From interacting particle systems to random matrices

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1 + 1 dimension. We discuss the large time distribution and processes and their dependence on the class of initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of Hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connection between growth models and random matrices is only partial.