Universality of the geodesic tree in last passage percolation

Abstract

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.

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

“…Compared to the best previous upper bound [24, Theorem 2.8], the result offers only a slight improvement mainly by extending the scope from L-shaped paths to arbitrary down-right paths. However, our upper bound is not uniform in the base vertex pi 0 , j 0 q, although uniform versions of it slightly weaker than the one in [24] can be readily stated as in Remark 4.2 below. Note also that, as in the prior bounds from [17,18,24], the scaling pm `nq 2{3 in our result is in terms of the distance between the vertex pi 0 , j 0 q and an endpoint of the geodesic π " π 1,0 0,0 pm `i0 , n `j0 q rather than the length of π. Theorem 4.1.…”

“…Some applications are described in this paper. Other applications utilized an earlier preprint [39] of this paper, are already in the literature in [24,39,85], and will be mentioned below in Section 1.6. Further applications and extensions of these results to other models are forthcoming [37].…”

“…Compared with our speed bounds on the LPP increments (Theorem 4.2), their result provides bounds with respect to a stronger metric but with a weaker decay rate. More recently, utilizing the exit point bounds from the previous version [39] of this article, [24,Theorem 2.8] established optimal-order fluctuation upper bounds for the bulk geodesics. In comparison, our exit point upper bound for the bulk geodesics (Theorem 4.1) is proved somewhat differently, is also of optimal order, and applies more generally to the exit points from arbitrary down-right paths, but lacks uniformity over these paths.…”

Diffusive scaling limit of the Busemann process in Last Passage Percolation

“…The fact that large W 0 results in constant values of the Busemann process around a fixed density ρ was first used in [3] to obtain strong local stationary results of the point to point geodesics. In [12], the authors used the same techniques to obtain the exact exponent for the probability that the point to point geodesic does not coalesce with the stationary one. In this sense, this work is a natural progression as we consider the constant times of the Busemann process across a continuum of densities around a fixed density.…”

“…They underlie many of the techniques used to study infinite geodesics [5,38] and have deep connections to the set of exceptional directions at which uniqueness of the geodesics fail [26]. Busemann functions are also important in the study of point-to-point geodesics [3], their coalescence [12] and the regularity of the passage time profile around a point [3,12,[32][33][34]. Busemann fucntions are usually associated with a direction on the first quadrant of the lattice -for ρ ∈ (0, 1) one defines the Busemann function B ρ by essentially setting B ρ (x, y) to be the difference in passage time between two infinite geodesics starting from x and y and going in a direction that is parametrised by ρ ∈ (0, 1).…”