Introduction to Random Matrix Theory and its various applications (3/5)
Satya Majumdar
LPTMS
Vendredi 04/12/2015, 10:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
\noindent Outline of the course: \\ 1) Brief historical introduction to RMT: 1applications. \vspace{-8pt} \begin{itemize} \addtolength{\itemsep}{-8pt} \item[-] Discussion of basic properties of matrices, different random matrix ensembles, rotationally invariant ensembles such as Gaussian ensembles etc. \end{itemize} \vspace{-8pt} 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: \vspace{-8pt} \begin{itemize} \addtolength{\itemsep}{-8pt} \item[-] Average density of eigenvalues --- Wigner semi-circle law \item[-] Counting statistics, spacings between eigenvalues etc \item[-] Distribution of the extreme (maximum or minimum eigenvalues) \end{itemize} \vspace{-8pt} 4) Two complementary approaches to study spectral statistics: \vspace{-8pt} \begin{itemize} \addtolength{\itemsep}{-8pt} \item[-] Large N (for an NxN matrix) method by the Coulomb gas approach: saddle point method \item[-] Finite N method: for Gaussian unitary ensemble: orthogonal polynomial method (essentially quantum mechanics of free fermions at zero temperature). \end{itemize} \vspace{-8pt} 5) Tracy-Widom distribution: prob. distribution of the top eigenvalue. Its appearance in a large number of problems, universality and an associated third order phase transition. \\ 6) Perspectives and summary.
Contact : Loic BERVAS

 

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