Log Gases Seminar - Summer 2026

Summary

Talk 1 Overview

  • Title of Talk. Joint Distribution of eigenvalues of Gaussian random matrices

Definitions.

  • Definition (Gaussian Unitary Ensemble GUE).

  • Definition (Gaussian Orthogonal Ensemble GOE).

Propositions.

  • Theorem 14.3 (Kemp (2022)). This Theorem gives a complete description of the joint law of eigenvalues and eigenvectors of a \(GOE_n\). (Hence the title of the talk).
    1. The proof will be covered.
    2. What does the result say? It says two things. First, the distribution of the eigenvectors is Haar measure on the orthogonal matrix group \(O(n)\). Second, the joint law of eigenvalues is the product of a Gaussian density term and a Vandermonte determinant term.

Where is the statement in section 2.5.1/2.5.2 of Anderson, Guionnet, and Zeitouni (2010)?

  • Theorem 14.4 (Kemp (2022)). This theorem gives a description of the joint law of the eigenvalues and eigenvectors of a \(GUE_n\).

Talk 2 Overview (My Talk)

Title of talk.

Definitions.

  • Definition (\(\beta\) ensemble).

Propositions.

  • Theorem 4.5.35 (Anderson, Guionnet, and Zeitouni (2010)).

Talk 3 Overview

More Details

Anderson, G. W., A Guionnet, and O Zeitouni. 2010. “An Introduction to Random Matrices.” Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2010. https://www.wisdom.weizmann.ac.il/~zeitouni/cupbook.pdf.
Kemp, T. 2022. “Introduction to Random Matrix Theory.” Lecture Notes. mathweb.ucsd.edu/~tkemp/RMT.Notes.pdf.