For the exactly solvable zero-temperature KPZ models treated by Busani–Seppäläinen–Sorensen — Poisson last-passage percolation, the Poisson lines model, the Seppäläinen–Johansson model (at directions of curvature), and geometric last-passage percolation — the multi-type invariant measures (equivalently, Busemann processes), under KPZ scaling with drift parameters separated on the N−1/3N^{-1/3} scale, are known to converge to the stationary horizon (SH) only in finite-dimensional projections in the drift parameter. Likewise, for the ASEP speed process, whose multi-type marginals converge to the SH by work of Aggarwal–Corwin–Hegde, only finite-dimensional convergence is known. The problem is to upgrade these statements to weak convergence of the full drift-indexed process on the Skorokhod-type path space D(R,C(R))D(\mathbb{R},C(\mathbb{R})), i.e. to prove tightness on that path space. Full-process convergence is currently known only for exponential LPP and for the TASEP speed process. No resolution is known in the literature to date.

Background

The Kardar–Parisi–Zhang (KPZ) universality class comprises random growth and transport models — such as last-passage percolation (LPP, where one maximizes the total weight of up-right lattice paths through i.i.d. random weights) and one-dimensional interacting particle systems like the totally asymmetric simple exclusion process (TASEP, particles hopping only rightwards on the integer lattice with exclusion) and its partially asymmetric generalization, the asymmetric simple exclusion process (ASEP, hops allowed in both directions with unequal rates) — whose fluctuations grow on the anomalous N1/3N^{1/3} scale with transverse correlations of order N2/3N^{2/3}. Under this scaling the bulk geometry of the six classical exactly solvable LPP models converges to the directed landscape (DL), a random directed metric on space-time constructed by Dauvergne and Virág; the associated height functions converge to the KPZ fixed point, the universal Markovian scaling limit of KPZ-class height functions.

For an LPP model, the Busemann function in a direction is the limit of differences of last-passage times from a faraway point in that direction; it is the natural stationary object of the model. A collection of Busemann functions across all directions is a jointly invariant measure: a coupling of the ergodic invariant measures of the model at different densities. These couplings describe the multi-type (multi-species) invariant measures that govern stationary shocks, and for particle systems they are the multi-class stationary distributions (e.g. for multi-type TASEP they are given by the Ferrari–Martin queueing construction, in which the multi-species stationary distribution is produced by a tandem system of queues fed by independent particle walks).

The stationary horizon (SH) is a stochastic process G=(Gμ)μ∈RG=(G_\mu)_{\mu\in\mathbb{R}} living on the path space D(R,C(R))D(\mathbb{R},C(\mathbb{R})) of càdlàg (right-continuous with left limits) functions of a real drift parameter μ\mu with values in C(R)C(\mathbb{R}), the continuous real functions equipped with uniform convergence on compact sets. Marginally each GμG_\mu is a two-sided Brownian motion with diffusivity 2\sqrt{2} and drift 2μ2\mu; the SH was introduced by Busani as the diffusive scaling limit of the exponential-LPP Busemann process and constructed independently from Brownian LPP by Seppäläinen–Sorensen, and its law is described through mappings of independent Brownian motions with drift. Busani–Seppäläinen–Sorensen proved that the SH is the unique multi-type invariant measure of the KPZ fixed point (under asymptotic spatial slope conditions) and that it describes the Busemann process of the directed landscape. It is conjectured to be the universal scaling limit of multi-type invariant measures across the KPZ class.

Busani–Seppäläinen–Sorensen (IMRN 2024) developed an abstract criterion — combining marginal tightness, joint invariance of the coupled initial data, and convergence of the evolution to the KPZ fixed point coupled through the DL — that implies convergence of multi-type invariant measures to the SH, and they verified it for the six exactly solvable LPP models. However, the convergence they obtain is weak convergence of finite-dimensional projections: for fixed μ1<⋯<μk\mu_1<\dots<\mu_k the rescaled jointly invariant measures converge to (Gμ1,…,Gμk)(G_{\mu_1},\dots,G_{\mu_k}) on C(R)kC(\mathbb{R})^k. Convergence of the full drift-indexed process requires tightness on D(R,C(R))D(\mathbb{R},C(\mathbb{R})), which at the time was established in only two cases: exponential LPP (Busani) and the TASEP speed process (Busani–Seppäläinen–Sorensen). For Brownian LPP the Busemann process is exactly the SH in the prelimit (Seppäläinen–Sorensen), so the full-process statement there follows from exact diffusive scaling invariance.

The particle-system side is analogous. The TASEP speed process of Amir–Angel–Valkó records the joint empirical velocities of all species of a multi-species TASEP; its multi-type marginals are exactly the multi-type invariant measures, and it converges to the SH on the full path space. The ASEP speed process, constructed by Aggarwal–Corwin–Ghosal, is the general-asymmetry analogue. Aggarwal–Corwin–Hegde proved that colored ASEP and stochastic six-vertex height functions converge to the DL, and used the abstract criterion of Busani–Seppäläinen–Sorensen (IMRN 2024) to deduce that the ASEP multi-type invariant measures converge to finite-dimensional projections of the SH — but the convergence of the ASEP speed process to the SH on the full path space remained out of reach, and the stochastic six-vertex speed process was subsequently constructed by Drillick–Haunschmid-Sibitz, again without path-space convergence to the SH.

Problem Statement

For the exactly solvable zero-temperature KPZ models treated in Busani–Seppäläinen–Sorensen — Poisson last-passage percolation (Hammersley), the Poisson lines model, the Seppäläinen–Johansson model (centered at a direction of curvature of its shape function), and geometric LPP (LPP with geometrically distributed weights) — prove that the KPZ-rescaled multi-type invariant measures (equivalently, Busemann processes), with drift parameters separated on the N−1/3N^{-1/3} scale and centered around an arbitrary direction of curvature, converge weakly to the stationary horizon G=(Gμ)μ∈RG=(G_\mu)_{\mu\in\mathbb{R}} as processes on the Skorokhod-type path space D(R,C(R))D(\mathbb{R},C(\mathbb{R})). Equivalently, establish tightness of these processes on D(R,C(R))D(\mathbb{R},C(\mathbb{R})), upgrading the known convergence of the finite-dimensional projections (Gμ1,…,Gμk)(G_{\mu_1},\dots,G_{\mu_k}) on C(R)kC(\mathbb{R})^k to convergence of the full drift-indexed process. Furthermore, prove that the ASEP speed process — and its stochastic six-vertex analogue, whose multi-type invariant measures coincide with those of ASEP — converges to the SH on D(R,C(R))D(\mathbb{R},C(\mathbb{R})) under the KPZ scaling of Aggarwal–Corwin–Hegde, for which currently only finite-dimensional convergence of the multi-type invariant measures is known. (For Brownian LPP the corresponding full-process statement follows from the exact prelimit identity between the BLPP Busemann process and the SH.) Prove this convergence for each of the models listed above; the objective is complete when all of these cases are covered.

References

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