Random Matrices Hieu D. Nguyen Rowan University Rowan Math Seminar

Statistics of Nuclear Energy Levels Historical Motivation - Excited states of an atomic nucleus

Level Spacings – Successive energy levels – Nearest-neighbor level spacings

Wigner’s Surmise

Level Sequences of Various Number Sets

Basic Concepts in Probability and Statistics – Data set of values – Mean – Continuous random variable on [a,b] – Probability density function (p.d.f.) Statistics Probability – Variance – Total probability equals 1

– Mean – Variance – Probability of choosing x between a and b Examples of P.D.F.

Wigner’s Surmise – Successive energy levels – Nearest-neighbor level spacings – Mean spacing – Relative spacings Notation 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 first 1000 prime numbers

Distribution of Zeros of Riemann Zeta Function 5. Zeros of Non-Trivial Zeros (RH): Trivial Zeros: can be analytically continued to all Fun Facts is irrational (Apery’s constant) (critical line) 4.(functional equation)

Distribution of Zeros and Their Spacings First 200 Zeros First 10 5 Zeros

Asymptotic Behavior of Spacings for Large Zeros Question: Is there a Hermitian matrix H which has the zeros of as its eigenvalues?

Quantum Mechanics Statistical Approach – Hamiltonian (Hermitian operator) – Hermitian matrix – Bound state (eigenfunction) – Energy level (eigenvalue) Model of The Nucleus (Matrix eigenvalue problem)

Basics Concepts in Linear Algebra n x n square matrix Matrices Special Matrices Symmetric: Hermitian: Orthogonal:

Eigensystems – Eigenvalue – Eigenvector Similarity Transformations (Conjugation) Diagonalization

Gaussian Orthogonal Ensembles (GOE) – random N x N real symmetric matrix Distribution of eigenvalues of 200 real symmetric matrices of size 5 x 5 Entries of each matrix is chosen randomly and independently from a Gaussian distribution with Level spacing Eigenvalues

500 matrices of size 5 x matrices of size 5 x 5

10 x 10 matrices 20 x 20 matrices

Why Gaussian Distribution? Uniform P.D.F.Gaussian P.D.F.

Statistical Model for GOE 1.Probability of choosing H is invariant under orthogonal transformations 2. Entries of H are statistically independent – random N x N real symmetric matrix – p.d.f. for choosing – j.p.d.f. for choosing Assumptions Joint Probability Density Function (j.p.d.f.) for H

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(H 2 ), …, tr(H N ).

Observation (Sum of eigenvalues of H)

Statistical Independence Assume Then

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 C k = 0. This allows us to separate variables once again: (constant) Solving these differential equations yields our desired result: (Gaussian)

Theorem Assumption 2 implies that P(H) can be expressed in terms of tr(H) and tr(H 2 ), i.e.

J.P.D.F. for the Eigenvalues of H Change of variables for j.p.d.f.

Joint P.D.F. for the Eigenvalues

Lemma Corollary Standard Form

Density of Eigenvalues 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

Two-Point Correlation We define the probability density of finding a level (regardless of labeling) around each of the points x 1 and x 2, 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

Level Spacings We define the probability density of finding a level spacing s = 2t between two successive levels y 1 = -t and y 2 = t 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 y 1 and y 2 to be P.D.F. of Level Spacings

Multiple Integration of Key Idea Write as a determinant: (Oscillator wave functions) (Hermite polynomials)

Harmonic OscillatorHarmonic Oscillator (Electron in a Box) NOTE: Energy levels are quantized (discrete)

Formula for Level Spacings? The derivation of this formula very complicated! - Eigenvalues of a matrix whose entries are integrals of functions involving the oscillator wave functions

Wigner’s Surmise

Random Matrices and Solitons Korteweg-de Vries (KdV) equation Soliton Solutions

Cauchy Matrices - Cauchy matrices are symmetric and positive definite Eigenvalues of A: Logarithms of Eigenvalues:

Level Spacings of Eigenvalues of Cauchy Matrices The values k n are chosen randomly and independently on the interval [0,1] using a uniform distribution Assumption 1000 matrices of size 4 x 4 Log distribution Distribution of spacings

Level Spacings First-Order Log Spacings 1000 matrices of size 4 x 410,000 matrices of size 4 x 4 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.