Diffusive scaling limit of the Busemann process in Last Passage Percolation

Abstract

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].

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

“…This is the central result that gives access to the properties of the Busemann process. It verifies the universality of SH conjectured by [Bus21]. Furthermore, it provides us with computational tools for studying the geometric features of DL, in particular, its semi-infinite geodesics.…”

“…The main theorem of [Bus21] is the following: The first author [Bus21] first proved this finite-dimensional convergence and then showed tightness of the process to conclude the existence of a limit taking values in D(R, C(R)). The last two authors [SS21b] discovered that the stationary horizon is also the Busemann process of Brownian last-passage percolation, up to an appropriate scaling and reflection (see Theorem 5.3 in [SS21b]).…”

“…The parameterization here is different from the one used in [Bus21], because in the present paper, G ξ is a Brownian motion with diffusivity √ 2 and drift 2ξ, while in [Bus21], G µ has diffusivity 2 and drift µ. Specifically, G ξ (x) = G 4ξ (x/2), where G denotes the parameterization used in the present paper, and G denotes the parameterization from [Bus21]. Using the scaling relations of Theorem D.5(ii), one can see that this formulation of the limit is the same.…”

“…The stationary horizon (SH) is a cadlag process indexed by the real line whose states are Brownian motions with drift (Definition D.1 in Appendix D). SH was constructed by the first author [Bus21] as the diffusive scaling limit of the Busemann process of exponential last-passage percolation from [FS20]. Remarkably, the last two authors also discovered in [SS21b] that the stationary horizon, restricted to nonnegative drifts, is the Busemann process of Brownian last-passage percolation.…”

“…. , I ρn ) ∼ µ ρ n .Appendix D. The stationary horizonTo describe the stationary horizon, we introduce some notation from[Bus21]. The map Φ :C(R) × C(R) → C(R) is defined as Φ(f, g)(x) =    f (x) + W 0 (f − g) + inf 0≤y≤x (f (y) − g(y)) − x ≥ 0 f (x) − W x (f − g) + inf x<y≤0 f (y) − f (x) − [g(y) − g(x)]− x < 0, where W x (f ) = sup y≤x [f (x) − f (y)].…”