Shocks, instability, and the twenty networks of infinite geodesics in the Directed Landscape

Abstract

For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another fundamental structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we study the KPZ fixed point, the central object of the KPZ universality class, which can be viewed as a prototype--albeit degenerate--of an inviscid SHJ equation in one spatial dimension. We describe the geometric structure of the instability region and give a detailed and precise analysis of its interplay with the shock structures of the two eternal solutions. We show that these shock structures allow one to reconstruct the instability region. Along the way, we obtain a complete classification of all possible configurations of semi-infinite geodesics emanating from arbitrary space-time points, in the directed landscape--the random environment in which the KPZ fixed point evolves.