PhD Course: Introduction to Random Matrices – Theory and Practice


Dr. Pierpaolo Vivo (King’s College of London, UK)

Period and Duration

9 lectures, 2 hours each, from Monday, May 6th to Friday, May 10th 2019

Syllabus of the course

  1. Simple classification of random matrix models. Gaussian and Wishart ensembles. Warmup calculations: semicircle and Marcenko-Pastur laws via resolvent method.
  2. Level spacing statistics: Poisson vs Wigner-Dyson.
  3. Coulomb gas method. Singular integral equation for the spectral density.
  4. Orthogonal polynomial technique and numerical checks.
  5. 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.
  6. The Replica method. Edwards-Jones formalism. Applications to full and sparse matrices (random graphs).
  7. Cavity method for single instances. Spectral density and extreme eigenpairs.
  8. Free probability. Sum of random matrices. Blue’s function.


  • Monday 10-12 (Aula Denina, DISAT – PoliTo)
  • Monday 14-16 (Aula 2F, DISAT – PoliTo)
  • Tuesday 10-12 and 14-16 (Aula Perucca, DISAT – PoliTo)
  • Wednesday 10-12 and 14-16 (Aula Perucca, DISAT – PoliTo)
  • Thursday 10-12 and 14-16 (Aula Perucca, DISAT – PoliTo)
  • Friday 10-12  (Aula Perucca, DISAT – PoliTo)


Pierpaolo Vivo studied Physics at the Università degli Studi di Parma (Italy), where he also graduated in Theoretical Physics cum laude in 2005 under the supervision of Enrico Onofri with the thesis “Modifications of gravity for low accelerations”. He then moved to Brunel University (West London), where he obtained his PhD in 2008 under the supervision of Gernot Akemann with the thesis “From Wishart to Jacobi ensembles: Statistical properties and applications”. He was then awarded a three-year Postdoctoral Fellowship at Abdus Salam ICTP – Trieste (Italy), where he worked in the Condensed Matter and Statistical Physics group led by Vladimir Kravtsov. There he collaborated mainly with Antonello Scardicchio and Matteo Marsili on applications of Random Matrix Theory and Statistical Mechanics to the stability of financial markets. During the period 2011-2014 he worked as a research scientist at the Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS) in Orsay, (France). He is a permanent member of the Disordered System group at KCL since September 2014.