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Permutation invariance in last-passage percolation and the distribution of the Busemann process
Erik Bates, Elnur Emrah, James Martin, Timo Sepp\"al\"ainen, Evan Sorensen
https://arxiv.org/abs/2506.12641

Permutation invariance in last-passage percolation and the distribution of the Busemann process
In i.i.d. exponential last-passage percolation, we describe the joint distribution of Busemann functions, over all edges and over all directions, in terms of a joint last-passage problem in a finite inhomogeneous environment. More specifically, the Busemann increments within a $k\times\ell$ grid, and associated to $d$ different directions, are equal in distribution to a particular collection of last-passage increments inside a $(k+d-1)\times(\ell+d-1)$ grid. The joint Busemann distribution was …