Random Matrix Theory: Replica Method

This lecture introduces random matrix theory and the replica approach to compute quantities related to non-hermitic series. The history of the theory is discussed, starting from the origins in studying heavy nuclei and scattering experiments. The lecture covers the computation of the Stieltjes transform using the replica method, involving formulas like the Schochosti formula and the manipulation of logarithms. The process includes steps such as decoupling replicas, introducing deltas, and computing extremums to derive the Stieltjes transform. The lecture concludes by explaining how the Stieltjes transform helps in computing the probability distribution of eigenvalues, leading to the observation of a compact support known as the semicircle law.