Chapter
2.3 Concentration for functionals of random matrices and logarithmic Sobolev inequalities
2.3.1 Smoothness properties of linear functions of the empirical measure
2.3.2 Concentration inequalities for independent variables satisfying logarithmic Sobolev inequalities
2.3.3 Concentration for Wigner-type matrices
2.4 Stieltjes transforms and recursions
2.4.1 Gaussian Wigner matrices
2.4.2 General Wigner matrices
2.5 Joint distribution of eigenvalues in the GOE and the GUE
2.5.1 Definition and preliminary discussion of the GOE and the GUE
2.5.2 Proof of the joint distribution of eigenvalues
2.5.3 Selberg’s integral formula and proof of (2.5.4)
2.5.4 Joint distribution of eigenvalues: alternative formulation
2.5.5 Superposition and decimation relations
2.6 Large deviations for random matrices
2.6.1 Large deviations for the empirical measure
A large deviation upper bound
A large deviation lower bound
2.6.2 Large deviations for the top eigenvalue
2.7 Bibliographical notes
3 Hermite polynomials, spacings and limit distributions for the Gaussian ensembles
3.1 Summary of main results: spacing distributions in the bulk and edge of the spectrum for the Gaussian ensembles
3.1.1 Limit results for the GUE
3.1.2 Generalizations: limit formulas for the GOE and GSE
3.2 Hermite polynomials and the GUE
3.2.1 The GUE and determinantal laws
3.2.2 Properties of the Hermite polynomials and oscillator wave-functions
3.3 The semicircle law revisited
3.3.1 Calculation of moments of LN
3.3.2 The Harer–Zagier recursion and Ledoux’s argument
3.4 Quick introduction to Fredholm determinants
3.4.1 The setting, fundamental estimates and definition of the Fredholm determinant
3.4.2 Definition of the Fredholm adjugant, Fredholm resolvent and a fundamental identity
Multiplicativity of Fredholm determinants
3.5 Gap probabilities at 0 and proof of Theorem 3.1.1
3.5.1 The method of Laplace
3.5.2 Evaluation of the scaling limit: proof of Lemma 3.5.1
3.5.3 A complement: determinantal relations
3.6 Analysis of the sine-kernel
3.6.1 General differentiation formulas
3.6.2 Derivation of the differential equations: proof of Theorem 3.6.1
3.6.3 Reduction to Painlev´e V
3.7 Edge-scaling: proof of Theorem 3.1.4
3.7.1 Vague convergence of the largest eigenvalue: proof of Theorem 3.1.4
3.7.2 Steepest descent: proof of Lemma 3.7.2
3.7.3 Properties of the Airy functions and proof of Lemma 3.7.1
3.8 Analysis of the Tracy–Widom distribution and proof of Theorem 3.1.5
3.8.1 The first standard moves of the game
3.8.2 The wrinkle in the carpet
3.8.3 Linkage to Painlev´e II
3.9 Limiting behavior of the GOE and the GSE
3.9.1 Pfaffians and gap probabilities
Pfaffian integration formulas
Determinant formulas for squared gap probabilities
3.9.2 Fredholm representation of gap probabilities
Matrix kernels and a revision of the Fredholm setup
Statements of main results
3.9.4 Differential equations
Block matrix calculations
3.10 Bibliographical notes
4.1 Joint distribution of eigenvalues in the classical matrix ensembles
4.1.1 Integration formulas for classical ensembles
Laguerre ensembles and Wishart matrices
Jacobi ensembles and random projectors
The classical compact Lie groups
4.1.2 Manifolds, volume measures and the coarea formula
4.1.3 An integration formula of Weyl type
4.1.4 Applications of Weyl’s formula
4.2 Determinantal point processes
4.2.1 Point processes: basic definitions
4.2.2 Determinantal processes
4.2.3 Determinantal projections
4.2.4 The CLT for determinantal processes
4.2.5 Determinantal processes associated with eigenvalues
4.2.6 Translation invariant determinantal processes
4.2.7 One-dimensional translation invariant determinantal processes
The biorthogonal ensembles
Birth–death processes conditioned not to intersect
4.3 Stochastic analysis for random matrices
4.3.1 Dyson’s Brownian motion
4.3.2 A dynamical version of Wigner’s Theorem
4.3.3 Dynamical central limit theorems
4.3.4 Large deviation bounds
4.4 Concentration of measure and random matrices
4.4.1 Concentration inequalities for Hermitian matrices with independent entries
Entries satisfying Poincare’s
inequality
Matrices with bounded entries and Talagrand’s method
4.4.2 Concentration inequalities for matrices with dependent entries
The setup with M = Rm and µ=Lebesgue measure
The setup with M a compact Riemannian manifold
Applications to random matrices
4.5 Tridiagonal matrix models and the β ensembles
4.5.1 Tridiagonal representation of β ensembles
4.5.2 Scaling limits at the edge of the spectrum
4.6 Bibliographical notes
5.1 Introduction and main results
5.2 Noncommutative laws and noncommutative probability spaces
5.2.1 Algebraic noncommutative probability spaces and laws
5.2.2 C-probability spaces and the weak*-topology
5.2.3 W-probability spaces
Laws of self-adjoint operators
5.3.1 Independence and free independence
5.3.2 Free independence and combinatorics
Basic properties of non-crossing partitions
Free cumulants and freeness
5.3.3 Consequence of free independence: free convolution
Multiplicative free convolution
5.3.4 Free central limit theorem
5.3.5 Freeness for unbounded variables
5.4 Link with random matrices
5.5 Convergence of the operator norm of polynomials of independent GUE matrices
5.6 Bibliographical notes
A Linear algebra preliminaries
A.1 Identities and bounds
A.2 Perturbations for normal and Hermitian matrices
A.3 Noncommutative matrix Lp-norms
A.4 Brief review of resultants and discriminants
B Topological preliminaries
B.2 Topological vector spaces and weak topologies
B.3 Banach and Polish spaces
B.4 Some elements of analysis
C Probability measures on Polish spaces
D Basic notions of large deviations
E The skew field H of quaternions and matrix theory over F
E.1 Matrix terminology over F and factorization theorems
E.2 The spectral theorem and key corollaries
E.3 A specialized result on projectors
E.4 Algebra for curvature computations
F.1 Manifolds embedded in Euclidean space
Lie groups and Haar measure
F.2 Proof of the coarea formula
F.3 Metrics, connections, curvature, Hessians, and the Laplace–Beltrami operator
Curvature of classical compact Lie groups
G Appendix on operator algebras
G.3 States and positivity
G.5 Noncommutative functional calculus
H Stochastic calculus notions
General conventions and notation
An Introduction to Random Matrices
( Cambridge Studies in Advanced Mathematics )
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