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

  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},
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… 

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

. 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

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

— 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

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

The stationary horizon and semi-infinite geodesics in the directed landscape

. The stationary horizon is a stochastic process consisting of coupled Brownian motions, indexed by their real-valued drifts. It was recently constructed by the first author as the scaling limit of

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

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.

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

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

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

Last passage isometries for the directed landscape

Consider the restriction of the directed landscape L(x, s;y, t) to a set of the form {x1, . . . , xk} × {s0} ×R × {t0}. We show that on any such set, the directed landscape is given by a last passage



Random coalescing geodesics in first-passage percolation

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

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

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

Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation

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

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

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

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


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

Coalescence of geodesics in exactly solvable models of last passage percolation

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

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

Euclidean models of first-passage percolation

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