Stochastic processes and random matrices : lecture notes of the Les Houches Summer School.

Summary The field of stochastic processes and random matrix theory (RMT) has been a rapidly evolving subject during the past fifteen years where 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 past 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 ensemble of random matrices. These chapters not only cover this topic in detail but also present more recent developments that have emerged from these discoveries, for instance in the context of low-dimensional heat transport (on the physics side) or in the context of integrable probability (on the mathematical side).