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The study of Kadar-Parsi-Zhang (KPZ) universality class has been a subject ofgreat 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 conditionsdefined by Busemann functions. To advance in this direction, we investigate theindependence property of the Busemann function across multiple directions invarious exactly solvable KPZ models. These models encompass the corner growthmodel, the inverse-gamma polymer, Brownian last-passage percolation, theO'Connell-Yor polymer, the KPZ equation, and the directed landscape. In thecontext of the corner growth model, our result states that disjoint Busemannincrements in different directions along a down-right path are independent, aslong as their associated semi-infinite geodesics have nonempty intersectionsalmost surely. The proof for the independence utilizes the queueingrepresentation of the Busemann process developed by Sepp\"al\"ainen et al. Asan application, our independence result yields a near-optimal probability upperbound (missing by a logarithmic factor) for the rare event where the endpointof a point-to-line inverse-gamma polymer is close to the diagonal.

Independence property of the Busemann function in exactly solvable KPZ models