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

@article{Balazs2020NonexistenceOB,
  title={Non-existence of bi-infinite geodesics in the exponential corner growth model},
  author={Marton E. Balazs and Ofer Busani and Timo Sepp{\"a}l{\"a}inen},
  journal={Forum of Mathematics, Sigma},
  year={2020},
  volume={8}
}
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. 

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

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

Coalescence estimates for the corner growth model with exponential weights

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

Non-existence of bi-infinite polymer Gibbs measures.

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise

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

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 $\mathbb{Z}^d$

Moderate Deviation and Exit Time Estimates for Stationary Last Passage Percolation

We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $\rho\in (0,1)$ and points $v_N=((1-\rho)^2 N,\rho^2 N)$ going to infinity along

Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension

We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic

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

Mixing times for the TASEP in the maximal current phase

We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size N with open boundaries. We focus on the maximal current phase, and prove that the mixing time is

Joint distribution of Busemann functions in the exactly solvable corner growth model

We describe the joint distribution of the Busemann functions of the corner growth model with exponential weights. The marginals of this measure are identified as the unique spatially ergodic fixed

FRACTAL STRUCTURE IN THE DIRECTED LANDSCAPE

. This short article surveys recent progress in investigations of fractal structure in the directed landscape, which is a putative universal scaling limit within the KPZ universality class and a

References

SHOWING 1-10 OF 42 REFERENCES

Geodesics in two-dimensional first-passage percolation

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

Geodesics and the competition interface for the corner growth model

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

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

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

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

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

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

Duality between coalescence times and exit points in last-passage percolation models

In this paper we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence

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

Joint distribution of Busemann functions in the exactly solvable corner growth model

We describe the joint distribution of the Busemann functions of the corner growth model with exponential weights. The marginals of this measure are identified as the unique spatially ergodic fixed

Characterizing stationary 1+1 dimensional lattice polymer models

Motivated by the study of directed polymer models with random weights on the square integer lattice, we define an integrability property shared by the log-gamma, strict-weak, beta, and inverse-beta

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,