Diffusive scaling limit of the Busemann process in last passage percolation

In exponential last passage percolation, we consider the rescaled Busemann process x↦ N-1/2B^ρ0,[xN]e₁ \,\, (x), 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 c\`adl\`ag 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 Sepp in {FS18}, and a sorting algorithm of random walks introduced by O'Connell and Yor in {OcY02}.