The directed landscape in half-space
Duncan Dauvergne, Lingfu Zhang
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.
AI Impact Assessments
(1 model)Scientific Impact Assessment
Core Contribution
This paper constructs the directed landscape in half-space, a random directed metric on the half-plane parameterized by a boundary interaction strength ρ, and proves it is the universal scaling limit for half-space KPZ models. Specifically, the authors prove convergence of two distinct models—half-space exponential last-passage percolation (LPP) and multi-level Poisson-avoiding metrics (generalizing colored TASEP)—to this common limit. The half-space directed landscape Lρ is the complete scaling limit for random metric models in the KPZ universality class with a boundary, analogous to the full-space directed landscape constructed by Dauvergne-Ortmann-Virág (2022).
The paper achieves three intertwined major results: (1) explicit construction of joint stationary measures ("horizons") for multiple half-space KPZ models (log-gamma polymer, KPZ equation, exponential/geometric LPP, and the directed landscape itself); (2) optimal moderate deviation estimates for half-space exponential LPP; and (3) a characterization theorem identifying the half-space directed landscape uniquely from its KPZ fixed point marginals, triangle inequality, and independence of time increments.
Methodological Rigor
The paper demonstrates exceptional methodological depth and introduces several novel techniques:
Novel proof strategy: Rather than following the Airy line ensemble approach of DOV22 (which would require overcoming formidable obstacles in half-space due to lack of translation invariance and richer boundary behavior), the authors develop an entirely different framework: construct stationary measures → prove tightness → characterize the limit via KPZ fixed point marginals.
Parameter permutation symmetry (Lemma 2.3): A new invariance property for exactly solvable half-space KPZ models under permutations of parameters, extending techniques from Barraquand-Corwin. This is the key algebraic engine for constructing joint stationary measures.
Moderate deviation bounds (Theorem 1.12): The one-point tail bounds are optimal up to constants and reveal a fascinating phase transition in the supercritical regime: Gaussian behavior in the shallow tail (controlled by boundary weights) transitions to Tracy-Widom behavior in the deep tail (controlled by bulk weights). The proof cleverly combines the Barraquand-Wang identity relating half-space point-to-line passage times to full-space point-to-point problems with novel geometric arguments.
Spatial two-point bounds via stationarity comparison: The Gaussian spatial tail bound (Proposition 3.20) uses a novel method of comparing with stationary initial conditions, requiring the extended half-space stationary measures as input. The authors note this is the first instance of such a method yielding sharp two-point bounds, and it should be portable to other models.
Characterization theorem (Theorem 1.15): Adapting the Lindeberg exchange strategy from the authors' prior full-space work (DZ24), but with substantially different quantitative estimates. The branching estimate (Proposition 5.10) requires handling the more complex structure of the half-space stationary horizon, and the Brownian absolute continuity arguments must be replaced by analysis of stationary initial and final conditions.
Potential Impact
Within KPZ universality: This paper completes a major milestone by establishing the universal scaling limit for half-space KPZ models at the metric level, paralleling what DOV22 achieved for full-space. The characterization theorem (Theorem 1.15) provides a practical tool for proving convergence of other half-space models—one only needs KPZ fixed point marginal convergence, which is a one-dimensional statement.
Random geometry: The geodesic convergence results (Theorem 1.10) and the uniqueness proof (Proposition 7.5, using a novel resampling argument avoiding Brownian absolute continuity) open the door to studying the random geometry of half-space KPZ models.
Broader applicability: The stationary measure constructions work for multiple models simultaneously (log-gamma polymer, KPZ equation, geometric/exponential LPP), and the comparison-with-stationarity method for spatial tail bounds should transfer to other settings where stationary measures are available but exact solvability is limited.
Open problems: The paper raises compelling questions about whether the boundary contributes any randomness in the scaling limit (Problem 8.3), and whether there exists a local time representation for the boundary effect (Problem 8.4).
Timeliness & Relevance
This work addresses a long-standing gap: while the full-space directed landscape was constructed in 2022, the half-space version—predicted to exhibit qualitatively different behavior with phase transitions in the boundary parameter since Kardar (1985)—remained open. The paper builds on several recent breakthroughs: Xincheng Zhang's half-space KPZ fixed point formulas (2024), the Barraquand-Wang identity (2023), and the authors' own full-space characterization (2024). The timing is ideal, as these ingredients have only recently become available.
Strengths & Limitations
Strengths: Comprehensive treatment covering construction, characterization, geodesic convergence, and stationary measures across multiple models. The proof strategy is fundamentally new relative to DOV22. The moderate deviation estimates have independent value for studying prelimiting models.
Limitations: The subcritical case ρ = −∞ receives a weaker characterization (Corollary 5.2 rather than Theorem 1.15). The off-diagonal tail bounds could be sharpened for α < 1/2 − n^{-1/3}. The paper does not resolve whether the boundary contributes randomness in the limit, though this is posed as an open problem.
This is a landmark paper that will likely shape the direction of half-space KPZ research for years to come.
Generated Jun 5, 2026
Comparison History (47)
Paper 1 establishes a fundamental scaling limit (the half-space directed landscape) unifying multiple models in the KPZ universality class, a central topic in mathematical physics and probability. It resolves convergence of geodesics, characterizes the limit via the KPZ fixed point, and constructs joint stationary measures across multiple models. This breadth of results and the foundational nature of the directed landscape construction give it higher impact. Paper 2 is a strong result proving first exponential mixing for inviscid stochastic Euler, but its scope is narrower, addressing a specific regime (large damping) of one equation class.
Paper 2 establishes a fundamental scaling limit (the directed landscape in half-space) for the KPZ universality class, connecting multiple models and proving convergence of geodesics. This has broader impact across probability theory, mathematical physics, and statistical mechanics. It resolves a central object in a highly active research area (KPZ universality), provides explicit constructions of stationary measures across multiple models, and connects to the directed landscape framework which has been transformative. Paper 1 makes solid contributions to Poisson-Voronoi tessellations and solves open problems, but addresses a more specialized topic with narrower cross-field impact.
Paper 2 establishes a fundamental scaling limit (the half-space directed landscape) for the KPZ universality class, resolving a major open problem in probability theory and mathematical physics. It unifies multiple models, proves geodesic convergence, and constructs explicit stationary measures across several important systems. This represents a deep structural advance in a highly active research area with broad mathematical significance. Paper 1, while ambitious in scope and technically rich, reads more as a synthesis and extension of existing signature/rough path theory with applications to Hawkes processes—impactful but more incremental relative to the field-defining nature of Paper 2.
While Paper 1 makes profound theoretical contributions to the KPZ universality class, Paper 2 exhibits higher potential for broad scientific impact. Mean field games (MFGs) have extensive real-world applications across economics, finance, epidemiology, and multi-agent reinforcement learning. By introducing a generic, versatile approach to establish MFG master equations using a novel representation of the Lions derivative, Paper 2 provides adaptable tools for multiple disciplines. Its broader interdisciplinary relevance and applicability to complex, large-scale systems give it a significantly wider footprint than the highly specialized mathematical physics focus of Paper 1.
Paper 2 establishes a fundamental scaling limit (the directed landscape in half-space) for the KPZ universality class, unifying multiple models and proving convergence of geodesics along with explicit constructions of stationary measures. This extends the landmark full-space directed landscape work to the half-space setting, which is a major open problem with broad implications across probability, statistical mechanics, and integrable systems. Paper 1 is technically impressive but addresses a more specialized question about stochastic regularization of a modified (logarithmically hyperdissipative) Navier-Stokes model, which is further from the actual Navier-Stokes regularity problem.
Paper 1 establishes the scaling limit (directed landscape) for half-space KPZ models, resolving a fundamental open problem in integrable probability and statistical mechanics. It unifies multiple models, proves geodesic convergence, and constructs explicit stationary measures across several key systems. This represents a major advance in the KPZ universality program, which has been central to mathematical physics for decades. Paper 2, while technically solid in extending mean-field theory to noisy edge weights via dissociated arrays, addresses a more incremental generalization of propagation of chaos with narrower impact scope.
Paper 1 establishes a major scaling limit (the half-space directed landscape) unifying multiple models in the KPZ universality class, a central topic in mathematical physics and probability. It proves convergence of geodesics, characterizes the limit via the KPZ fixed point, and constructs joint stationary measures across multiple models. This represents a foundational contribution to an intensely active field. Paper 2 provides a useful transfer principle for hyperuniform systems with nice applications, but its scope is more technical and incremental compared to the broad unifying framework established in Paper 1.
Paper 1 establishes a fundamental scaling limit (the half-space directed landscape) for the KPZ universality class, resolving a major open problem by proving convergence of multiple models to a common limit. It provides deep structural characterizations and explicit constructions of stationary measures across multiple important models. This represents a landmark contribution to probability theory and mathematical physics. Paper 2, while solid and useful, proves a relatively expected mean-field limit result (law of large numbers) for graphon-based interacting particle systems, which is a more incremental extension of existing mean-field theory.
Paper 2 likely has higher impact: it advances KPZ universality by establishing a common half-space scaling limit (directed landscape in half-space) for multiple integrable models, characterizes it via the half-space KPZ fixed point, and proves geodesic convergence. The explicit construction of joint stationary measures across several core models broadens applicability and provides foundational tools for probability, statistical physics, and integrable systems. The topic is timely and central, with wide cross-field relevance. Paper 1 is rigorous and novel but more specialized to stochastic flow dualities.
Paper 1 introduces a temporal stochastic block model with direct applications to modeling infection spreading in inhomogeneous populations. Its relevance to epidemiology, network science, and sociology provides broader cross-disciplinary impact and significant real-world applicability. In contrast, Paper 2 is highly theoretical, focusing on the KPZ universality class, which primarily impacts specialized subfields of mathematical physics and pure probability theory.