The KPZ class is a very large set of 1+1 models that are meant to describe random growth interfaces. It is believed that upon scaling, the long time behavior of members in this class is universal and is described by a limiting random object, a Markov process called the KPZ fixed-point. The (one-type) stationary measures for the KPZ fixed-point as well as many models in the KPZ class are known - it is a family of distributions parametrized by some set I_ind that depends on the model. For k\in \mathbb{N} the k-type stationary distribution with intensities \rho_1,...,\rho_k \in I_ind is a coupling of one-type stationary measures of indices \rho_1,...,\rho_k that is stationary with respect to the model dynamics. In this talk we will present recent progress in our understanding of the multi-type stationary measures of the KPZ fixed-point as well as the scaling limit of multi-type stationary measures of two families of models in the KPZ class: metric-like models (e.g. last passage percolation) and particle systems (e.g. exclusion process).
Based on joint work with Timo Seppalainen and Evan Sorensen.