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

A co-publication of the AMS and IAS/Park City Mathematics Institute Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory. This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research. 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-N asymptotics and large deviations for the Ginibre ensemble Counting multiple equilibria via Kac-Rice formulas Mean number of equilibria: asymptotic analysis for large deviations Appendix: Supersymmetry and characteristic polynomials of real Ginibre matrices. Exercises with hints A Short Introduction to Operator Limits of Random Matrices The Gaussian Ensembles The Gaussian Orthogonal and Unitary Ensembles. Tridiagonalization and spectral measure. β-ensembles. Graph Convergence and the Wigner semicircle law Graph convergence. Wigner’s semicircle law. The top eigenvalue and the Baik-Ben Arous-Péché transition The top eigenvalue. Baik-Ben Arous-Péché transition. The Stochastic Airy Operator Global and local scaling. The heuristic convergence argument at the edge. The bilinear form SAOβ. Convergence to the Stochastic Airy Operator. Tails of the Tracy Widomβ distribution. Related Results The Bulk Limit The Hard–edge Limit Universality of local processes Properties of the limit processes Spiked matrix models and more on the BBP transition Sum rules via large deviations The Stochastic Airy semigroup From the totally asymmetric simple exclusion process to the KPZ fixed point The totally asymmetric simple exclusion process The growth process. Distribution function of TASEP Proof of Schütz’s formula using Bethe ansatz. Direct check of Schütz’s formula. Determinantal point processes Probability of an empty region. L-ensembles of signed measures. Conditional L-ensembles. Biorthogonal representation of the correlation kernel Non-intersecting random walks. The correlation kernel of the signed measure. Explicit formulas for the correlation kernel Finite initial data. Correlation kernel as a transition probability. Path integral formulas. Proof of the TASEP path integral formula. The KPZ fixed point State space and topology. Auxiliary operators. The KPZ fixed point formula. Symmetries and invariance. Markov property. Regularity and local Brownian behavior. Variational formulas and the Airy sheet The 1:2:3 scaling limit of TASEP From one-sided to two-sided formulas. Continuum limit. Delocalization of eigenvectors of random matrices Introduction Reduction of no-gaps delocalization to invertibility of submatrices From no-gaps delocalization to the smallest singular value bounds The ε-net argument. Small ball probability for the projections of random vectors Density of a marginal of a random vector. Small ball probability for the image of a vector. No-gaps delocalization for matrices with absolutely continuous entries. Decomposition of the matrix The negative second moment identity B is bounded below on a large subspace E+ G is bounded below on the small complementary subspace E− Extending invertibility from subspaces to the whole space. Applications of the no-gaps delocalization Erdős-Rényi graphs and their adjacency matrices Nodal domains of the eigenvectors of the adjacency matrix Spectral gap of the normalized Laplacian and Braess’s paradox Microscopic description of Log and Coulomb gases Introduction and motivations Fekete points and approximation theory Statistical mechanics Two component plasmas Random matrix theory Complex geometry and theoretical physics Vortices in condensed matter physics Equilibrium measure and leading order behavior The macroscopic behavior: empirical measure Large Deviations Principle at leading order Further questions Splitting of the Hamiltonian and electric approach The splitting formula Electric interpretation The case d =1 The electric energy controls the fluctuations Consequences for the energy and partition function Consequence: concentration bounds CLT for fluctuations in the logarithmic cases Reexpressing the fluctuations as a ratio of partition functions Transport and change of variables Energy comparison Computing the ratio of partition functions Conclusion in the one-dimensional one-cut regular case Conclusion in the two-dimensional case or in the general one-cut case The renormalized energy Definitions Scaling properties Partial results on the minimization of W, crystallization conjecture Renormalized energy for point processes Lower bound for the energy in terms of the empirical field Large Deviations Principle for empirical fields Specific relative entropy Statement of the main result Proof structure Screening and consequences Generating microstates and conclusion Random matrices and free probability Introduction. Lecture 0: Non-commutative probability spaces. Executive summary. Non-commutative measure spaces. Non-commutative probability spaces. Summary: non-commutative measure spaces. Exercises. Lecture 1: Non-commutative Laws. Classical and Free Independence. Executive summary. Non-commutative laws. Examples of non-commutative probability spaces and laws. Notions of independence. The free Gaussian Functor. Exercises Lecture 2: R-transform and Free Harmonic Analysis. Executive summary. Additive and multiplicative free convolutions. Computing ⊞: R-transform. Combinatorial interpretation of the R-transform. Properties of free convolution. Free subordination. Multiplicative convolution ⊠. Other operations. Multivariable and matrix-valued results. Exercises. Lecture 3: Free Probability Theory and Random Matrices. Executive summary. Non-commutative laws of random matrices. Random matrix models. GUE matrices: bound on eigenvalues. GUE matrices and Stein’s method: proof of Theorem 3.3.5(1’). On the proof of Theorem 3.3.5(2) and (3). Exercises. Lecture 4: Free Entropy Theory and Applications. Executive summary. More on free difference quotient derivations. Free difference quotients and free subordination. Free Fisher information and non-microstates free entropy. Free entropy and large deviations: microstates free entropy χ. χ vs χ*. Lack of atoms. Exercises Addendum: Free Analogs of Monotone Transport Theory. Classical transport maps. Non-commutative transport. The Monge-Ampère equation. The Free Monge-Ampère equation. Random Matrix Applications. Least singular value, circular law, and Lindeberg exchange 1. The least singular value 1.1 The epsilon-net argument 1.2 Singularity probability 1.3 Lower bound for the least singular value 1.4 Upper bound for the least singular value 1.5 Asymptotic for the least singular value 2. The circular law 2.1 Spectral instability 2.2 Incompleteness of the Moment Method 2.3 The logarithmic potential 3. The Lindeberg exchange method Back Cover