Random Matrix Theory and Its Applications - Lecture 5

Explore the fifth lecture in a series on Random Matrix Theory and its Applications, delivered by Satya Majumdar at the Bangalore School on Statistical Physics - X. Delve into advanced topics including rotational invariant Gaussian ensembles, finite N and large N approaches, coarse-graining techniques, and the famous Wigner semi-circular law. Learn about empirical density introduction, partial tracing, saddle point method, and the Cauchy singular value equation. Examine the scale of interparticle distance, asymptotic properties, and the work of Tracy & Widom from 1994. This 84-minute lecture is part of a comprehensive program aimed at bridging the gap between masters-level courses and cutting-edge research in statistical physics, suitable for PhD students, postdoctoral fellows, and interested faculty members.

Recap - Rot inv ensembles Gaussian
Two approaches - finite N approach
Large N continuum approach
What is the typical scale of lambda?
Large N - coarse graining
Step 1: Introduce an empirical density
Step 2: Coarse graining - Partial tracing
Saddle point method
Finite support of the charged density
Cauchy singular value equation
F.G. Tricomi, 1957
Famous Wigner semi-circular law
Scale of interparticle distance
Asymptotic properties
Tracy & Widom, 1994