An Introduction to Random Matrix Theory: Gaussian Ensembles

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An Introduction to Random Matrix TheoryGaussian
Ensembles

• Helena David
• 27/02/08

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Outline

• Brief history and introduction to random matrix
  theory (RMT)
• Gaussian ensembles
• Distribution of matrix elements
• Distribution of eigenvalues
• Distribution of level spacings
• Summary

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Brief History and Introduction

• Used by Wigner in the 1950s
• Dealt with complex many-body quantum systems
• More recent applications include
• Biological networks
• Climate correlations
• Zeros of the Riemann zeta function
• The RMT hypothesis
• RMT deals with three Gaussian ensembles

Level spacing distribution for 1726 nuclei of
same spin and parity.
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Gaussian ensembles

• Systems characterised by their Hamiltonian
• Hamiltonians represented by Hermitian matrices
• Gaussian orthogonal, unitary and symplectic
  ensembles (GOE,GUE,GSE)
• Probability distribution of matrix elements for
  GOE,GUE and GSE must
• be invariant under orthogonal, unitary and
  symplectic transformations respectively
• be such that the elements are statistically
  independent

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Hermitian Matrix - Definition

• A square matrix is defined as Hermitian
  if the following relationship holds.
• where denotes the conjugate transpose,
  or Hermitian conjugate, of .

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Gaussian ensembles

• Systems characterised by their Hamiltonian
• Hamiltonians represented by Hermitian matrices
• Gaussian orthogonal, unitary and symplectic
  ensembles (GOE,GUE,GSE)
• Probability distribution of matrix elements for
  GOE,GUE and GSE must
• be invariant under orthogonal, unitary and
  symplectic transformations respectively
• be such that the elements are statistically
  independent

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Eigenvalue Distributions

• Theorem
• The probability density function for the
  eigenvalues of matrices from a Gaussian
  orthogonal, Gaussian symplectic, or Gaussian
  unitary ensemble is given by
• where n 2 if the ensemble is orthogonal, n 5
  if it is symplectic and n 3 if it is unitary. C
  is a constant chosen so that P(E) is normalised
  to unity.

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Level Spacings

• Difference between successive eigenvalues
• i.e. for , the
  level spacings S are given by

Wigner Surmises proposed good approximate forms
of level spacing distributions of GOE, GUE and
GSE
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Summary

• Hamiltonians described by Hermitian matrices
• Three Gaussian ensembles
• Distributions of elements, eigenvalues and level
  spacings

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• ANY QUESTIONS?