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

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.