Intro; Preface; Contents; 1 Getting Started; 1.1 One-Pager on Random Variables; 2 Value the Eigenvalue; 2.1 Appetizer: Wigner's Surmise; 2.2 Eigenvalues as Correlated Random Variables; 2.3 Compare with the Spacings Between i.i.d.'s; 2.4 Jpdf of Eigenvalues of Gaussian Matrices; References; 3 Classified Material; 3.1 Count on Dirac; 3.2 Layman's Classification; 3.3 To Know More ... ; References; 4 The Fluid Semicircle; 4.1 Coulomb Gas; 4.2 Do It Yourself (Before Lunch); References; 5 Saddle-Point-of-View; 5.1 Saddle-Point. What's the Point?; 5.2 Disintegrate the Integral Equation. 5.3 Better Weak Than Nothing5.4 Smart Tricks; 5.5 The Final Touch; 5.6 Epilogue; 5.7 To Know More ... ; References; 6 Time for a Change; 6.1 Intermezzo: A Simpler Change of Variables; 6.2 ... that Is the Question; 6.3 Keep Your Volume Under Control; 6.4 For Doubting Thomases ... ; 6.5 Jpdf of Eigenvalues and Eigenvectors; 6.6 Leave the Eigenvalues Alone; 6.7 For Invariant Models ... ; 6.8 The Proof; References; 7 Meet Vandermonde; 7.1 The Vandermonde Determinant; 7.2 Do It Yourself; References; 8 Resolve(nt) the Semicircle; 8.1 A Bit of Theory; 8.2 Averaging; 8.3 Do It Yourself. 8.4 Localize the Resolvent8.5 To Know More ... ; References; 9 One Pager on Eigenvectors; References; 10 Finite N; 10.1 β=2 is Easier; 10.2 Integrating Inwards; 10.3 Do It Yourself; 10.4 Recovering the Semicircle; References; 11 Meet Andréief; 11.1 Some Integrals Involving Determinants; 11.2 Do It Yourself; 11.3 To Know More ... ; References; 12 Finite N Is Not Finished; 12.1 β=1; 12.2 β=4; References; 13 Classical Ensembles: Wishart-Laguerre; 13.1 Wishart-Laguerre Ensemble; 13.2 Jpdf of Entries: Matrix Deltas ... ; 13.3 ... and Matrix Integrals; 13.4 To Know More ... ; References. 14 Meet MarÄ#x8D;enko and Pastur14.1 The MarÄ#x8D;enko-Pastur Density; 14.2 Do It Yourself: The Resolvent Method; 14.3 Correlations in the Real World and a Quick Example: Financial Correlations; References; 15 Replicas ... ; 15.1 Meet Edwards and Jones; 15.2 The Proof; 15.3 Averaging the Logarithm; 15.4 Quenched versus Annealed; References; 16 Replicas for GOE; 16.1 Wigner's Semicircle for GOE: Annealed Calculation; 16.2 Wigner's Semicircle: Quenched Calculation; 16.2.1 Critical Points; 16.2.2 One Step Back: Summarize and Continue; References; 17 Born to Be Free. 17.1 Things About Probability You Probably Already Know17.2 Freeness; 17.3 Free Addition; 17.4 Do It Yourself; References.

Modern developments of Random Matrix Theory as well as pedagogical approaches to the standard core of the discipline are surprisingly hard to find in a well-organized, readable and user-friendly fashion. This slim and agile book, written in a pedagogical and hands-on style, without sacrificing formal rigor fills this gap. It brings Ph. D. students in Physics, as well as more senior practitioners, through the standard tools and results on random matrices, with an eye on most recent developments that are not usually covered in introductory texts. The focus is mainly on random matrices with real spectrum. The main guiding threads throughout the book are the Gaussian Ensembles. In particular, Wigner's semicircle law is derived multiple times to illustrate several techniques (e.g., Coulomb gas approach, replica theory). Most chapters are accompanied by Matlab codes (stored in an online repository) to guide readers through the numerical check of most analytical results.