  The study of Kadar-Parsi-Zhang (KPZ) universality class has been a subject of
great interest among mathematicians and physicists over the past three decades.
A notably successful approach for analyzing KPZ models is the coupling method,
which hinges on understanding random growth from stationary initial conditions
defined by Busemann functions. To advance in this direction, we investigate the
independence property of the Busemann function across multiple directions in
various exactly solvable KPZ models. These models encompass the corner growth
model, the inverse-gamma polymer, Brownian last-passage percolation, the
O'Connell-Yor polymer, the KPZ equation, and the directed landscape. In the
context of the corner growth model, our result states that disjoint Busemann
increments in different directions along a down-right path are independent, as
long as their associated semi-infinite geodesics have nonempty intersections
almost surely. The proof for the independence utilizes the queueing
representation of the Busemann process developed by Sepp\"al\"ainen et al. As
an application, our independence result yields a near-optimal probability upper
bound (missing by a logarithmic factor) for the rare event where the endpoint
of a point-to-line inverse-gamma polymer is close to the diagonal.
