Directed Landscape in Random Geometry

• Directed landscape is a universal scaling limit for KPZ last-passage percolation, defined as a random continuous function with fractal geometric properties.
• Its metric composition law and KPZ scaling yield a natural directed metric structure, ensuring unique and well-ordered geodesics with precise Hausdorff dimensions.
• The extended framework connects multi-point optimizers, Airy sheet marginals, and Brownian motion, providing deep insights into disordered space-time geometry.

The directed landscape is the canonical universal scaling limit for all last-passage percolation models in the Kardar-Parisi-Zhang (KPZ) universality class. It is a random continuous function $\mathcal{L} : \mathbb{R}^4_\uparrow \to \mathbb{R}$, where $$\mathbb{R}^4_\uparrow = \{ (x, s; y, t) \in \mathbb{R}^4 : s < t \}$$, encoding a disordered space-time geometry with highly nontrivial geodesic structure. Its construction, fractal geometric properties, and association with universal probabilistic and geometric exponents place it as a central object in modern probability theory and random geometry.

1. Definition and Metric Structure

The directed landscape $\mathcal{L}$ is a random continuous function $$\mathcal{L}: \mathbb{R}^4_\uparrow \rightarrow \mathbb{R}$$ characterized by three interlocking properties:

• Metric composition law (reverse triangle inequality): For any $(x,s; y,t) \in \mathbb{R}^4_\uparrow$ and $r \in (s,t)$,

$$\mathcal{L}(x,s; y,t) = \sup \left\{ \mathcal{L}(x,s; z,r) + \mathcal{L}(z,r; y,t) : z \in \mathbb{R} \right\}.$$

• KPZ scaling and stationarity: For any disjoint time-intervals, the fields

$$(t-s)^{-1/3}\mathcal{L}(x (t-s)^{2/3}, s; y (t-s)^{2/3}, t)$$

over each block are independent and identically distributed copies of $\mathcal{L}(\cdot,0;\cdot,1)$.

• Airy sheet marginals: For fixed $s<t$, $(x, y) \mapsto \mathcal{L}(x, s; y, t)$ is an independent Airy sheet of scale $(t-s)^{1/3}$.

These properties induce a natural “directed metric” structure and enforce scaling invariance: under KPZ $1{:}2{:}3$ scaling, for $\lambda > 0$,

$$\mathcal{L}(\lambda^{-2}x,\lambda^{-3}s;\lambda^{-2}y,\lambda^{-3}t) \stackrel{d}{=} \lambda \mathcal{L}(x,s;y,t).$$

The directed landscape is the scaling limit of Brownian last-passage percolation, as well as discrete exponential and log-gamma polymers, and can be explicitly constructed from an almost sure bijection involving collections of independent Brownian motions (Dauvergne et al., 2018, Dauvergne et al., 2024, Zhang, 8 May 2025).

2. Geodesics: Existence, Uniqueness, and Regularity

Given endpoints $(x,s)$ and $(y,t)$, a geodesic is a continuous path $\gamma: [s,t] \rightarrow \mathbb{R}$ maximizing the sum of increments:

$$\mathcal{L}(x,s; y,t) = \sup_{\gamma: \gamma(s)=x, \gamma(t)=y} \inf_{\text{partitions}} \sum_i \mathcal{L}(\gamma(t_{i-1}), t_{i-1}; \gamma(t_i), t_i).$$

Almost surely, there exists a unique geodesic between any two space-time points (Dauvergne et al., 2018, Bates et al., 2019). These paths exhibit several universal regularity properties:

• Planarity ordering: If $x_1 \le x_2$ and $y_1 \le y_2$, any $\gamma_1 \in G(x_1, s; y_1, t)$ lies below any $\gamma_2 \in G(x_2, s; y_2, t)$.
• Leftmost and rightmost geodesics exist almost surely; geodesics are Hölder-$2/3^{-}$ in time (Bates et al., 2019).
• The paths are not truly Hölder-$2/3$; instead, for every $\alpha > 2/3$, the supremum of $|\gamma(t)-\gamma(s)|/|t-s|^\alpha$ diverges as $|t-s|\to 0$ (Dauvergne et al., 2020). Geodesics have exact $3/2$-variation and their weight profiles have cubic variation.

3. Disjoint Geodesics and Hausdorff Dimension Results

A primary focus is the fractal geometry of points with disjoint geodesics:

• For fixed $(x_1,x_2)$, the set of $y$ such that geodesics from $(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$ only coalesce at time $1$ has Hausdorff dimension $1/2$.
• The set of $(x,y)$ pairs supporting two geodesics that intersect only at $0$ and $1$ also has Hausdorff dimension $1/2$ (Bates et al., 2019).

Letting

$$Z_y := \mathcal{L}(x_2,0;y,1) - \mathcal{L}(x_1,0;y,1)$$

defines a nondecreasing continuous function of $y$, corresponding to a singular measure. The support of this measure matches exactly the set of $y$ supporting disjoint geodesics, and its Hausdorff dimension is rigorously shown to be $1/2$, utilizing the local variation points of the difference profile (Bates et al., 2019). The two-variable measure is similarly constructed, and dimension proofs use box-covering arguments combined with disjoint geodesic tail estimates.

Key estimate: For small intervals $(x-\delta,x+\delta)$, $(y-\delta,y+\delta)$, the probability that there exist two disjoint geodesics is at most $O(\delta^{3/2})$. No-arbitrary-closeness results guarantee geodesics from nearby endpoints cannot remain uniformly close unless they intersect.

4. Extended Directed Landscape and Disjoint Optimizers

The extended directed landscape (Dauvergne et al., 2021) generalizes $\mathcal{L}$ to optimizers over $k$ disjoint paths, corresponding to multi-point last-passage percolation and the parabolic Airy line ensemble. For each $(x_1<x_2,\ldots<x_k)$, the directed landscape along $$\{x_1,\ldots,x_k\} \times \{s_0\} \times \mathbb{R} \times \{t_0\}$$ can be represented as a last-passage problem across $k$ locally Brownian functions $B_i$,

$$\mathcal{L}(x_i,s_0;y,t_0) = \sup_{x_1 = y_0 < y_1 < \cdots < y_i = y} \sum_{j=1}^i [B_j(y_j) - B_j(y_{j-1})].$$

Disjoint optimizers always exist, and their lengths can be used to characterize extended geodesic networks. Two-variable difference profiles and their scaling properties are locally absolutely continuous with respect to Brownian local time, explaining the universal Hausdorff exponent $1/2$ for their support (Dauvergne, 2021).

5. Fractal Networks and Exceptional Sets

Beyond pairs, the directed landscape supports $27$ distinct geodesic network types between pairs of points. Classification uses combinatorial parameters $(\deg(p), \deg(q))$, and the Hausdorff dimension of the endpoint sets for each network type can be computed explicitly in the $1{:}2{:}3$ metric (Dauvergne, 2023):

$$d_{1{:}2{:}3}((x,s;y,t),(x',s';y',t')) = |x-x'|^{1/2} + |y-y'|^{1/2} + |s-s'|^{1/3} + |t-t'|^{1/3}.$$

For generic endpoints, the geodesic is unique and the endpoints have full dimension $10$; exceptional sets for networks with multiple branches are lower-dimensional (e.g., $2$-edge parallel network endpoints have $d=7$ or $d=6$).

These exceptional structures encode the full local fractal geometry of the landscape and are intimately connected to analogous random surface structures in Liouville quantum gravity and the Brownian map.

6. Difference Profile, Local Variation, and No-Arbitrary-Closeness

For two fixed starting points, the difference profile $Z_y$ encapsulates the geometry of disjoint geodesics. Key ingredients include:

• If two geodesics intersect, $$y \mapsto \mathcal{L}(x_2, s; y, t) - \mathcal{L}(x_1, s; y, t)$$ is locally constant over intervals, and the difference measure vanishes there.
• Disjoint geodesics correspond to points of local variation, whose set has dimension $1/2$, matching the support of the difference measure.
• No-arbitrary-closeness: Geodesics from different nearby endpoints must intersect if they stay close, ruling out "almost disjoint, almost parallel" behavior.

Dimension upper bounds use a covering argument and sharp tail estimates for the number of disjoint geodesics originating and terminating in small intervals. The probability of two disjoint geodesics is controlled by a universal $O(\varepsilon^{3/2})$ bound, leading to a box-counting dimension calculation.

7. Connections, Applications, and Universality

• The directed landscape unifies the Tracy–Widom distributions, Airy processes, KPZ fixed-point evolutions, and universal fluctuation exponents.
• Its fractal structure allows for precise classification of multiplicity and coalescence events, with Hausdorff exponents serving as universal quantitative signatures.
• Extended objects (Airy sheet, line ensemble) and stochastic calculus identities (Brownian Gibbs resampling, difference profiles) permit fine control over geometric and fractal properties.

In summary, the directed landscape exhibits rigorous fractal geometry, uniquely characterizes exceptional sets of disjoint geodesics, and provides a universal framework for scaling limits in random geometry. Its Hausdorff dimension results are optimal and connect directly to intrinsic geometric features of last-passage percolation paths in planar random environments (Bates et al., 2019).