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.

In exponential last passage percolation, we consider the rescaled Busemann process, as a process parametrized by the scaled density ρ = 1/2+ µ 4 N −1/2 , and taking values in C(R). We show that these processes, as N → ∞, have a càdlàg scaling limit G = (G µ ) µ∈R , parametrized by µ and taking values in C(R). The limiting process G, which can be thought of as the perturbation of the stationary initial condition under the KPZ scaling, can be described as an ensemble of "sticky" lines. Our proof provides some insight into this limiting behaviour by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Seppäläinen in [16], and a sorting algorithm of random walks introduced by O'Connell and Yor in [31].

We consider the simple exclusion process on Z × {0, 1}, that is, an "horizontal ladder" composed of 2 lanes. Particles can jump according to a lane-dependent translation-invariant nearest neighbour jump kernel, i.e. "horizontally" along each lane, and "vertically" along the scales of the ladder. We prove that generically, the set of extremal invariant measures consists of (i) translation-invariant product Bernoulli measures; and, modulo translations along Z: (ii) at most two shock measures (i.e. asymptotic to Bernoulli measures at ±∞) with asymptotic densities 0 and 2; (iii) at most three shock measures with a density jump of magnitude 1. We fully determine this set for certain parameter values. In fact, outside degenerate cases, there is at most one shock measure of type (iii). The result can be partially generalized to vertically cyclic ladders with arbitrarily many lanes. For the latter, we answer an open question of [5] about rotational invariance of stationary measures.

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In this paper we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder of width o(N 2/3 ) and length o(N ) agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width δN 2/3 and length up to δ 3/2 N/(log(δ −1 )) 3 and provide lower and upper bound for the probability that the geodesics agree in that cylinder.