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This paper describes the geometric picture of where uniqueness fails for a central random growth model, and how that picture is tied to sudden jumps called shocks. The authors study the KPZ fixed point, a universal limit that describes many one-dimensional random growth processes, and the random environment it lives in, the directed landscape. They show how shock lines and optimal paths in the landscape determine the set of space-time points where two distinct long-running solutions with the same average slope differ. The title’s ‘‘twenty networks’’ refers to the complete list the authors give of possible ways infinite optimal paths can meet and branch in this model.
The researchers work in the inviscid (zero-viscosity) setting of stochastic Hamilton–Jacobi equations, for which the KPZ fixed point is a degenerate example. They focus on two related structures. Instability points are places in space and time where two ‘‘eternal’’ solutions with the same asymptotic velocity take different values. Shocks are places where the velocity field jumps. Using the directed landscape’s variational description—where values are given by the best passage times of continuous paths—they analyze how geodesics (maximizing paths) and shock lines interact and how that interaction builds up the instability region.
At a high level their method studies the web of semi-infinite geodesics that start from every space-time point. These geodesics are the maximizers of the same variational formula that defines the directed landscape. By classifying all the ways such semi-infinite geodesics can meet and split, the authors obtain a full catalogue of allowed local configurations. They then link these configurations to shock trees of the two eternal solutions and show that those shock trees let one reconstruct a distinguished subset of the instability region. That subset, the authors call the ‘‘skeleton’’ of the instability graph, is the union of countably many boundaries of what they call stability islands, and its closure equals the full instability set.
The paper gives several concrete geometric conclusions. The instability region is a closed, path-connected graph that is nowhere dense and has no isolated points. The classification of semi-infinite geodesic configurations is complete (the paper identifies the finite list of possible ‘‘networks’’). The authors also locate each type of configuration inside the instability region, so the work ties the local combinatorics of geodesics to the global geometry of instability. These results make the previously vague idea of ‘‘where uniqueness breaks’’ into a precise, geometric object in this model.
There are important caveats. All results are proved for the KPZ fixed point and the directed landscape, a specific and well-studied continuum model. The authors describe the KPZ fixed point as a degenerate inviscid prototype of stochastic Hamilton–Jacobi equations, and they expect similar phenomena in related models, but the paper does not claim a general theorem for all stochastic Hamilton–Jacobi equations. The work also raises open questions—for example about whether instability points are associated with a notion of energy dissipation at shocks—so some physical interpretations remain to be settled. Finally, much of the paper is technical and the reported figures illustrate the structures; the rigorous statements and proofs are given in the main text.