For directions xi in the random countable dense set Xi at which the Busemann process of the directed landscape is discontinuous, the set Split_{s,xi} of spatial points from which two disjoint xi-directed semi-infinite geodesics emanate is known to have Hausdorff dimension 1/2 at each fixed time level s, but on an s-dependent event of full probability, so the statement is known only level by level (and, by countable intersection, simultaneously for all rational s). The open problem, posed in the open-problems section of Busani-Seppalainen-Sorensen and restated as a belief by Rassoul-Agha-Sweeney in 2026, is to prove that on a single event of full probability the splitting set has Hausdorff dimension exactly 1/2 simultaneously for every time level s in R and every direction xi in Xi. This is the surviving open core of a three-item open-question bundle from Sorensen's dissertation on the stationary horizon: the item asking whether every semi-infinite geodesic is a Busemann geodesic was resolved by Busani (2024), while the directed-landscape universality item remains a field-wide program without a single completion boundary and is documented here as context.

Background

The Kardar-Parisi-Zhang (KPZ) universality class is the family of random growth and directed transport models whose fluctuation exponents are governed by the exponent triple (1/2, 1/3, 2/3), introduced through the physical scaling theory of Kardar, Parisi and Zhang. For exactly solvable members of the class the large-scale limit is understood in detail. Its two central objects are the directed landscape L\mathcal{L} of Dauvergne-Ortmann-Virag, a random directed metric on the plane in which L(p;q)\mathcal{L}(p;q), for p=(x,s)p=(x,s) and q=(y,t)q=(y,t) with s<ts<t, is the optimal passage cost from pp to qq, and the KPZ fixed point of Matetski-Quastel-Remenik, the Markov process on real-valued functions that is the 1:2:3-scaling limit of KPZ models and whose time evolution is generated by L\mathcal{L}. A geodesic is a path attaining L(p;q)\mathcal{L}(p;q); a semi-infinite geodesic is a continuous path π:[s,∞)→R\pi:[s,\infty)\to\mathbb{R}, each finite segment of which is a geodesic, and π\pi has direction ξ\xi if π(t)/t→ξ\pi(t)/t\to\xi as t→∞t\to\infty. The Busemann process {Wξ±}ξ∈R\{W^{\xi\pm}\}_{\xi\in\mathbb{R}} of the landscape records the almost-sure limits of landscape increments measured against a point escaping to infinity in direction ξ\xi; the sign ±\pm distinguishes the two possible one-sided limits. Busani-Seppalainen-Sorensen identified the law of this process as the stationary horizon (SH), the multi-type invariant measure of the KPZ class: the coupling of two-sided Brownian motions with drift indexed by tilt ξ\xi that describes the jointly stationary (eternal) solutions of the KPZ fixed point. Sorensen's dissertation, from which this problem is taken, synthesizes that program. The process ξ↦Wξ±\xi\mapsto W^{\xi\pm} has jumps precisely on a random countable dense set Ξ\Xi of directions. These are the instability directions: for ξ∈Ξ\xi\in\Xi there are two distinct eternal solutions of the KPZ fixed point with asymptotic velocity ξ\xi (the profiles Wξ−W^{\xi-} and Wξ+W^{\xi+}), and from every space-time point there emanate two eventually disjoint ξ\xi-directed semi-infinite geodesics, the leftmost ξ−\xi^- geodesics and the rightmost ξ+\xi^+ geodesics, each family coalescing internally; for ξ∉Ξ\xi\notin\Xi all ξ\xi-directed semi-infinite geodesics coalesce. By a theorem of Busani, building on Bhatia and Dauvergne, every semi-infinite geodesic in the directed landscape is a Busemann geodesic, that is, it is eventually obtained by successively maximizing one of the Busemann functions Wξ±W^{\xi\pm}. For ξ∈Ξ\xi\in\Xi and a time level ss, the splitting set Splits,ξ⊂R\mathrm{Split}_{s,\xi}\subset\mathbb{R} is the set of spatial locations xx such that from (x,s)(x,s) there emanate two ξ\xi-directed semi-infinite geodesics that are disjoint except for the shared initial point; equivalently, it is where the ξ−\xi^- and ξ+\xi^+ geodesics separate immediately instead of first running together. Up to the removal of an at most countable set, Splits,ξ\mathrm{Split}_{s,\xi} is the set of points of increase of the nondecreasing Busemann difference profile fs,ξ(x)=Wξ+(x,s;0,s)−Wξ−(x,s;0,s)f_{s,\xi}(x)=W^{\xi+}(x,s;0,s)-W^{\xi-}(x,s;0,s), and the union of these sets over (s,ξ)(s,\xi) is the space-time locus where the two geodesic families come apart, i.e. the shock set of the KPZ fixed point. Busani-Seppalainen-Sorensen proved: for each fixed ss, on an ss-dependent event of full probability, Splits,ξ\mathrm{Split}_{s,\xi} has Hausdorff dimension 1/21/2 for every ξ∈Ξ\xi\in\Xi at once (countability of Ξ\Xi makes the ξ\xi-uniformity immediate); on a single event of full probability, Splits,ξ\mathrm{Split}_{s,\xi} is simultaneously nonempty and unbounded in both directions for all (s,ξ)(s,\xi); and the dimension value comes from the identification, under Palm conditioning on ξ∈Ξ\xi\in\Xi (a conditioning on the zero-probability event of discontinuity, made sense of by disintegration), of fs,ξf_{s,\xi} with the running maximum of a two-sided Brownian motion, i.e. Brownian local time, which reduces the dimension of its support to the classical fact that the zero set of Brownian motion has Hausdorff dimension 1/21/2. In the exactly solvable prelimiting models, exponential and Brownian last-passage percolation, the analogous geodesic and shock structures are understood through the same circle of ideas, and several global statements that remain open in the directed landscape, such as the all-time-level analogues of Rassoul-Agha-Sweeney's instability questions, admit affirmative answers there. The gap between the fixed-level and the global statement is the problem. The dimension statement is currently available level by level, and by countable intersection simultaneously for all rational ss. Rassoul-Agha-Sweeney, who re-derived the fixed-level statement in their study of shocks and instability in the landscape and in Brownian last-passage percolation, record that it is believed to hold for all s∈Rs\in\mathbb{R} on one event of full probability. Related fractal geometry of the landscape, such as the dimension-4/34/3 non-uniqueness sets (Bhatia), Dauvergne's classification of the 27 geodesic network types, and Ganguly-Zhang's analysis of the space-time difference profile, does not address simultaneity in the time variable. No continuity, monotonicity or measurability of Splits,ξ\mathrm{Split}_{s,\xi} in ss is known (Busemann-process monotonicity holds only along horizontal directions), which is why the countable-intersection argument cannot be closed.

Problem Statement

Prove or disprove the following statement about the directed landscape L\mathcal{L} of Dauvergne-Ortmann-Virag. Let Ξ\Xi be the random countable dense set of directions at which the Busemann process of L\mathcal{L} is discontinuous, and for s∈Rs\in\mathbb{R} and ξ∈Ξ\xi\in\Xi let Splits,ξ={x∈R: there exist two disjoint semi-infinite geodesics emanating from (x,s) in direction ξ}\mathrm{Split}_{s,\xi}=\{x\in\mathbb{R}:\ \text{there exist two disjoint semi-infinite geodesics emanating from }(x,s)\text{ in direction }\xi\} be the splitting set of Busani-Seppalainen-Sorensen, definitions (2.6)-(2.7) (disjointness allows the shared initial point; equivalently, up to the removal of an at most countable set, Splits,ξ\mathrm{Split}_{s,\xi} is the set of points of increase of the nondecreasing Busemann difference profile fs,ξ(x)=Wξ+(x,s;0,s)−Wξ−(x,s;0,s)f_{s,\xi}(x)=W^{\xi+}(x,s;0,s)-W^{\xi-}(x,s;0,s)). Prove, or disprove, that there exists an event Ω0\Omega_0 of probability one on which, simultaneously for every time level s∈Rs\in\mathbb{R} and every direction ξ∈Ξ\xi\in\Xi, the set Splits,ξ\mathrm{Split}_{s,\xi} has Hausdorff dimension exactly 1/21/2. This is item (ii) of the open-problems Section 9 of Busani-Seppalainen-Sorensen ('Does Ss,ξS_{s,\xi} have Hausdorff dimension 1/2 simultaneously for all s∈Rs\in\mathbb{R} and ξ∈Ξ\xi\in\Xi?'), recorded as a stated expectation in the remark following the splitting-points theorem of Sorensen's dissertation; Rassoul-Agha-Sweeney state the equivalent belief that their window version, 'Ssθ∩(a,b)\mathcal{S}^{\theta}_{s}\cap(a,b) is empty or has Hausdorff dimension 1/21/2 for a<ba<b', holds on an event of full probability for all s∈Rs\in\mathbb{R}, whereas it is currently proven only for rational ss. A disproof must exhibit, with proof, an event of positive probability on which the simultaneous statement fails. An answer establishing the conclusion only for each fixed ss, only for a countable dense set of ss, or only for a substitute object does not answer the question.

References

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