Random Matrix Theory and its Applications - Lecture 4

Explore the fourth 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 in statistical physics, including rotationally invariant ensembles, Gaussian cases, joint distributions, and the Ising model. Examine natural observables, n-point correlation functions, and spacing distributions between neighboring eigenvalues. Investigate counting statistics for Gaussian ensembles and extreme value statistics of correlated random variables. Analyze R. M. May's work on the stability of large complex systems and explore the effects of pairwise interactions. This 93-minute lecture is part of a comprehensive program aimed at bridging the gap between masters-level courses and cutting-edge research in statistical physics.

Recap - Rotationally invariant ensemble
1. Focus on Rotation Invariant Ensemble Gaussian Case
Joint Distribution - Gibbs -Boltzmann prob
Ising model
Distribution of the matrix entries
Exercise
Natural observables
2. n-point correlation function
3. Spacing distribution bet n neighbouring eigenvalues
Wigner surmise
4. Counting Statistics
For Gaussian ensembles
Extreme value statistics of correlated random variables
R. M. May Nature, 238, 413 1972 - "Will a large complex system be stable"?
Switch on pairwise interaction
Stable