The directed landscape in half-space

Abstract: We prove that two half-space models in the KPZ universality class, exponential last-passage percolation and a family of Poisson-avoiding metrics generalizing colored TASEP, converge to a common scaling limit. This scaling limit is the directed landscape in half-space, a random directed metric in the half-plane indexed by a parameter which determines the strength of the boundary interaction. As part of our analysis, we characterize the half-space directed landscape in terms of the half-space KPZ fixed point, and prove convergence of geodesics. We also give an explicit construction of joint stationary measures (or horizons) in half-space for the log-gamma polymer, the KPZ equation, exponential and geometric last passage percolation, and the directed landscape itself.

• The paper establishes the convergence of half-space exponential LPP and Poisson-avoiding metrics to a unique directed landscape under KPZ 1:2:3 scaling.
• It characterizes the landscape using the triangle inequality, independence of increments, and KPZ fixed point properties, revealing a boundary phase transition.
• The work introduces innovative coupling via stationary horizons and sharp moderate deviation bounds to ensure geodesic convergence and tightness.

The Directed Landscape in Half-Space: Summary and Analysis

Introduction and Context

This work rigorously constructs the directed landscape in half-space, providing the canonical scaling limit for a broad class of stochastic growth models and last-passage percolation (LPP) models in domains with a boundary. It extends the KPZ universality framework from full-space (the entire real line or plane) to half-space geometries, capturing the role of boundary interactions in integrable probabilistic systems. The analysis is focused primarily on two models: exponential LPP in half-space and the half-space Poisson-avoiding (colored TASEP) metric, and provides a new proof approach based on stationary measures, tightness, and limit characterizations.

Main Results

Convergence to Half-Space Directed Landscape

The paper proves that both half-space exponential LPP (with boundary parameter $\alpha$) and half-space Poisson-avoiding metrics (generalizing colored TASEP) admit the same scaling limit as the mesh size vanishes or system size diverges. This limit exists as a continuous, random, directed metric $L^\rho$ on the half-plane indexed by a boundary parameter $\rho=\rho(\alpha)$, which determines the interaction strength at the boundary. The main convergence is uniform-on-compact in the topology of functions on the directed half-plane.

Key Statements:

• For any subcritical or critical boundary regime ($\alpha \geq 1/2$ or properly tuned), after the correct $1:2:3$ KPZ scaling, both models converge in law to the half-space directed landscape $L^\rho$ [(2604.10020), Thm 1.1].
• The scaling limit is unique and independent of regularization details, and the dependence on the boundary parameter undergoes a phase transition.

Characterization of the Half-Space Directed Landscape

The directed landscape is characterized via three properties:

1. Triangle inequality (metric composition law).
2. Independence of increments (Markov property).
3. Half-space KPZ fixed point marginals (variational formulas specified via the fixed point process).

This extends and adapts the Matetski-Quastel-Remenik/Dauvergne-O'Connell-Virág approach from the full-space directed landscape to half-space.

Stationary Horizons and Couplings

An explicit construction of the joint stationary horizon (also called horizon or Busemann field) is provided for the half-space directed landscape and key solvable models: log-gamma polymer, geometric and exponential LPP, and the stochastic KPZ equation. The stationary horizon jointly couples all one-sided stationary measures (stationary for the boundary parameter–indexed slope), and is given by a high-dimensional last-passage structure involving Brownian motions and exponential random variables.

Tightness and Quantitative Estimates

Strong quantitative tightness and moderate deviation bounds are proved using combinatorial identities and geometric inequalities:

• Optimal (up to constants) moderate deviation and two-point estimates for exponential LPP in half-space, crucial for tightness in function space.
• Extension of path-crossing (quadrangle) inequalities in both half-space and symmetric models.
• Substantial advances in the control of geodesic convergence and modulus of continuity, establishing Hölder-regularity (in space and time), shape theorems, and geodesic uniqueness/continuity in the half-space directed landscape.

Geodesic Convergence and Random Geometry

As in the full-space setting, the limit admits a metric composition law for geodesics, and geodesics converge in the scaling limit, inheriting properties such as Hölder-$2/3^-$ continuity and almost-sure uniqueness. The landscape provides a universal metric geometry for half-space random growth and LPP, capturing the KPZ geodesic geometry with boundary.

Technical Innovations

• New construction strategy: The approach bypasses the Airy line ensemble and Pfaffian point process technologies that are formidable in half-space. Instead, it leverages coupling via stationary horizons, sharp moderate deviation bounds, and a characterization theorem based on KPZ fixed-point variational structure.
• Symmetry and parameter permutation identities: The construction of the stationary horizon hinges on new invariance properties under weight parameter permutations, extending techniques from full-space LPP to half-space symmetrizations.
• Coupling techniques for tightness: The development of probabilistic comparison arguments, adapted path-crossing, and two-point fudge estimates, enables tighter control (compared to RSK-based approaches) on prelimit models.
• Extensive handling of multiple models: The framework not only provides the universal scaling limit but also carries over explicit constructions and stationarity for a wide variety of exactly solvable KPZ-class models in half-space, including the log-gamma polymer, geometric LPP, and the stochastic six-vertex model.

Implications and Significance

Theoretical Implications

• Canonical scaling limit with boundaries: The half-space directed landscape is the universal object governing fluctuations and geodesics for a large class of 1+1-dimensional random growth, polymer, and percolation models with boundary.
• Boundary phase transition: The rigorous demonstration of the scaling limit and its four-equivalence-class structure (subcritical, critical, supercritical, and Gaussian) provides clarity regarding KPZ phase diagrams in restricted domains.
• Framework for random geometry: The extension of geodesic existence, uniqueness, and convergence shows that the rich random metric structure discovered in the full-plane persists (with modifications) in the presence of a boundary, including new effects such as changes in geodesic attraction to the boundary.
• Coupling with stationary measures: The explicit, universal coupling of all stationary measures materializes the Busemann function perspective in the presence of boundaries, opening the way for further analysis of infinite-volume geometry and ergodic properties.

Practical/Applied Implications

• Modeling with boundaries: For statistical physics, queueing theory, and interacting particle systems (e.g., TASEP, ASEP, stochastic vertex models), the behavior at and near domain boundaries is now governed by a precise and universal random object.
• Robustness of scaling exponents and shape theorems: The KPZ $1:2:3$ scaling extends, with explicit universal constants, to a broad range of models with different types of boundary interactions, enabling prediction and analysis for further models in stochastic transport and growth.

Future Directions

• Characterization of randomness at the boundary: An open question remains whether the boundary contributes extra randomness to limit objects; resolving this may have implications for higher-dimensional or multi-boundary problems.
• Extensions to strips or wedge domains: The techniques can potentially be adapted to more complex geometries, such as strips (multiple boundaries) and wedges.
• Integrable probability and stochastic analysis: The framework developed here could help to rigorously connect half-space integrable stochastic models to universality conjectures in non-integrable settings, offering a platform for further physical and mathematical exploration.
• Busemann functions in half-space: The detailed behavior, ergodicity, and geometric interpretations of the stationary horizon process (Busemann functionals) in the half-space context, especially for log-gamma and KPZ, point to a rich vein for future ergodic and geometric investigation.

Strong Numerical Results and Notable Claims

• Sharp estimates: One-point and two-point moderate deviation probabilities match optimal exponents, indicating precise control over fluctuations (Gaussian, Tracy-Widom, deep tails).
• Explicit horizon marginals: The explicit description of k-point marginals for the horizon process in terms of maximal functionals of independent Brownian motions and exponentials demonstrates tractability and computational potential [(2604.10020), Prop 1.4].

Conclusion

The results of "The directed landscape in half-space" (2604.10020) establish a robust, explicit, and universal structure for KPZ-class models in the presence of boundaries. By constructing the half-space directed landscape and its stationary horizon, and proving strong convergence and regularity properties, the authors provide the foundational object that governs both fluctuations and geodesics in half-space LPP, polymer, and random growth models. The methods eschew the complexity of line ensemble techniques, opting instead for coupling via stationary measures, and achieve powerful technical innovations in tightness and coupling arguments that may be applicable in broader settings. This work will serve as a central reference in probabilistic KPZ theory and random geometry with boundaries.