Non-existence of non-trivial bi-infinite geodesics in Geometric Last Passage Percolation

TL;DR

This paper proves that in a specific solvable last-passage percolation model with geometric weights, there are no non-trivial bi-infinite geodesics, marking a significant advancement in understanding the model's geometric structure.

Contribution

It provides the first proof of non-existence of non-trivial bi-infinite geodesics in a discrete-weight model, using the structure of increment-stationary versions and identifying key properties for generalization.

Findings

No non-trivial bi-infinite geodesics in the geometric last-passage percolation model.

Results extend to a general class of weight distributions under certain properties.

Framework relies on increment-stationary structure and recent methodological approaches.

Abstract

We show non-existence of non-trivial bi-infinite geodesics in the solvable last-passage percolation model with i.i.d. geometric weights. This gives the first example of a model with discrete weights where non-existence of non-trivial bi-infinite geodesics has been proven. Our proofs rely on the structure of the increment-stationary versions of the model, following the approach recently introduced by Bal\'azs, Busani, and Sepp\"al\"ainen. Most of our results work for a general weights distribution and we identify the two properties of the stationary distributions which would need to be shown in order to generalize the main result to a non-solvable setting.