# Random Matrices and the Statistical Theory of Energy Levels

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Random Matrices and the Statistical Theory of Energy Levels focuses on the processes, methodologies, calculations, and approaches involved in random matrices and the statistical theory of energy levels, including ensembles and density and correlation functions. The publication first elaborates on the joint probability density function for the matrix elements and eigenvalues, including the Gaussian unitary, symplectic, and orthogonal ensembles and time-reversal invariance. The text then examines the Gaussian ensembles, as well as the asymptotic formula for the level density and partition function. The manuscript elaborates on the Brownian motion model, circuit ensembles, correlation functions, thermodynamics, and spacing distribution of circular ensembles. Topics include continuum model for the spacing distribution, thermodynamic quantities, joint probability density function for the eigenvalues, stationary and nonstationary ensembles, and ensemble averages. The publication then examines the joint probability density functions for two nearby spacings and invariance hypothesis and matrix element correlations. The text is a valuable source of data for researchers interested in random matrices and the statistical theory of energy levels.

## Table of Contents

Preface

1 / Introduction

1.1. The Need to Study Random Matrices

1.2. A Summary of Statistical Facts about Nuclear Energy Levels

1.3. Definition of a Suitable Function for the Study of Level Correlations

1.4. Wigner Surmise

1.5. Electromagnetic Properties of Small Metallic Particles

2 / Gaussian Ensembles. The Joint Probability Density Function of the Matrix Elements

2.1. Preliminaries

2.2. Time-Reversal Invariance

2.3. Gaussian Orthogonal Ensemble

2.4. Gaussian Symplectic Ensemble

2.5. Gaussian Unitary Ensemble

2.6. Joint Probability Density Function for Matrix Elements

3 / Gaussian Ensembles. The Joint Probability Density Function of the Eigenvalues

3.1. Orthogonal Ensemble

3.2. Symplectic Ensemble

3.3. Unitary Ensemble

4 / Gaussian Ensembles

4.1. The Partition Function

4.2. The Asymptotic Formula for the Level Density

5 / The Gaussian Orthogonal Ensemble

5.1. General Remarks

5.2. The Method of Integration over Alternate Variables

5.3. The Normalization Constant

5.4. One- and Two-Level Correlation Functions

5.5. Level Spacings

5.6. Bounds for the Distribution Function F(t)

6 / The Gaussian Unitary and Symplectic Ensembles

6.1. The Gaussian Unitary Ensemble

6.2. Gaussian Symplectic Ensemble

7 / Brownian Motion Model

7.1. Stationary Ensembles

7.2. Nonstationary Ensembles

7.3. Some Ensemble Averages

8 / Circular Ensembles

8.1. General Remarks

8.2. The Orthogonal Ensemble

8.3. Symplectic Ensemble

8.4. Unitary Ensemble

8.5. The Joint Probability Density Function for the Eigenvalues

9 / Circular Ensembles. Correlation Functions, Spacing Distribution, etc.

9.1. Orthogonal Ensemble

9.2. Symplectic Ensemble, ß = 4

9.3. Unitary Ensemble, ß = 2

9.4. Browning Motion Model

10 / Circular Ensembles. Thermodynamics

10.1. General Remarks

10.2. The Partition Function

10.3. Thermodynamic Quantities

10.4. Statistical Interpretation of U and C

10.5. Continuum Model for the Spacing Distribution

11 / The Orthogonal Circular Ensemble. Wigners Method

11.1. General Remarks

11.2. The Method of Integration

11.3. Spacing Distribution

12 / Matrices with Gaussian Element Densities But with No Unitary or Hermitian Condition Imposed

12.1. Complex Matrices

12.2. Quaternion Matrices

12.3. Real Matrices

13 / Gaussian Ensembles. Level Density in the Tail of the Semicircle

Text

14 / Bordered Matrices

14.1. Random Linear Chain

14.2. Bordered Matrices

15 / Invariance Hypothesis and Matrix Element Correlations

Text

16 / The Joint Probability Density Functions for Two Nearby Spacings

16.1. Integrations

16.2. An Integral Equation with a Boundary Condition and A2m(θ, α)

16.3. The Limit of A2m(θ, α)

16.4. Power Series Expansion and Numerical Results

16.5. The Distribution of Spacings between Next-Nearest Neighbors

17 / Restricted Trace Ensembles. Ensembles Related to the Classical Orthogonal Polynomials

17.1. Fixed Trace Ensemble

17.2. Bounded Trace Ensembles

17.3. Matrix Ensembles and Classical Orthogonal Polynomials

Appendices

A.1. Proof of Equation (2.52)

A.2. Counting the Dimensions of TßG and T'ßG (Chapter 3) and of Tßc and T'ßc (Chapter 8)

A.3. Two Proofs of Equation (4.5) for the Case N = 3

A.4. The Minimum Value of W, Equation (4.6)

A.5. Proof of Equation (4.15)

A.6. Proof of Equations (5.4), (5.51), and (5.52)

A.7. Proof of Equation (5.20). Expansion of a Pfaffian along Its Principal Pseudodiagonal

A.8. The Limit of ΣN0-1 φi2(x)

A.9. The Limits of ΣN0-1 φ1(x) φi(y) etc

A.10. The Fourier Transforms of the Two-Point Cluster Functions

A.11. Proof of Equations (5.84) and (9.39)

A.12. Various Probability Distribution and Probability Density Functions

A.13. Some Applications of Gram's Result

A.14. Power-Series Expansion of Im(θ)

A.15. Proof of the Inequalities (5.130)

A.16. The Confluent Alternant

A.17. Proof of Equations (8.12) and (8.28)

A.18. Proof of Equation (11.29)

A.19. Wilson's Proof of Equation (10.11)

A.20. Proof That the Second Term in Equation (7.28) Drops Out on Summation

A.21. Proof of the Inequality (10.5)

A.22. The Probability Density of the Spacings Resulting from a Random Superposition of n Unrelated Sequences of Energy Levels

A.23. Some Properties Connected with Symmetric and Antisymmetric Unitary Matrices

A.24. Evaluation of the Interval (12.9) for Complex Matrices

A.25. A Few Remarks about the Eigenvalues of a Quaternion Real Matrix and Its Diagonalization

A.26. Evaluation of the Integral (12.46)

A.27. The Proof of Equation (12.70)

A.28. The Case of Random Real Matrices

A.29. The Density of Eigenvalues of a Random Matrix Whose Elements All Have the Same Mean Square Value

A.30. Values of the Functions B(x1 , x2) (Table A.30.1), and P(x1 , x2) (Table A.30.2)

A.31. Proof of Equations (6.25") and (16.69)

References

Index

## Product details

- No. of pages: 270
- Language: English
- Copyright: © Academic Press 1967
- Published: January 1, 1967
- Imprint: Academic Press
- eBook ISBN: 9781483258560

## About the Author

### M. L. Mehta

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