Strong existence and uniqueness of the tilt-indexed Busemann process in the planar corner growth model

TL;DR

This work resolves the long-standing questions of strong existence and strong uniqueness for the tilt-indexed Busemann process in planar last-passage percolation by working directly on the canonical space with i.i.d. weights. The authors develop a framework based on generalized Busemann functions (cocycles) and covariant coalescing geodesics, constructing a tilt-indexed family B^h that is forward-measurable, recovering, and L^1 with deterministic tilt h under ergodicity. They prove that for every tilt h in the relevant super-differential, there is a unique B^h, and that the whole tilt-indexed process is well-defined and measurable across the base space; moreover, Busemann cocycles with coalescing geodesics decompose into this tilt-indexed family, yielding an ergodic decomposition. The results connect cocycle methods with covariant geodesic systems, extend ergodicity/mixing properties of the Busemann process, and leverage limit-shape curvature results to achieve a robust, canonical construction with potential for future stochastic-analytic applications in the KPZ class.

Abstract

We show that the Busemann process indexed by tilts in the super-differential of the limit shape exists and is unique in the strong sense in the i.i.d.\ planar corner growth model. This means that every probability space that supports the field of i.i.d.~weights supports a copy of the process and any two realizations of the process are equal almost surely.

Figures (2)

• Figure 2.1: A level set $\{x\in\mathbb{R}_+^2:\textup{g}(x)=1\}$ of a limit shape (left) and its associated super-differential curve $\partial \textup{g}(\mathcal{U})$ (right). The full level set of the limit shape is depicted in the inset image on the left and the dashed lines indicate the enlarged portion. This shape has four linear segments and a cusp on the diagonal. It is differentiable at the endpoints of the two inner segments (green) and non-differentiable at five points: the four endpoints of the outer segments (blue) and the cusp on the diagonal (red). Non-differentiability points of the shape correspond to line segments in the super-differential, which represent the intervals of slopes of supporting lines at those points, while linear segments of the shape correspond to non-differentiability points in the super-differential.
• Figure 4.1: A simulation of an approximation to the tree $\mathcal{G}_0^\omega$ of infinite geodesics rooted at the origin in the i.i.d. Exponential(1) model. Right-isolated geodesics are those which cannot be approximated from the right (in the topology in which paths converge if they are eventually equal on finite sets) in this tree. Graphically, such geodesics appear as the left boundary of one of the connected components of the quadrant $\mathbb{Z}_+^2$ which contain no vertices on any geodesics of $\mathcal{G}_0^\omega$. Left-isolated geodesics are the right boundary of such a region.

Theorems & Definitions (92)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Theorem 2.7
• Remark 2.8
• Remark 2.9
• Definition 2.10