Scaling limit of multi-type invariant measures via the directed landscape

TL;DR

This work addresses the scaling limits of multi-type invariant measures and Busemann functions within KPZ-class stochastic growth. By introducing a general framework that couples last-passage percolation (LPP) models to the directed landscape (DL), the authors prove convergence to the stationary horizon (SH), the universal multi-type invariant for the KPZ fixed point, under broad hypotheses. They verify these hypotheses for six exactly solvable LPP models (Poisson LPP, Poisson lines, SJ, exponential/Geometric LPP, Brownian LPP), establishing finite-dimensional convergence of joint Busemann distributions to SH and connecting them to the DL. A second, more general convergence theorem extends applicability to polymer and particle-system settings, highlighting SH as a universal scaling limit in the KPZ class. Together, these results bolster the conjecture that SH governs the universal multi-type invariant-measure scaling in KPZ models and advance a framework independent from explicit prelimit-measure structures.

Abstract

This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of multi-type invariant measures of last-passage percolation (LPP) models to the stationary horizon (SH), which is the unique multi-type stationary measure of the KPZ fixed point. Our limit theorem utilizes conditions that are expected to hold broadly in the Kardar-Parisi-Zhang class, including convergence of the scaled last-passage process to the directed landscape. We verify these conditions for the six exactly solvable models whose scaled bulk versions converge to the directed landscape, as shown by Dauvergne and Virág. We also present a second, more general, convergence theorem with potential future applications to polymer models and particle systems. Our paper is the first to show convergence to the SH without relying on information about the structure of the multi-type invariant measures of the prelimit models. These results are consistent with the conjecture that the SH is the universal scaling limit of multi-type invariant measures in the KPZ class.

Figures (1)

• Figure 1: An illustration of the construction of the down-left path. The horizontal axis and the ray $\{y\} \times (0,\infty)$ are in black/light. The particle trajectories are in black/medium thickness. There are two particles crossing the line $\{y\} \times (0,\infty)$and three particles emanating from the horizontal axis in the window shown. The path we construct is red/dashed. In this case, the procedure terminates at the point $(y_{-i},t_{-(i+1)})$, where $t_{-(i+1)} = 0$. The point $(x^\star,0)$ is the particle along the horizontal axis that generated the last trajectory travelled by the red path. There can be no other particles $(x,0)$ for $x \in (y_{-i},x^\star)$; otherwise, the red path would follow another particle trajectory before terminating.

Theorems & Definitions (69)

• Theorem 1.1
• Lemma 2.1
• Theorem 2.2: Busa-Sepp-Sore-22a
• Remark
• Theorem 2.3
• Remark
• proof
• Theorem 2.4
• proof
• Lemma 3.1