DIFFUSIVE SCALING LIMIT OF THE BUSEMANN PROCESS IN LAST PASSAGE PERCOLATION

In exponential last passage percolation, we consider the rescaled Busemann process x bar right arrow N-(1/3) B-0,[xN2/3]e1(rho), x is an element of R, as a process parametrized by the scaled density rho = 1/2+ mu/4 N-1/3, and taking values in C(R). We show that these processes, as N -> infinity, have a right continuous left limit scaling limit G = (G(mu))(mu is an element of R), parametrized by mu and taking values in C(R). The limiting process G, which can be thought of as the Busemann process under the KPZ scaling, can be described as an ensemble of "sticky" lines of Brownian regularity. We believe G is the universal scaling limit of Busemann processes in the KPZ universality class. Our proof provides insight into this limiting behavior by highlighting a connection between the joint distribution of Busemann functions obtained by Fan and Seppalainen (in Probab. Math. Phys. 1 (2020) 55-100), and a sorting algorithm of random walks introduced by O'Connell and Yor (in Electron. Commun. Probab. 7 (2002) 1-12).