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

@inproceedings{Groathouse2021NonexistenceON, title={Non-existence of non-trivial bi-infinite geodesics in Geometric Last Passage Percolation}, author={Sean Groathouse and Christopher Janjigian and Firas Rassoul-Agha}, year={2021} }

— 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 nontrivial 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 Balázs, Busani, and Seppäläinen. Most of our results work for a general weights distribution…

## Figures from this paper

## One Citation

### Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation

- Mathematics, Computer Science
- 2022

New probabilistic arguments are introduced to derive optimal-order central moment bounds in planar directed last-passage percolation in i.i.d. exponential weights for both zero and near-stationary boundary conditions.

## References

SHOWING 1-10 OF 47 REFERENCES

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

- MathematicsForum of Mathematics, Sigma
- 2020

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…

### Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation

- Mathematics, Physics
- 2018

Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics,…

### Geometry of geodesics through Busemann measures in directed last-passage percolation

- MathematicsJournal of the European Mathematical Society
- 2022

We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of…

### Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation

- Mathematics
- 2012

We study first-passage percolation on $${\mathbb{Z}^2}$$Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of…

### Random coalescing geodesics in first-passage percolation

- Mathematics, Physics
- 2016

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an…

### Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

- Mathematics
- 2021

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal…

### Absence of backward infinite paths for first-passage percolation in arbitrary dimension

- Mathematics
- 2020

In first-passage percolation (FPP), one places nonnegative random variables (weights) $(t_e)$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on $\mathbb{Z}^d$…

### Euclidean models of first-passage percolation

- Mathematics
- 1997

Summary. We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝd. Compared to standard FPP on ℤd, these models have some technical…

### Geodesics in two-dimensional first-passage percolation

- Mathematics
- 1996

We consider standard first-passage percolation on Z 2 . Geodesics are nearest-neighbor paths in Z 2 , each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite…

### Stationary cocycles and Busemann functions for the corner growth model

- Mathematics
- 2015

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside of the class of exactly solvable…