Explore a comprehensive lecture on random matrices and Dyson-Schwinger equations delivered by Alice Guionnet from ENS, Lyon, France. Delve into advanced mathematical concepts, beginning with an introduction to the topic and progressing through first asymptotics. Discover the applications in operator algebra and examine rational states. Follow the detailed proof and study covariances, inversion theorems, and limits. Analyze various strategies and conclude with an exploration of the Central Limit Theorem. Gain valuable insights into this complex area of mathematics through this in-depth presentation, part of the School and Workshop on Random Matrix Theory and Point Processes.