Random Matrices

Hieu D. NguyenRowan University

Rowan Math Seminar12-10-03

Statistics of Nuclear Energy Levels

Historical Motivation

- Excited states of an atomic nucleus

Level Spacings

1 2 3{ , , ,...}E E E – Successive energy levels

1i i iS E E – Nearest-neighbor level spacings

Wigner’s Surmise

Level Sequences of Various Number Sets

Basic Concepts in Probability and Statistics

1 2 3{ , , ,..., }Nx x x x – Data set of values

1

1 N

ii

xN

x

– Mean

– Continuous random variable on [a,b]

( ) 0f x – Probability density function (p.d.f.)

Statistics

Probability

2 2

1

1( )

N

ii

xN

– Variance

( ) 1b

af x dx – Total probability equals 1

( )xf x dx

2 2 ( )x f x dx

– Mean

– Variance

( ) ( )b

aP a x b f x dx – Probability of choosing x

between a and b

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

( ) 1f x 2

21

( )2

x

f x e

Examples of P.D.F.

Wigner’s Surmise

1 2 3{ , , ,...}E E E – Successive energy levels

1i i iS E E – Nearest-neighbor level spacings

iD S – Mean spacing

ii

Ss

D – Relative spacings

Notation

Wigner’s P.D.F. for Relative Spacings

2( ) exp ,2 4W

s Sp s s s

D

Are Nuclear Energy Levels Random?

Poisson Distribution (Random Levels)

0.2 0.4 0.6 0.8 1

5

10

15

20

25

30

35

0.001 0.002 0.003 0.004 0.005 0.006

25

50

75

100

125

150

175

Distribution of 1000 random numbers in [0,1]

Let's generate random numbers between 0 and 1 Nu 1000;

spacings;rn TableRandomReal,0, 1,i, 1, Nu;rn Sortrn;DoAppendTospacings,rni1 rni,i, 1, Nu 1HistogramrnHistogramspacings

0.001 0.002 0.003 0.004 0.005 0.006

25

50

75

100

125

150

175

How should we model the statistics of nuclear energylevels if they are not random?

2000 4000 6000 8000

10

20

30

40

50

5 10 15 20 25 30 35

50

100

150

200

250

Distribution of first 1000 prime numbers

In[138]:= Let's generate prime numbers Nu 1000;

DoAppendTospacings, Primei1 Primei,i, 1, Nu 1HistogramTablePrimei,i, 1, NuHistogramspacings

Distribution of Zeros of Riemann Zeta Function

1

1( ) (Re 1)

zn

z zn

5. Zeros of

Non-Trivial Zeros (RH):

{ 2, 4,...}z

1

2 nz i

( )z

Trivial Zeros:

can be analytically continued to all 1z

Fun Facts2

2 2 2

1 1 1(2) ...

1 2 3 6

(3) is irrational (Apery’s constant)

1.

2.

3. ( )z

(critical line)

(1 ) 2(2 ) cos( / 2) ( ) ( )zz z z z 4. (functional equation)

100 200 300 400

5

10

15

20

25

30

1 2 3 4 5 6 7

10

20

30

40

50

Distribution of Zeros and Their Spacings

First 200 Zeros First 105 Zeros

Asymptotic Behavior of Spacings for Large Zeros

Question: Is there a Hermitian matrix H which hasthe zeros of as its eigenvalues?( )z

Quantum Mechanics

Statistical Approach

i i iH E

– Hamiltonian (Hermitian operator)H

H – Hermitian matrix *( )H H

i – Bound state (eigenfunction)

– Energy level (eigenvalue)iE

Model of The Nucleus

i i iH E (Matrix eigenvalue problem)

Basics Concepts in Linear Algebra

11 12 1

21 22 2

1 2

...

...( )

... ... ... ...

...

n

njk

n n nn

a a a

a a aA a

a a a

n x n square matrix

Matrices

Special Matrices

Symmetric:

Hermitian:

Orthogonal:

TA A

*A A

1( )

T

T

A A I

A A

1 2

2 1A

1

1

iA

i

cos sin

sin cosA

EigensystemsA x x

1 2

2 1A

x

– Eigenvalue

– Eigenvector

1 2

1 11, 3,

1 1

x x

Similarity Transformations (Conjugation) 1A A UAU

Diagonalization1

1 2( , ,..., )nUAU D

Gaussian Orthogonal Ensembles (GOE)

( )jkH h – random N x N real symmetric matrix

Distribution of eigenvalues of 200 real symmetricmatrices of size 5 x 5

Entries of each matrix is chosen randomly andindependently from a Gaussian distribution with

-4 -2 0 2 4

10

20

30

40

1 2 3 4

10

20

30

40

50

Level spacingEigenvalues

0, 1

500 matrices of size 5 x 5

1000 matrices of size 5 x 5

-6 -4 -2 0 2 4 6

25

50

75

100

125

150

1 2 3 4 5

20

40

60

80

100

120

140

-4 -2 0 2 4 6

20

40

60

80

100

1 2 3 4 5

25

50

75

100

125

150

10 x 10 matrices

20 x 20 matrices

-5 -2.5 0 2.5 5 7.5

50

100

150

200

1 2 3 4

50

100

150

200

250

300

350

-10 -5 0 5 10

50

100

150

200

250

300

1 2 3 4

100

200

300

400

Why Gaussian Distribution?

Uniform P.D.F. Gaussian P.D.F.

-2 -1 0 1 2

50

100

150

0.2 0.4 0.6 0.8 1 1.2

50

100

150

200

250

-5 -2.5 0 2.5 5 7.5

50

100

150

200

1 2 3 4

50

100

150

200

250

300

350

0 0, 1

Statistical Model for GOE

1. Probability of choosing H is invariant underorthogonal transformations

2. Entries of H are statistically independent

( )jkH h – random N x N real symmetric matrix

( )jk jkf h – p.d.f. for choosing jkh

( ) ( )N

jk jkj k

p H f h

– j.p.d.f. for choosing H

Assumptions

Joint Probability Density Function (j.p.d.f.) for H

Lemma (Weyl, 1946)

All invariant functions of an (N x N) matrix Hunder nonsingular similarity transformations

1H H AHA

can be expressed in terms of the traces of the first Npowers of H.

Corollary

Assumption 1 implies that P(H) can beexpressed in terms of tr(H), tr(H2), …, tr(HN).

Observation

1

2 2

1

1

tr( )

tr( )

...

tr( )

N

ii

N

ii

NN N

ii

H

H

H

(Sum of eigenvalues of H)

Statistical Independence

Assume

1

cos sin 0 ... 0

sin cos 0 ... 0

, 0 0 1 ... 0

... ... ... ... ...

0 0 0 ... 1

H UHU U

Then

1

(*)

TT

TT

T

HU HU

U UHU U H

U UUH HU

AH HA

0 1 0 ... 0

1 0 0 ... 0

... ... ... ... ...

0 0 0 ... 0

TUA U

Now, P(H) being invariant under U means that itsderivative should vanish:

( ) ( )

0

10

kj kjj k

kj kj

kj kj

p H f h

p

f h

f h

We now apply (*) to the equation immediately aboveto ‘separate variables’, i.e. divide it intogroups of expressions which depend on mutually exclusive sets of variables:

It follows that say

1 22 1

1 1 2 2

1 2

1 1 1 2 2 2 1 2

1 1

1 1

k kk k k

k k k k

k k k

k k k k k k k k

f fh h C

f h f h

f f C

h f h h f h h h

(constant)

1 22 1

3 1 1 2 2

1 10

Nk k

k kk k k k k

f fh h

f h f h

11 22 1212 11 22

11 11 22 22 12 12

1 1 1(2 ) ( )

f f fh h h

f h f h f h

11 12 22 1 2{ , , },{ , }k kh h h h h

It can be proven that Ck = 0. This allows us toseparate variables once again:

1

1 1 1

2

2 2 2

12

12

k

k k k

k

k k k

fa

h f h

fa

h f h

(constant)

Solving these differential equations yields ourdesired result:

21 1 1

22 2 2

( ) exp( )

( ) exp( )

k k k

k k k

f h ah

f h ah

(Gaussian)

Theorem

Assumption 2 implies that P(H) can be expressedin terms of tr(H) and tr(H2), i.e.

2( ) exp( tr( ) tr( ) )p H a H b H c

J.P.D.F. for the Eigenvalues of H

1

: ( ) Sym( )

( , ) , ( ,..., )

N

TN

F O N N

U H UDU

Change of variables for j.p.d.f.

1

2

0 ... 0

0 ... 0, ,

... ... ... ...

0 0 0

T T

N

D U HU U U I D

11 12

1 11 12

( , ,..., )Jac( )

( ,..., , , ,... )NN

N NN

H H HF

U U U

1

1

1

( )

( )( )

( )

( ) ( ) , Sym( )

( )Jac( )

( ) Jac( )

( )

N

N

F

O NF

N

F

P H p H dH N

p F dUd

p F dU d

p d

( )

( ) ( ) Jac( )N

O N

p p F dU

Joint P.D.F. for the Eigenvalues

LemmaJac( ) ( ) k j

j k

F g U

Corollary

21( ,..., ) exp( )N N i i k j

j k

p a b c

21

1( ,..., ) exp

2N N N i k jj k

p x x C x x x

Standard Form

1

22j j

bx

aa

Density of Eigenvalues

Level Density

1 2 2( ) ( ) ... ( , ,..., ) ...N N N nx R x N p x x x dx dx

We define the probability density of finding a level(regardless of labeling) around x, the positions of the remaining levels being unobserved, to be

Asymptotic Behavior for Large N (Wigner, 1950’s)

212 , 2

( )

0 2N

N x x Nx

x N

-10 -5 0 5 10

50

100

150

200

250

300

20 x 20 matrices

Two-Point Correlation

2 1 2 1 3

!( , ) ... ( ,..., ) ...

( 2)! N N n

NR x x p x x dx dx

N

We define the probability density of finding a level(regardless of labeling) around each of the points x1 and x2, the positions of the remaining levels beingunobserved, to be

1 2 1 3

out

!( ; , ) ... ( ,..., ) ...

2!( 2)!N N N N

NA x x p x x dx dx

N

We define the probability density for findingtwo consecutive levels inside an interval to be

( , )

, 1,2

, 3,...,

j

j

x j

x j N

Level Spacings

( ) 2 ( ; , )p s B t t t

We define the probability density of finding alevel spacing s = 2t between two successive levelsy1 = -t and y2 = t to be

21 2 1 2( ; , ) lim ( ; , ), / , /N j jN

B t y y A x x t y x

mean spacing

Limiting Behavior (Normalized)

We define the probability density that in an infiniteseries of eigenvalues (with mean spacing unity)an interval of length 2t contains exactly two levelsat positions around the points y1 and y2 to be

P.D.F. of Level Spacings

Multiple Integration of 1( ,..., )N Np x x

Key Idea

Write 1( ,..., )N Np x x as a determinant:

21exp det ( )

2 i k j i jj k

x x x c x

21

1( ,..., ) exp

2N N N i k jj k

p x x C x x x

21( ) exp ( )

22 !j j

j

xx H x

j

(Oscillator wave functions)

2

( ) [ ]j

xj j

dH x e

dx (Hermite polynomials)

Harmonic Oscillator (Electron in a Box)

22 2

2

1 1( ) ( ), ( )

2 2n n n n

dm x x E x E n

m dx

NOTE: Energy levels are quantized (discrete)

Formula for Level Spacings?

The derivation of this formula very complicated!

2

22( ) (1 )

dp s

ds

2 - Eigenvalues of a matrix whose entries are integrals of functions involving the oscillator wave functions

Wigner’s Surmise

2( ) exp ,2 4W

s Sp s s s

D

Random Matrices and Solitons

6 0t x xxxu uu u

Korteweg-de Vries (KdV) equation

2

2( , ) 2 logdet( ( , ))u x t I A x t

x

Soliton Solutions

3 3( ) 4( )

, 1

( , ) m n m n

N

k k x k k tm n

m n m n

c cA x t e

k k

Cauchy Matrices

, 1

1( 0)

N

nm n m n

A kk k

1 1 1

2 2 2 3 2 51 1 1

3 2 3 3 3 51 1 1

5 2 5 3 5 5

A

- Cauchy matrices are symmetric and positive definite

0.502136, 0.0142635, 0.000267133Eigenvalues of A: 0.688884, 4.25005, 8.22776Logarithms of Eigenvalues:

Level Spacings of Eigenvalues of Cauchy Matrices

2 4 6 8 10

200

400

600

800

1000

1200

-15 -10 -5 0 5

50

100

150

200

250

300

350

The values kn are chosen randomly and independentlyon the interval [0,1] using a uniform distribution

Assumption

1000 matrices of size 4 x 4

Log distributionDistribution of spacings

Level Spacings

First-Order Log Spacings

2 4 6

20

40

60

80

100

120

140

1000 matrices of size 4 x 4

2 4 6 8

100

200

300

400

500

600

700

10,000 matrices of size 4 x 4

-6 -4 -2 0 2

10

20

30

40

50

60

70

-6 -4 -2 0 2

50

100

150

200

250

300

Second-Order Log Spacings

Open Problem

Mathematically describe the distributions of these first- and higher-order log spacings

References

1. Random Matrices, M. L. Mehta, Academic Press, 1991.