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Basic Concepts in Probability and Statistics

Statistics

– Data set of values

– Mean

– Variance

Probability

– Continuous random variable on [a,b]

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

– Total probability equals 1

Notation

– Successive energy levels

– Nearest-neighbor level spacings

– Mean spacing

– Relative spacings

Wigner’s P.D.F. for Relative Spacings

Are Nuclear Energy Levels Random?

Poisson Distribution (Random Levels)

Distribution of 1000 random numbers in [0,1]

How should we model the statistics of nuclear energy

levels if they are not random?

Distribution of Zeros of Riemann Zeta Function

Fun Facts

1.

is irrational (Apery’s constant)

2.

can be analytically continued to all

3.

4.

(functional equation)

5. Zeros of

Trivial Zeros:

(critical line)

Non-Trivial Zeros (RH):

Asymptotic Behavior of Spacings for Large Zeros

Question: Is there a Hermitian matrix H which has

the zeros of as its eigenvalues?

Quantum Mechanics

– Hamiltonian (Hermitian operator)

– Bound state (eigenfunction)

– Energy level (eigenvalue)

Statistical Approach

– Hermitian matrix

(Matrix eigenvalue problem)

Basics Concepts in Linear Algebra

Matrices

n x n square matrix

Special Matrices

Symmetric:

Hermitian:

Orthogonal:

Gaussian Orthogonal Ensembles (GOE)

– random N x N real symmetric matrix

Distribution of eigenvalues of 200 real symmetric

matrices of size 5 x 5

Level spacing

Eigenvalues

Entries of each matrix is chosen randomly and

independently from a Gaussian distribution with

1000 matrices of size 5 x 5

20 x 20 matrices

– random N x N real symmetric matrix

Assumptions

- Probability of choosing H is invariant under
- orthogonal transformations
- 2. Entries of H are statistically independent

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

– p.d.f. for choosing

– j.p.d.f. for choosing

Lemma (Weyl, 1946)

All invariant functions of an (N x N) matrix H

under nonsingular similarity transformations

can be expressed in terms of the traces of the first N

powers of H.

Corollary

Assumption 1 implies that P(H) can be

expressed in terms of tr(H), tr(H2), …, tr(HN).

(Sum of eigenvalues of H)

Now, P(H) being invariant under U means that its

derivative should vanish:

We now apply (*) to the equation immediately above

to ‘separate variables’, i.e. divide it into

groups of expressions which depend on mutually

exclusive sets of variables:

It follows that say

(constant)

It can be proven that Ck = 0. This allows us to

separate variables once again:

(constant)

Solving these differential equations yields our

desired result:

(Gaussian)

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

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

Level Density

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)

20 x 20 matrices

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 being

unobserved, to be

We define the probability density for finding

two consecutive levels inside an interval

to be

Limiting Behavior (Normalized)

We define the probability density that in an infinite

series of eigenvalues (with mean spacing unity)

an interval of length 2t contains exactly two levels

at positions around the points y1 and y2 to be

P.D.F. of Level Spacings

We define the probability density of finding a

level spacing s = 2t between two successive levels

y1 = -t and y2 = t to be

Key Idea

as a determinant:

Write

(Oscillator wave functions)

(Hermite polynomials)

Harmonic Oscillator (Electron in a Box)

NOTE: Energy levels are quantized (discrete)

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

The derivation of this formula very complicated!

- Cauchy matrices are symmetric and positive definite

Eigenvalues of A:

Logarithms of Eigenvalues:

Level Spacings of Eigenvalues of Cauchy Matrices

Assumption

The values kn are chosen randomly and independently

on the interval [0,1] using a uniform distribution

1000 matrices of size 4 x 4

Log distribution

Distribution of spacings

First-Order Log Spacings

1000 matrices of size 4 x 4

10,000 matrices of size 4 x 4

Second-Order Log Spacings

Mathematically describe the distributions of

these first- and higher-order log spacings

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

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