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

@article{Janjigian2022GeometryOG, title={Geometry of geodesics through Busemann measures in directed last-passage percolation}, author={Christopher Janjigian and Firas Rassoul-Agha and Timo Seppalainen}, journal={Journal of the European Mathematical Society}, year={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 the random environment. The structure of the geodesics is studied through the properties of the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model, we give the first complete characterization of the uniqueness and…

## Figures from this paper

## 17 Citations

Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

- Mathematics
- 2022

. We consider ﬁrst-passage percolation on Z 2 with independent and identically distributed weights whose common distribution is absolutely continuous with a ﬁnite exponential moment. Under the…

Busemann process and semi-infinite geodesics in Brownian last-passage percolation

- Mathematics
- 2021

We prove the existence of semi-infinite geodesics for Brownian last-passage percolation (BLPP). Specifically, on a single event of probability one, there exist semi-infinite geodesics started from…

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

- Mathematics
- 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…

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…

Coalescence estimates for the corner growth model with exponential weights

- Mathematics
- 2019

We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds…

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2022

We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary…

Empirical distributions, geodesic lengths, and a variational formula in first-passage percolation.

- Mathematics
- 2020

This article resolves, in a dense set of cases, several open problems concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our primary interest is in the empirical measures of…

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…

Diffusive scaling limit of the Busemann process in Last Passage Percolation

- Mathematics
- 2021

In exponential last passage percolation, we consider the rescaled Busemann process x 7→ N−1/2B 0,[xN ]e1 (x ∈ R), as a process parametrized by the scaled density ρ = 1/2+μ4N −1/2, and taking values…

Dynamic polymers: invariant measures and ordering by noise

- MathematicsProbability Theory and Related Fields
- 2021

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at…

## References

SHOWING 1-10 OF 64 REFERENCES

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…

Geodesics and the competition interface for the corner growth model

- Mathematics
- 2015

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…

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,…

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 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…

Coalescence of geodesics in exactly solvable models of last passage percolation

- MathematicsJournal of Mathematical Physics
- 2019

Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite…

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…

Geodesics in first passage percolation

- Mathematics
- 2005

We consider a wide class of ergodic first passage percolation processes on I? and prove that there exist at least four one-sided geodesies a.s. We also show that coexistence is possible with positive…

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…