Classification of the eternal solutions and multiple coalescing shocks in the KPZ fixed point

Abstract

We give a complete classification of the eternal solutions for the KPZ fixed point. Each of these is a (possibly infinite) max-plus convolution of the known eternal solutions, called Busemann functions. Specifically, we show that the space of eternal solutions is homeomorphic to a certain space of upper semicontinuous functions that encodes the weights of each of the Busemann functions. As a result, we show that the Busemann process gives a spectral decomposition of eternal solutions of the KPZ fixed point analogous to that for Hamilton-Jaccobi equations. The resulting evolution of the KPZ fixed point exhibits a shock at each of the boundaries between the different Busemann functions. Moving forward in time, the shocks coalesce, while moving backwards in time, additional shocks can form. We describe several geometric properties of this tree of shocks. This completes the study of eternal solutions initiated in earlier work of the second and third authors with Sepp\"al\"ainen and continued in the previous work of the authors and in recent work of Rassoul-Agha and Sweeney.