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...Dynamical approach to random matrix theory l aszl o erdos horng tzer yau may partially supported by erc advanced grant ranmat the nsf dms and a simons investigator award amssubject classication b keywords local semicircle law dyson sine kernel wigner mehta conjecture tracy widom distribution brownian motion preface this book is concise self contained introduction of recent techniques prove spectral universality for large matrices fast expanding research area mainly focuses on methods we participated in developing over past few years many other interesting topics are not included nor several new developments within framework these have choseninsteadtopresentkeyconceptsthatwebelievearethecoreofthesemethodsandshouldberelevantfor future applications keep technicalities minimum make accessible graduate students with mind include basic notions tools high dimensional analysis such as deviation entropy dirichlet form logarithmic sobolev inequality thematerial originates from our joint works gr...