Non-existence of bi-infinite geodesics in the exponential corner growth model

TL;DR

This paper proves that in a specific probabilistic model of growth, there are no infinite paths extending infinitely in both directions, using coupling and planarity techniques to establish this non-existence.

Contribution

It provides a self-contained proof of the non-existence of bi-infinite geodesics in exponential last-passage percolation, advancing understanding of the model's structure.

Findings

No bi-infinite geodesics exist in the model.

Techniques include couplings, coarse graining, and planarity-based estimates.

The proof is self-contained and rigorous.

Abstract

This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.