Georgia Institute of Technology College of Sciences

In the study of random growth models belonging to the Kardar--Parisi--Zhang (KPZ) universality class, a notably successful approach has been to analyze stationary initial conditions defined by the Busemann functions. Recently, this perspective has been extended to handle multiple asymptotic directions simultaneously, but the joint distribution of the Busemann process is more difficult to access, and many aspects of this process remain elusive. In particular, the remarkable independence property present in the exactly solvable setting fails when considering Busemann functions across different directions. In the corner growth model, also known as exponential last-passage percolation (LPP), we prove that, regardless of their different directions, Busemann functions along a down-right path are always negatively associated across each individual direction. In other words, increasing the value of Busemann functions in one direction tends to probabilistically decrease the values of neighboring ones. As an application, we obtain an exponential concentration inequality on the diffusive scale for Busemann functions along a down-right path, in the absence of independence. Joint work with Erik Bates.