The Martin boundary of the Directed Landscape

Abstract: In the directed landscape, the Martin boundary coincides with the horofunction boundary. We show that functions in this boundary are precisely the eternal solutions possessing a spatial growth rate, and that the minimal Martin boundary is given by the Busemann functions. Moreover, every eternal solution can be expressed as a max-plus convex combination of countably many Busemann functions. Horofunctions are exactly those eternal solutions that admit a representation in terms of at most two Busemann functions with a common growth rate. As a consequence of instability, not all horofunctions are Busemann functions, and the Martin boundary is strictly larger than its minimal part.

• The paper proves that every eternal solution decomposes into a countable max-plus combination of minimal Busemann functions, establishing a Choquet-type theorem for the directed landscape.
• It shows that the Martin boundary equals the horofunction boundary, while horofunctions have at most two Busemann components with a common spatial growth rate of 2α.
• Geodesic instability in exceptional directions produces distinct Busemann functions with the same slope, proving that the full Martin boundary is strictly larger than its minimal part.

The paper studies the max-plus potential theory of the directed landscape $\mathcal L$, the random field that drives the KPZ fixed point (2605.21366). Eternal solutions—functions $h:\mathbb R^2\to\mathbb R$ satisfying the dynamic programming equation $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$ for all $s<t$—are precisely the harmonic functions of $\mathcal L$ in the max-plus algebra, and they form a semimodule over $(\mathbb R\cup\{-\infty\},\max,+)$. The paper identifies the Martin boundary of this structure, connects it to the horofunction boundary of metric geometry, and shows that both are generated by Busemann functions.

Setting and background

The directed landscape satisfies the composition law $$\mathcal L(x,s;y,t)=\sup_z(\mathcal L(x,s;z,r)+\mathcal L(z,r;y,t))$$ for $t>r>s$ and admits a variational (last-passage) representation over continuous space-time paths, so that point-to-point geodesics exist and semi-infinite geodesics arise as limits as $t\to\infty$. Via the Hopf–Lax–Oleinik formula for the KPZ fixed point, geodesics play the role of characteristic curves, and eternal solutions correspond to conserved quantities along characteristics.

A central object is the family of Busemann functions: for a direction $\alpha\in\mathbb R$ and sign $h:\mathbb R^2\to\mathbb R$0,

$h:\mathbb R^2\to\mathbb R$1

where $h:\mathbb R^2\to\mathbb R$2 is an $h:\mathbb R^2\to\mathbb R$3-directed semi-infinite geodesic from $h:\mathbb R^2\to\mathbb R$4; the limit exists by coalescence and is independent of the choice among leftmost/middle/rightmost geodesics. Each $h:\mathbb R^2\to\mathbb R$5 is an eternal solution with spatial growth rate $h:\mathbb R^2\to\mathbb R$6 and satisfies the cocycle identity. Crucially, for almost every $h:\mathbb R^2\to\mathbb R$7 there exists a countable dense exceptional set $h:\mathbb R^2\to\mathbb R$8 such that if $h:\mathbb R^2\to\mathbb R$9 then the leftmost and rightmost $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$0-directed geodesics separate immediately from certain starting points, producing distinct functions $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$1 with the same slope $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$2. This instability phenomenon is what ultimately forces the Martin boundary to be non-minimal.

Main results

Three theorems constitute the core contribution, all holding on events of full probability.

Choquet-type decomposition. Every eternal solution normalized at $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$3 admits the representation

$h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$4

with coefficients $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$5 having supremum zero; moreover only countably many pairs $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$6 contribute. The coefficient attached to $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$7 is read off from any $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$8-geodesic coalescing with the corresponding Busemann geodesic. Combined with minimality of each $h(x,s)=\sup_y\{\mathcal L(x,s;y,t)+h(y,t)\}$9 (proved via a lemma showing that two Busemann functions bounded above by a constant multiple of one another must share slope, and sign when the slope is exceptional), this identifies the minimal Martin boundary with the set of Busemann functions and realizes all harmonic functions as their max-plus convex combinations—a Choquet theorem for the max-plus cone of eternal solutions.

Characterization of horofunctions. Horofunctions, defined as local uniform limits of $s<t$0 with $s<t$1, are exactly the eternal solutions possessing a spatial growth rate $s<t$2 (which then holds at every time slice). Any such function has the two-component form

$s<t$3

and conversely every function of this form arises as such a limit, with terminal points satisfying $s<t$4. Hence the Martin boundary coincides with the horofunction boundary. A notable corollary of the converse construction: because $s<t$5 is dense and nonempty, there is a continuum of distinct eternal solutions with any given growth rate $s<t$6, parameterized by pairs $s<t$7.

Instability implies non-minimality. Since $s<t$8 for some $s<t$9 whenever $\mathcal L$0, the two-component combinations with both $\mathcal L$1 finite are horofunctions that are not Busemann functions. Thus the full Martin boundary strictly contains its minimal part—an answer, in this random zero-temperature setting, to the classical question of whether all horofunctions are Busemann functions (affirmative for Hadamard spaces, negative e.g. for Teichmüller spaces). The paper emphasizes that in a hypothetical model without instability the entire Martin boundary would be minimal; randomness is the mechanism producing the strict inclusion.

Geometric description and interfaces

The decomposition is given geometric content through generalized competition interfaces. For each time $\mathcal L$2, the map $\mathcal L$3 records which Busemann geodesic the leftmost/rightmost $\mathcal L$4-geodesic out of $\mathcal L$5 eventually follows; these maps are monotone step functions whose jump sets $\mathcal L$6 shrink as $\mathcal L$7 decreases backward in time. On intervals between consecutive jumps, $\mathcal L$8 coincides exactly with a single shifted Busemann function, and the interfaces $\mathcal L$9 separating regions coalesce pairwise in finite backward time—the alternative would be a bi-infinite geodesic, excluded by known results. This yields a tree of coalescing interfaces partitioning space-time into countably many regions where $(\mathbb R\cup\{-\infty\},\max,+)$0 agrees with one Busemann component. In the two-component case the interface is a single bi-infinite curve, recovering and extending constructions based on competition interfaces in prior work, though here the interface machinery serves mainly as bookkeeping for proving that convex combinations are horofunctions uniformly on arbitrary compact subsets of $(\mathbb R\cup\{-\infty\},\max,+)$1, not merely on single time slices.

Technically, the proofs rest on a sub-quadratic growth bound for eternal solutions (derived from the moderate-deviation bound on $(\mathbb R\cup\{-\infty\},\max,+)$2), which guarantees boundedness of maximizers and existence/continuity of leftmost and rightmost $(\mathbb R\cup\{-\infty\},\max,+)$3-geodesics; a max-plus dominated convergence theorem (supremum commutes with locally uniform limits under a dominating superlinearly decaying function) supplies spatial continuity of the KPZ fixed point run from rough terminal data.

Limitations and open questions

Several caveats are explicit. Finiteness of a general max-plus mixture must be checked case by case; the paper cannot guarantee it abstractly, though it holds for finite mixtures of finite solutions. The universality of the conclusions beyond the KPZ fixed point—to viscous and inviscid stochastic Hamilton–Jacobi equations and to planar first-passage percolation—is conjectural rather than proved, since those models lack an established directed-landscape limit or the required geodesic infrastructure. The continuity of the interfaces $(\mathbb R\cup\{-\infty\},\max,+)$4 (which would follow from the competition-interface theory) is noted but not proved, as it is not needed. Whether an analogous Choquet decomposition holds in positive-temperature polymer models, where the max-plus algebra is replaced by ordinary convexity, remains outside the scope of the paper.

Conclusion

The paper establishes that for the directed landscape the Martin boundary equals the horofunction boundary, that the minimal Martin boundary consists exactly of the Busemann functions $(\mathbb R\cup\{-\infty\},\max,+)$5, and that every eternal solution is a countable max-plus convex combination of these extremal elements, with horofunctions characterized as the at-most-two-component combinations sharing a common slope. The strictness of the inclusion of the minimal part within the full boundary is a direct consequence of geodesic instability at exceptional directions, giving a clean instance in a random degenerate Hamilton–Jacobi setting where the classical identification of horofunctions with Busemann functions fails.