Random Matrices Hieu D Nguyen Rowan University Rowan
- Slides: 45
Random Matrices Hieu D. Nguyen Rowan University Rowan Math Seminar 12 -10 -03
Historical Motivation Statistics of Nuclear Energy Levels - 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 – Variance Probability – Continuous random variable on [a, b] – Probability density function (p. d. f. ) – Total probability equals 1
Examples of P. D. F. – Probability of choosing x between a and b – Mean – Variance
Wigner’s Surmise 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 first 1000 prime numbers
Distribution of Zeros of Riemann Zeta Function Fun Facts 1. 2. is irrational (Apery’s constant) 3. 4. can be analytically continued to all (functional equation) 5. Zeros of Trivial Zeros: Non-Trivial Zeros (RH): (critical line)
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 has the zeros of as its eigenvalues?
Model of The Nucleus 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:
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 Eigenvalues Level spacing Entries of each matrix is chosen randomly and independently from a Gaussian distribution with
500 matrices of size 5 x 5 1000 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 – random N x N real symmetric matrix Assumptions 1. 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(H 2), …, tr(HN).
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 Ck = 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 Limiting Behavior (Normalized) We define the probability density that in an infinite series of eigenvalues (with mean spacing unity) an interval of length 2 t contains exactly two levels at positions around the points y 1 and y 2 to be P. D. F. of Level Spacings We define the probability density of finding a level spacing s = 2 t between two successive levels y 1 = -t and y 2 = t to be
Multiple Integration of Key Idea Write as a determinant: (Oscillator wave functions) (Hermite polynomials)
Harmonic Oscillator (Electron in a Box) NOTE: Energy levels are quantized (discrete)
Formula for Level Spacings? - Eigenvalues of a matrix whose entries are integrals of functions involving the oscillator wave functions The derivation of this formula very complicated!
Wigner’s Surmise
Random Matrices and Solitons Korteweg-de Vries (Kd. V) 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 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 Distribution of spacings Log distribution
Level Spacings First-Order Log Spacings 1000 matrices of size 4 x 4 Second-Order Log Spacings 10, 000 matrices of size 4 x 4
Open Problem Mathematically describe the distributions of these first- and higher-order log spacings
References 1. Random Matrices, M. L. Mehta, Academic Press, 1991.
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