KPZ universality, from Kardar, Parisi and Zhang, describes the dynamics of large scale fluctuations in a variety of settings, such as growing interfaces, one-dimensional classical and quantum fluids, or random geometry, which exhibit characteristic long range correlations generated by local interactions. In the past decade, it has become a prominent topic at the interface between non-equilibrium statistical physics and probability theory, with recent experimental observations in a few classical and quantum systems. KPZ fluctuations have in particular been observed in integrable spin chains, classical and quantum, in the early time regime where the correlation length is small compared to the system size. A goal of the proposed internship would be to explore the longer time scale associated with the relaxation of fluctuations to a non-equilibrium stationary state, when correlations eventually span the whole system. There, KPZ fluctuations in finite volume, taking into account boundary effects responsible for the spin current flowing into the system, are expected. During the internship, after familiarizing herself / himself with integrable spin chains and their relation with KPZ fluctuations, the student will explore a quantum spin chain with dissipative boundaries, and especially analytical representations for its stationary density matrix, from which stationary spin fluctuations may be extracted.