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

TL;DR

This paper proves the strong existence and uniqueness of the tilt-indexed Busemann process in the planar corner growth model, ensuring a well-defined, consistent process across probability spaces.

Contribution

It establishes the strong existence and uniqueness of the Busemann process in the i.i.d. planar corner growth model, a significant theoretical advancement.

Findings

The Busemann process exists and is unique in the strong sense.

Any realization of the process on the same probability space is almost surely identical.

Supports the robustness of the Busemann process in the model.

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.