Invariant measures for half-space geometric LPP: classification and the one force--one solution principle

TL;DR

This paper fully characterizes extremal invariant measures for half-space geometric last-passage percolation, introducing the Busemann process and confirming conjectures about invariant measures and geodesic directions in this boundary-influenced KPZ model.

Contribution

It provides the first complete classification of extremal invariant measures for half-space geometric LPP and constructs the Busemann process, confirming related conjectures.

Findings

Complete classification of extremal invariant measures.

Construction of the Busemann process for half-space models.

Confirmation of conjectures on invariant measures and geodesic directions.

Abstract

We prove a complete characterization of the extremal invariant measures for half-space geometric last-passage percolation with an arbitrary boundary parameter. This is the first result of its kind for a model in the KPZ universality class that has boundary effects and an unbounded domain. A description of a class of invariant measures was previously given in a work of Barraquand and Corwin, where it was conjectured that these should comprise all extremal invariant measures. To complete the classification, we prove a one force--one solution principle: when started in the distant past from an arbitrary initial condition with a given asymptotic slope at $\infty$, the recentered solution at time $0$ converges to a process which is distributed as the associated invariant measure with the specified slope. This limiting process is called the Busemann process, the first of its kind constructed for a half-space model. The Busemann process across all slopes is distributed as the joint invariant measure for geometric half-space LPP, recently constructed by Dauvergne and Zhang. There, it was conjectured that the constructed family of jointly invariant measures comprises all extremal jointly invariant measures; our analysis also confirms this conjecture. When the model has a strong (attractive) boundary, the collection of slopes for the invariant measures has a discontinuity, which does not arise in the full-space case. To handle this difficulty, we combine the control of the directions of semi-infinite geodesics with techniques from the theory of half-space Gibbsian line ensembles. Along the way, we classify the set of directions of semi-infinite geodesics for half-space geometric LPP, confirming a recent conjecture of Dauvergne and Zhang.