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.