Permutation invariance in last-passage percolation and the distribution of the Busemann process

• 成果类型: Article; Early Access
• 署名作者: Bates, Erik; Emrah, Elnur; Martin, James; Seppalainen, Timo; Sorensen, Evan
• 署名单位: North Carolina State University; University College Dublin; University of Oxford; University of Wisconsin System; University of Wisconsin Madison; Columbia University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01516-7
• 发表日期: 2026-07-22
• 关键词: Exponential last-passage percolation; busemann functions; Permutation invariance; Burke property; DIMENSIONAL DIRECTED POLYMER; geodesics; queues; INTERCHANGEABILITY; covariance; particle

摘要

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 previously described along a horizontal line by Fan and the fourth author, using certain queueing maps. By contrast, our new description explicitly gives the joint distribution for any collection of edges (not just along a horizontal line) using only finitely many random variables. Our result thus provides an exact and accessible way to sample from the joint distribution. In the proof, we rely on one-directional marginal distributions of the inhomogeneous Busemann functions recently studied by Janjigian and the second and fourth authors. The second ingredient of our proof is a novel joint invariance of inhomogeneous last-passage times under permutations of the inhomogeneity parameters. Our proof of the invariance is different from earlier proofs of such results, using the Burke property instead of the RSK correspondence, and leading to an explicit coupling of the weights before and after the permutation of the parameters.