Abstract:Année de publication : 2015
(1) Brief historical introduction to RMT: applications Discussion of basic properties of matrices, different random matrix ensembles, rotationally invariant ensembles such as Gaussian ensembles etc. (2) Gaussian ensembles: derivation of the joint probability distribution of eigenvalues, starting from the joint distribution of matrix entries. (3) Analysis of the spectral properties of eigenvalues: given the joint distribution of eigenvalues, how to calculate various observables such as: (i) Average density of eigenvalues ----Wigner semi-circle law (ii) Counting statistics, spacings between eigenvalues etc. (iii) Distribution of the extreme (maximum or minimum eigenvalues) (4) Two complementray approaches to study spectral statistics: (a) Large N (for an NxN matrix) method by the Coulomb gas approach: saddle point method (b) finite N method: for Gaussian unitary ensemble: orthogonal polynomial method (essentially quantum mechanics of free fermions at zero temperature). (5) Tracy-Widom distribution: prob. distribution of the top eigenvalue. Its appearence in a large number of problems, universality and an associated third order phase transition. (6) Perspectives and summary.
lectures_notes--handwritten.pdf Slides:History+Cold_atoms.pdf Slides:Tracy-Widom.pdf References.pdf Poster.pdf