),
(λ1 , . . . , λm ) ∝ |∆(λ)|β
m
Y
i=1
m
1X
λa−p
exp
−
λi
i
2
!
.
i=1
• (q1 , . . . , qm ) is distributed as (χβ , . . . , χβ ) normalized to unit length,
independent of (λ1 , . . . , λm ).
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Spectral measures of beta ensembles
2016/03/21
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Decomposition of Wishart beta ensemble matrices
• m, n → ∞, m/n → γ ∈ (0, 1), cj ∼ χβ(n−j+1) , dj ∼ χβ(m−j)
c12
c1 d1
2 + d2

c
d
c
1 1
2
1
fm = 1 
W

.
.
βn 
.


=
√
1
γ
√
√
γ 1+γ γ
.. .. ..
. . .
!

c2 d2


..
..

.
.
2
2
cm−1 dm−1 cm + dm−1

2c̄1
1  √γ c̄1 +d¯1
+√
βn
√

γ c̄1 +d¯1
√
√
2(c̄2 + γ d¯1 ) γ c̄2 +d¯2
..
..
.
.
1
¯
• c̄i , dj : i.i.d. ∼ N (0, 2 ).

√
√ √

c̄j , dj ≈ βn γ + d¯j ,
cj ≈ βn + √
√ √
cj2 ≈ βn + 2 βnc̄j , dj2 ≈ βnγ + 2 βn γ d¯j ,

√
√
√

cj dj ≈ βn γ + βn( γ c̄j + d¯j ).
..
,
.
• For more details, see arXiv:1601.01146
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
31 / 32
Thank you very much for your attention!
Trinh Khanh Duy (IMI, Kyushu University)
Spectral measures of beta ensembles
2016/03/21
32 / 32