Random Matrix Theory and Its Applications - Lecture 3

Explore the third lecture in the Random Matrix Theory and its Applications series, delivered by Satya Majumdar at the Bangalore School on Statistical Physics - X. Delve into advanced topics such as complex quaternionic matrices, rotationally invariant ensembles, and joint distribution of eigenvalues. Learn about specific orthonormal transits, tricks for changing variables, and special properties of rotationally invariant ensembles. Examine real symmetric, complex Hermitian, and complex quaternionic cases, understanding how eigenvalues and eigenvectors decouple in rotationally invariant ensembles. Conclude with an exploration of 1D disordered models, including GOE and Anderson models. This comprehensive 1 hour 32 minute lecture is part of a broader program aimed at bridging the gap between masters-level courses and cutting-edge research in statistical physics.

Recap: Random matrices with real spectrum
Complex quaternionic matrices
Other examples of rotationally invariant ensembles
Exercise: 2x2 - generic orthonormal transit
Specific orthonormal transit
Joint distribution of eigenvalues
Trick - Right set of variables to change
Remarks
Rotational Invariant Ensemble
3x3 - Gaussian case
Counting degrees of fraction
Special property of rotationally invariant ensembles
Real symmetric rotationally invariant ensemble
Complex Hermitian Case
Complex Quaternionic
For rotationally Invariant ensemble =eigenvalues & eigenvectors decouple
1 d disordered model - GOE and Anderson model