Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School: Volume 104, July 2015Gregory Schehr, Yan V. Fyodorov, Alexander Altland, Neil O'Connell, Leticia F. Cugliandolo Oxford University Press, 2017 - 613 Seiten The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side). |
Inhalt
stochastic vertex models and symmetric functions | 26 |
3 Free probability | 132 |
a statistical physics perspective | 177 |
5 Random matrix theory and quantum chromodynamics | 228 |
6 Random matrix theory and big data analysis | 283 |
7 Random matrices and loop equations | 304 |
some recent themes | 348 |
a playground for physicists? | 382 |
13 Quantum spin chains and classical integrable systems | 578 |
131 Introduction | 580 |
132 The master Toperator for spin chains | 584 |
133 From the master Toperator to the classical RS model and back | 596 |
134 Spectrum of the spin chain Hamiltonians from the classical RS model | 601 |
135 The QC correspondence via nested Bethe ansatz | 605 |
136 Concluding remarks | 608 |
Hamiltonian formulation of the RS model | 609 |
10 Random matrix approaches to open quantum systems | 409 |
11 Impurity models and products of random matrices | 474 |
12 Gaussian multiplicative chaos and Liouville quantum gravity | 548 |
Acknowledgements | 610 |