# Geodesics and the competition interface for the corner growth model

@article{Georgiou2015GeodesicsAT, title={Geodesics and the competition interface for the corner growth model}, author={Nicos Georgiou and Firas Rassoul-Agha and Timo Sepp{\"a}l{\"a}inen}, journal={Probability Theory and Related Fields}, year={2015}, volume={169}, pages={223-255} }

We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. In Georgiou et al. (Probab Theory Relat Fields, 2016, doi:10.1007/s00440-016-0729-x) we constructed stationary cocycles and Busemann functions for this model. Using these objects, we prove new results on the competition interface, on existence, uniqueness, and coalescence of directional semi…

## 52 Citations

Busemann functions, geodesics, and the
competition interface for directed last-passage
percolation

- MathematicsProceedings of Symposia in Applied Mathematics
- 2018

In this survey article we consider 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…

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…

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…

Stationary random walks on the lattice

- Mathematics
- 2016

We consider translation invariant measures on configurations of nearest-neighbor arrows on the integer lattice. Following the arrows from each point on the lattice produces a family of semi-infinite…

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…

Existence, uniqueness and coalescence of directed planar geodesics: Proof via the increment-stationary growth process

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2020

We present a proof of the almost sure existence, uniqueness and coalescence of directed semi-infinite geodesics in planar growth models that is based on properties of an increment-stationary version…

Existence, uniqueness and coalescence of directed planar geodesics: proof via the increment-stationary growth process

- Mathematics
- 2018

We present a proof of the almost sure existence, uniqueness and coalescence of directed semi-infinite geodesics in planar growth models that is based on properties of an increment-stationary version…

Stationary coalescing walks on the lattice

- MathematicsProbability Theory and Related Fields
- 2018

We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In 2d, we classify the collective…

Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^{2}$

- Mathematics
- 2018

We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the…

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…

## References

SHOWING 1-10 OF 59 REFERENCES

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…

A shape theorem and semi-infinite geodesics for the Hammersley model with random weights

- Mathematics
- 2010

In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also…

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…

MULTIPLE GEODESICS WITH THE SAME DIRECTION

- Mathematics
- 2011

The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and…

ABSENCE OF GEODESICS IN FIRST-PASSAGE PERCOLATION ON A HALF-PLANE

- Mathematics
- 1998

An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are…

On the number of infinite geodesics and ground states in disordered systems

- Mathematics
- 1997

We study first-passage percolation models and their higher dimensional analogs—models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the…

Limiting shape for directed percolation models

- Mathematics
- 2004

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z d + , d ≥ 2, with i.i.d. weights at the vertices. Under certain moment conditions on the common…

Busemann functions and equilibrium measures in last passage percolation models

- Mathematics
- 2009

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a…

Coexistence in two-type first-passage percolation models

- Mathematics
- 2003

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Zd or on the infinite cluster in Bernoulli percolation. We prove for a large class of…

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…