Random Matrices

1. Random Matrices Hieu D. Nguyen Rowan University Rowan Math Seminar 12-10-03

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

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

4. Wigner’s Surmise

5. Level Sequences of Various Number Sets

6. 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

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

8. Wigner’s Surmise Notation – Successive energy levels – Nearest-neighbor level spacings – Mean spacing – Relative spacings Wigner’s P.D.F. for Relative Spacings

9. Are Nuclear Energy Levels Random? Poisson Distribution (Random Levels) Distribution of 1000 random numbers in [0,1]

10. How should we model the statistics of nuclear energy levels if they are not random?

11. Distribution of first 1000 prime numbers

12. 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):

13. Distribution of Zeros and Their Spacings First 105 Zeros First 200 Zeros

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

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

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

17. Eigensystems – Eigenvalue – Eigenvector Similarity Transformations (Conjugation) Diagonalization

18. 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

19. 500 matrices of size 5 x 5 1000 matrices of size 5 x 5

20. 10 x 10 matrices 20 x 20 matrices

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

22. Statistical Model for GOE – 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

23. 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).

24. Observation (Sum of eigenvalues of H)

25. Statistical Independence Assume Then

26. Now, P(H) being invariant under U means that its derivative should vanish:

27. 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)

28. 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)

29. Theorem Assumption 2 implies that P(H) can be expressed in terms of tr(H) and tr(H2), i.e.

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

31. Joint P.D.F. for the Eigenvalues

32. Lemma Corollary Standard Form

33. 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

34. Two-Point Correlation 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

35. 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 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

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

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

38. 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!

39. Wigner’s Surmise

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

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

42. 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

43. Level Spacings First-Order Log Spacings 1000 matrices of size 4 x 4 10,000 matrices of size 4 x 4 Second-Order Log Spacings

44. Open Problem Mathematically describe the distributions of these first- and higher-order log spacings

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