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).

About the Author:
Gregory Schehr, Senior Researcher, CNRS - Universite Paris-Sud, France, Alexander Altland, Professor of Theoretical Condensed Matter Physics, Institute for Theoretical Physics, University of Koln, Germany, Yan V. Fyodorov, Professor, Department of Mathematics, King's College London, UK, Neil O'Connell, Professor of Mathematics, Mathematics Institute, University of Warwick, UK, Leticia F. Cugliandolo, Professor, Universite Pierre et Marie Curie, Laboratoire de Physique Theorique et Hautes Energies, CNRS, Paris, France

Gregory Schehr received his PhD in theoretical physics at the Ecole Normale Superieure in Paris in 2003. After a postdoc at the University of Saarland (Germany), he obtained a CNRS position in 2006. Currently, he is a CNRS research director at the Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris-Saclay. His research interests are in statistical mechanics. His main topics include in particular random matrix theory, extreme value statistics, non-equilibrium dynamics, first-passage problems and disordered systems. He received the CNRS bronze medal in 2010.

Alexander Altland received his PhD at the Max-Planck-Institute for Nuclear Physics in Heidelberg in 1991. After postdocs at the Weizmann Institute, Israel, and Cambridge University, he was appointed assistant professor at Bochum University, before he became full professor at Cologne University in 2001. His research interests include the physics of random systems, topological condensed matter, and quantum field theory in general.

Yan V. Fyodorov received his Ph.D. in theoretical & mathematical physics from Petersburg Nuclear Physics Institute, Russia, in 1988. After holding post-doctoral fellowships at University of Essen, Germany and Weizmann Institute, Israel he spent some time as a researcher at the University of Essen, before moving to the UK where he hold professorial positions subsequently at Brunel University, University of Nottingham, and Queen Mary University of London. He is currently a full professor at King's College London. His current research focuses on theory of random matrices and its applications to physics of disordered systems, and beyond. He received Prix Institut Henri Poincare-Gauthier Villars (1999), Bessel Research Award of Alexander von Humboldt foundation (2006) and Leverhulme Research Fellowship (2008).

Neil O'Connell received his Ph.D. in Statistics from the University of California at Berkeley, in 1993. He is currently professor at the University of Bristol. His research focuses on probability, random matrices, representation theory and statistical physics. He received the Ito prize, and the Rollo Davidson Prize.

Leticia F. Cugliandolo received her Ph.D. in theoretical physics from the Universidad Nacional de La Plata, Argentina, in 1991. After post-docs in Universita di Roma I, La Sapienza, and CEA/Saclay she joined the Physics Department at Ecole Normale Superieure de Paris in 1997. She is currently a full professor at Universite Pierre et Marie Curie in Paris and the director of Ecole de Physique des Houches since 2007. Her research focuses on statistical physics and condensed matter problems. She received the Marie Curie Excellence Award, the Guggenheim Fellowship and the Prix Langevin of the French Physical Society.