Introduction to Random Matrices

by P. Vivo (2 credits, type B)

(source: WikiCommons)

Main Topics

  • Simple classification of random matrix models. Gaussian and Wishart ensembles. Warmup calculations: semicircle and Marcenko–Pastur laws.
  • Level spacing statistics: Poisson vs Wigner–Dyson.
  • Coulomb gas method.
  • Orthogonal polynomial technique and numerical checks.
  • Largest eigenvalue of a random matrix. Comparison with Extreme Value Statistics for i.i.d. random variables. Tracy–Widom distribution and third-order phase transitions.
  • The Replica method. Edwards–Jones formalism. Applications to full and sparse matrices (random graphs).
  • Free probability. Sum of random matrices. Blue’s function.

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