Geodesics and the competition interface for the corner growth model

Abstract

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-infinite geodesics, and on nonexistence of doubly infinite geodesics.

A Auxiliary technical results

Cocycles satisfy a uniform ergodic theorem. The following is a special case of Theorem 9.3 of [24]. Note that a one-sided bound suffices for a hypothesis. Recall Definition 3.1 of stationary $L^1(\mathbb {P})$ cocycles. Let $h({B})\in \mathbb {R}^2$ denote the vector that satisfies

$$\begin{aligned} \mathbb {E}[{B}(0,e_i)]=-h({B})\cdot e_i \qquad \text {for }i\in \{1,2\}. \end{aligned}$$

Theorem A.1

Assume $\mathbb {P}$ is ergodic under the group $\{T_x\}_{x\in \mathbb {Z}^2}$. Let B be a stationary $L^1(\mathbb {P})$ cocycle. Assume there exists a function V such that for $\mathbb {P}$-a.e. $\omega$

$$\begin{aligned} \varlimsup _{\varepsilon \searrow 0}\;\varlimsup _{n\rightarrow \infty } \;\max _{x: \vert x\vert _1\le n}\;\frac{1}{n} \sum _{0\le k\le \varepsilon n} \vert V(T_{x+ke_i}\omega )\vert =0\qquad \text {for }i\in \{1,2\} \end{aligned}$$

(A.1)

and $\max _{i\in \{1,2\}} B(\omega ,0,e_i)\le V(\omega )$. Then

$$\begin{aligned} \lim _{n\rightarrow \infty }\;\max _{\begin{array}{c} x=z_1+\cdots +z_n\\ z_{1,n}\in \{e_1, e_2\}^n \end{array}} \;\frac{\vert B(\omega ,0,x)+h(B)\cdot x\vert }{n}=0 \qquad \text {for } \mathbb {P}\hbox {-}\hbox {a.e.}\ \omega . \end{aligned}$$

If the process $\{V(T_x\omega ):x\in \mathbb {Z}^2\}$ is i.i.d., then a sufficient condition for (A.1) is $\mathbb {E}(\vert V\vert ^p)<\infty$ for some $p>2$ [40, Lemma A.4].

The following is a deterministic fact about gradients of passage times. This idea has been used profitably in planar percolation, and goes back at least to [1, 2]. See Lemma 6.3 of [23] for a proof.

Lemma A.2

Fix $\omega \in \varOmega$. Let $u,v\in \mathbb {Z}^2_+$ be such that $\vert u\vert _1=\vert v\vert _1\ge 1$ and $u\cdot e_1\le v\cdot e_1$. Then

$$\begin{aligned}&G_{0,u}-G_{e_1,u}\ge G_{0,v}-G_{e_1,v}\quad \text {and}\end{aligned}$$

(A.2)

$$\begin{aligned}&G_{0,u}-G_{e_2,u}\le G_{0,v}-G_{e_2,v}. \end{aligned}$$

(A.3)