מטריצות אקראיות

תשע״ז

• הרצאה 1 ←

  introduction and motivation
  the Wigner model
  method of moments
  moments of semi - circle law

• הרצאה 2 ←

  basic properties of Catalan numbers
  proof of Wigner's theorem

• הרצאה 3 ←

  proof of lemma from previous lesson
  generalization of the Wigner theorem to:
  
  complex Wigner matrices
  reducing the moment restriction

• הרצאה 4 ←

  theorem on maximal eigenvalue
  logarithmic Sobolev inequalities
  Herbst's lemma
  implication on random matrices

• הרצאה 5 ←

  alternative approach to Wigner's theorem using Stieltjes transform
  relations with convergance of measures

• הרצאה 6 ←

  Gaussian Orthogonal/ Unitary ensemble
  density of GOE, GUE on matrices
  joint density of the eigenvalues

• הרצאה 7 ←

  Hermite polynomials - defintion and properties
  oscillator wave function
  formula for joint density of p unordered eigenvalues
  integration of these densities
  introduction of Fredholm determinants
  Gaudin - Mehta theorem

• הרצאה 8 ←

  Hadamard inequality
  continuity of Fredholm determinants
  additional properties of Hermite polynomials
  proof of main lemma - finishing the proof of the Gaudin - Mehta theorem
  Laplace method (Saddle point method) - idea
  Laplace theorem