Description: This course will give a brief introduction to random matrix theory. Some topics we plan to cover are: Wigner semicircle law, the moment method, the resolvent method, invariant ensembles, Wigner matrices, sample covariance matrices, bulk universality, edge universality, rigidity of eigenvalues, Dyson Brownian motion, Tracy-Widom law, and free probability.Prerequisite:Probability, S...
Description: The central-limit type results are universal in many random walk models: they are known as Donsker’s theorem for the classical Zd random walk, and also hold for some random walks in random environment, even when the environment is degenerate like percolation. In their proofs, the homogenization theory, which comes from PDE, plays an important role. This course will cover the follo...