Random Matrices (Ias/Park City Mathematics Series) (IAS/Park City Mathematics, 26)

Cover
Title page
Preface
Introduction
Riemann–Hilbert Problems
	Lecture 1
	Lecture 2
	Lecture 3
	Lecture 4
The Semicircle Law and Beyond: The Shape of Spectra of Wigner Matrices
	Introduction
	First Results: the Weak Semicircle Law
		One name, many possible assumptions.
		Method of Moments.
		From moments to graphs.
		Additional notes and context.
		Problems.
	Stronger results, weaker assumptions
		Convergence in Probability; Proof of Theorem 2.1.5.
		Removal of Moment Assumptions.
		Additional notes and context.
		Problems.
	Beyond the Semicircle Law: A Central Limit Theorem
		Additional notes and context.
		Problems.
	One more dimension: minor processes and the Gaussian Free Field
		The Gaussian Free Field, the Height Function, and a Pullback.
		Additional Notes and Context.
		Problems.
	Acknowledgements
The Matrix Dyson Equation and its Applications for Random Matrices
	Introduction
		Random matrix ensembles
		Eigenvalue statistics on different scales
	Tools
		Stieltjes transform
		Resolvent
		The semicircle law for Wigner matrices via the moment method
	The resolvent method
		Probabilistic step
		Deterministic stability step
	Models of increasing complexity
		Basic setup
		Wigner matrix
		Generalized Wigner matrix
		Wigner type matrix
		Correlated random matrix
		The precise meaning of the approximations
	Physical motivations
		Basics of quantum mechanics
		The “grand” universality conjecture for disordered quantum systems
		Anderson model
		Random band matrices
		Mean field quantum Hamiltonian with correlation
	Results
		Properties of the solution to the Dyson equations
		Local laws for Wigner-type and correlated random matrices
		Bulk universality and other consequences of the local law
	Analysis of the vector Dyson equation
		Existence and uniqueness
		Bounds on the solution
		Regularity of the solution and the stability operator
		Bound on the stability operator
	Analysis of the matrix Dyson equation
		Properties of the solution to the MDE
		The saturated self-energy matrix
		Bound on the stability operator
	Ideas of the proof of the local laws
		Structure of the proof
		Probabilistic part of the proof
		Deterministic part of the proof
Counting equilibria in complex systems via random matrices
	May model of a complex system: an introduction
	Large-