Introduction to Random Matrices: Theory and Practice

✍ Scribed by Giacomo Livan,Marcel Novaes,Pierpaolo Vivo (auth.)

• Publisher: Springer International Publishing
• Year: 2018
• Tongue: English
• Leaves: 122
• Series: SpringerBriefs in Mathematical Physics 26
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter ....Pages i-ix
Getting Started (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 1-5
Value the Eigenvalue (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 7-13
Classified Material (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 15-21
The Fluid Semicircle (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 23-31
Saddle-Point-of-View (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 33-43
Time for a Change (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 45-51
Meet Vandermonde (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 53-56
Resolve(nt) the Semicircle (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 57-63
One Pager on Eigenvectors (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 65-66
Finite N (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 67-74
Meet Andréief (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 75-79
Finite N Is Not Finished (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 81-87
Classical Ensembles: Wishart-Laguerre (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 89-95
Meet Marčenko and Pastur (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 97-103
Replicas... (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 105-108
Replicas for GOE (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 109-117
Born to Be Free (Giacomo Livan, Marcel Novaes, Pierpaolo Vivo)....Pages 119-124

✦ Subjects

Mathematical Methods in Physics